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On the sensations of tone as a physiological basis for the theory of music

THE

SENSATIONS OF TONE

Bibliographical Note.

First English Edition, June, 1875 ;

Second Edition, revised and Aj^pendix added, August, 1885;

Third Edition, reprinted from Second Edition, June, 1895.

ON THE

SENSATIONS OF TONE

AS A PHYSIOLOGICAL BASIS FOR THE

THEOEY OF MUSIC

BY

HERMANN L. F. HELMHOLTZ, M.D.

LATE FOREIGN MEMBEE OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH,

PROFESSOR OF PHYSIOLOGY IN THE UNIVERSITY OF HEIDELBERG, AND

PROFESSOR OF PHYSICS IN THE UNIVERSITY OF BERLIN

Trmisluted, thoroughly Revised ami Corrected, rendered conformable to the Fourth

(a7id last) German Edition of 1877, with numerous additional Notes and a

New additional Appendix bringing doicn information to 1885

and especially adapted to the use of 3Iusical Students

BY

ALEXANDER J. ELLIS

B.A., F.R.S., F.S.A., F.C.P.S., F.C.P.

T^\^CE PRESIDENT OF THE PHILOLOGICAL SOCIETY, MEMBER OF THE MATHEMATICAL SOCIETY,

FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE, AUTHOR OF ' EARLY ENGLISH PRONUNCIATION ' AND ' ALGEBRA IDENTIFIED WITH GEOMETRY '

THIRD £DITPON

LONDON LONGMANS, GREEN, AND CO.

AND NEW YORK 1895

All rights reserved

PHYSICS DEPT.

(JIL.^^Ua^<$>^ '

ABEEDEEN UNIVERSITY PRESS.

TRANSLATOR'S NOTICE

TO THE

SECOND ENGLISH EDITION.

Ix preparing a new edition of this translation of Professor Helmlioltz's great work on the Sensations of Tone, which was originally made from the third German edition of 1870, and was finished in June 1875, my first care was to make it exactly conform to i\\e fon,rth German edition of 1877 (the last which has appeared). The numerous alterations made in the fourth edition are specified in the Author's pre- face. In order that no merely verbal changes might escape me, every sentence of my translation was carefully re-read with the German. This has enabled me to correct several misprints and mistranslations which had escaped my previous very careful revision, and I have taken the opportunity of improving the language in many places. Scarcely a page has escaped such changes.

Professor Helmholtz's book having taken its place as a work which all candidates for musical degrees are expected to study, my next care was by supplementary notes or brief insertions, always carefully distinguished from the Author's by being inclosed in [ ], to explain any difficulties which the student might feel, and to shew him how to acquire an insight into the Author's theories, which were quite strange to musicians when they appeared in the first German edition of 1863, but in the twenty-two years which have since elapsed have been received as essentially valid by those competent to pass judgment.

For this purpose I have contrived the Harmonical, explained on pp. 466-469, by winch, as shewn in numerous footnotes, almost every point of theory can be illustrated ; and I have arranged for its being readily procurable at a moderate charge. It need scarcely be said that my interest in this instrument is purely scientific.

My own Appendix has been entirely re-written, much has been rejected and the rest condensed, but, as may be seen in the Contents, I have added a considerable amount of information about points hitherto little known, such as the Determi- nation and History of Musical Pitch, Non-Harmonic scales, Tuning, &c., and in especial I have given an account of the work recently done on Beats and Com- binational Tones, and on Vowel Analysis and Synthesis, mostly since the fourth German edition appeared.

Finally, I wish gratefully to acknowledge the assistance, sometimes very great, which I have received from Messrs. D. J. Blaikley,"R. H. M. Bosanquet, Colin Brown, A. Cavaille-Coll, A. J. Hipkins, W. Huggins, F.R.S., Shuji Isawa, H. Ward Poole, R. S. Rockstro, Hermann Smith, Steinway, Augustus Stroh, and James Paid White, as will be seen by referring to their names in the Index.

ALEXANDER J. ELLIS.

25 Argyll Road, Kensington July, 1885.

238213

AUTHOR'S PREFACE

TO THE

FIRST GERMAN EDITION.

In laying before the Public the result of eight years' labour, I must first pay a debt of gratitude. The following investigations could not have been accomplished without the construction of new instruments, which did not enter into the inventory of a Physiological Institute, and which far exceeded in cost the usual resources of a German philosopher. The means for obtaining them have come to me from unusual sources. The apparatus for the artificial construction of vowels, described on pp. 121 to 126, I owe to the munificence of his Majesty King Maximilian of Bavaria, to Avhom German science is indebted, on so many of its fields, for ever- ready sympathy and assistance. For the construction of my Harmonium in perfectly natural intonation, descriljcd on p. 316, I was able to use the Soemmering prize which had been awarded me by the Senckenberg Physical Society {die Senckenbergische nahirforscheiide Gesellschaft) at Frankfurt-on-the-Main. While publicly repeating the expression of my gratitude for this assistance in my investi- gations, I hope that the investigations themselves as set forth in this book will prove far better than mere words how earnestly I have endeavoured to make a worthy use of the means thus placed at my conmiand.

H. HELMHOLTZ. Heidelberg : October, 1862.

AUTHOR'S PREFACE

TO THE

THIRD GERMAN EDITION.

The present Third Edition has been much more altered in some parts than the second. Thus in the sixth chapter I have been able to make use of the new physiological and anatomical researches on the ear. This has led to a modification of my view of the action of Corti's ai'ches. Again, it appears that the pecviliar articulation between the auditory ossicles called 'hammer' and 'anvil' might easily cause within the ear itself the formation of harmonic upper partial tones for simple tones which are sounded loudly. By this means that pecidiar series of upper partial tones, on the existence of which the present theory of music is essentially founded, receives a new subjective value, entirely independent of external alterations in the quality of tone. To illustrate the anatomical descriptions, I have been able to add a series of new woodcuts, principally from Henle's Manual of Anatomy, with the author's permission, for which I here take the opportunity of publicly thankine: him.

PREFACE. vii

1 have made many changes iu re-editing the section on the History of Music, and hope that I have improved its connection. I must, however, request the reader to regard this section as a mere compilation from secondaiy sources ; I have neither time nor preliminary knowledge sufficient for original studies in this extremely difficult field. The older history of music to the commencement of Discant, is scarcely more than a confused heap of [secondary subjects, while we can only make hypotheses concerning the principal matters in question. Of course, however, every theoi-y of music must endeavour to bi-ing some order into this chaos, and it cannot be denied that it contains many important facts.

For the representation of pitch in just or natural intonation, I have abandoned the method originally proposed by Hauptmann, which was not sufficiently clear in involved cases, and have adopted the system of Herr A. von Oettingen [p. 276] , as had already been done in M. G. Gueroult's French translation of this book.

[A comparison of the Third with the Second editions, shewing the changes and additions individually, is here omitted.]

If 1 may be allowed in conclusion to add a few words on the reception expe- rienced by the Theory of Music here propounded, I should say that published objections almost exclusively relate to my Theory of Consonance, as if this were the pith of the matter. Those who prefer mechanical explanations express their regret at my having left any room in this field for the action of artistic invention and esthetic inclination, and they have endeavoured to complete my system by new numerical speculations. jT)*'^®^' critics with more metaphysical proclivities have rejected my Theory of Consonance, and with it, as they imagine, my whole Theory of Music, as too coarsely mechanical. ;

T hope my critics will excuse me if I conclude from the opposite nature of their objections, that I have struck out nearly the right path. As to my Theory of Consonance, I must claim it to be a mere systematisation of observed facts (with the exception of the functions of the cochlea of the ear, which is moreover an hypothesis that may be entirely dispensed with). But I consider it a mistake to make the Theory of Consonance the essential foundation of the Theory of Music, and I had thought that this opinion was clearly enough expressed in my book. The essential basis of Music is Melody. Harmony has become to \Vestem Euro- peans during the last three centuries an essential, and, to our present taste, indispensable means of strengthening melodic relations, but finely developed music existed for thousands of years and still exists in ultra-European nations, without any hannony at all. And to my metaphysico-esthetical opponents I must reply, that I cannot think I have undervalued the artistic emotions of the human mind in the Theory of Melodic Constmction, by endeavouring to establish the physiological facts on which esthetic feeling is based. But to those who think I have not gone far enough in my physical explanations, 1 answer, that in the first place a natural philosopher is never bound to construct systems about everything he knows and does not know ; and secondly, that I should consider a theory which claimed to have shewn that all the laws of modern Thorough Bass were natural necessities, to stand condemned as having proved too much.

Musicians have found most fault with the manner in which I have characterised the Minor Mode. I must refer in reply to those very accessible documents, the musical compositions of a.d. 1500 to a.d. 1750, dm-ing v/hich the modern Minor was developed. These will shew how slow and fluctuating was its development, and that the last traces of its incomplete state are still visible in the works of Sebastian Bach and Handel.

Heidelberg : May, 1870.

AUTHOR'S PREFACE

FOURTH GERMAN EDITION.

In the essential conceptions of musical relations I have found nothing to alter in this new edition. In this respect I can but maintain what I have stated in the chapters containing them and in my preface to the third [German] edition. In details, however, much has been remodelled, and in some parts enlarged. As a guide for readers of former editions, I take the liberty to enumerate the following places containing additions and alterations.*

P. 16d, note*. — On the French system of counting vibrations.

P. 18«. — Appunn and Preyer, limits of the highest audible tones.

Pp. 596 to 65b. — On the circumstances under which we distinguish compound sensations.

P. 16a, b, c. — Comparison of the upper partial tones of the strings on a new and an old grand pianoforte.

P. 83, note f. — Herr Clement Neumann's observations ou the vibrational form of nolin strings.

Pp. 89ft to 93&.— The action of blowing organ-pipes.

P. 1106.— Distinction of Ou from U.

Pp. 1116 to 116a. — The various modifications in the sounds of vowels.

P. 145a. — The ampulla? and semicircular canals no longer considered as parts of the organ of hearing.

P. 1476. — Waldeyer's and Preyer's measurements adopted.

Pp. 1506 to 151d. — On the parts of the ear which perceive noise.

P. 1596. — Koenig's observations on combinational tones with tuning-forks.

P. 176d, note. — Preyer's observations on deepest tones.

P. 179c.— Preyer's observation on the sameness of the quality of tones at the highest pitches.

Pp. 203t; to 204«. — Beats between upper partials of the same compound tone condition the preference of musical tones with hannonic upper partials.

Pp. 328c to 3296. — Division of the Octave into 53 degi-ees. Bosanquet's harmonivmi.

Pp. 338c to 3396. — j\Iodulations tluough chords composed of two major Thirds.

P. 365, note t. — Oettingen and Riemann's theory of the minor mode.

P. 372. — Improved electro-magnetic driver of the siren.

P. 373ft. — Theoretical formulte for the pitch of resonators.

P. 374c. — Use of a soap-bubble for seeing vibrations.

Pp. 389*:^ to 3966. — Later use of striking reeds. Theory of the blowing of pipes.

Pp. 403c to 4056. — Theoretical treatment of svmpathetic resonance for noises.

P. 417f^. — A. Mayer's experiments on the audibility of vibrations.

P. 428c. d. — Against the defenders of tempered intonation.

P. 429. — Plan of Bosanquet's Harmonium.

H. HELMHOLTZ. Berlin : A2Jril, 1877.

* [The pages of this edition are substituted first edition of this translation are mostly for the German throughout these prefaces, pointed out in footnotes as they arise. — Trans- and omissions or alterations as respects the lator.]

CONTENTS.

and notes in [ ] are due to the Translator, and the Author is in no way responsible for their contents.

Translatoe's Notice to the Second English Edition, p. v.

Author's Preface to the First German Edition, p. vi.

Author's Preface to the Third German Edition, pp. vi-vii.

Author's Preface to the Fourth German Edition, p. viii.

Contents, p. ix.

List of Figures, p. xv.

List of Passages in Musical Notes, p. xvi.

List of Tables, p. xvii.

INTRODUCTION, pp. 1-6.

Relation of Musical Science to Acoustics, 1

Distinction between Physical and Physiological Acoustics, 3

Plan of the Investigation, 4

PAET I. (pp. 7-151.)

ON THE COMPOSITION OF VIBRATIONS.

Upl^er Partial Tones, and Qualities of Tone.

CHAPTEK I. On the Sensation of Sound in General, pp. 8-25.

Distinction between Noise and Musical Tone, 8

Musical Tone due to Periodic, Noise to non-Periodic Llotions in the air, 8

General Property of Undulatory Motion : while Waves continually advance, the Particles

of the Medium through which they pass execute Periodic ]\Iotions, 9 Differences in Musical Tones due to Force, Pitch, and Quality, 10 Force of Tone depends on Amplitude of Oscillation, Pitch on the length of the Period of

Oscillation, 10-14 Simple relations of Vibrational Numbers for the Consonant Intervals, 14 Vibrational Numbers of Consonant Intervals calculated for the whole Scale, 17 Quality of Tone must depend on Vibrational Form, 19 Conception of and Graphical Representation of Vibrational Form, 20 Harmonic Upper Partial Tones, 22 Terms explained : Tone, Musical Tone, Simple Tone, Partial Tone, Compound Tone, Pitch

of Compound Tone, 23

CHAPTEE 11. On the Composition of Vibrations, pp. 25-36.

Composition of Vv'aves illustrated by waves of water, 25

The Heights of superimposed Waves of Water are to be added algebraically, 27

Corresponding Superimposition of Waves of Sound in the air, 28

CONTENTS.

A Composite Mass of Musical Tones will give rise to a Periodic Vibration wlien their Pitch

Numbers are Multiples of the same Number, 30 Every such Composite Mass of Tones may be considered to be composed of Simple

Tones, 33 This Composition corresponds, according to G. S. Ohm, to the Composition of a Musical

Tone from Simple Partial Tones, 33

CHAPTER III. Analysis of Musical Tones by Sympathetic Ee- SONANCE, pp. 36-49.

Explanations of the Mechanics of Sympathetic Vibration, 3G

Sympathetic Resonance occurs when the exciting vibrations contain a Simple Vibration

corresponding to one of the Proper Vibrations of the Sympathising Body, 33 Difference in the Sympathetic Resonance of Tuning-forks and Membranes, 40 Description of Resonators for the more accurate Analysis of Musical Tones, 43 Sympathetic Vibration of Strings, 45 Objective Existence of Partial Tones, 48

CHAPTEE IV. On the Analysis of Musical Tones by the Ear, pp. 49-65.

Slethods for observing Upper Partial Tones, 49

Proof of G. S. Ohm's Law by means of the tones of Plucked Strings, of the Simple Tones of Tuning-forks, and of Resonators, 51

Difference between Compound and Simple Tones, 56

Seebeck's Objections against Ohm's Law, 58

The Difficulties experienced in perceiving Upper Partial Tones analytically depend upon a peculiarity common to all human sensations, 59

We practise observation on sensation only to the extent necessary for clearly apprehend- ing the external world, 62

Analysis of Compound Sensations, 63

CHAPTEE Y. On the Differences in the Quality of Musical Tones, pp. 65-119.

Noises heard at the beginning or end of Tones, such as Consonants in Speech, or during Tones, such as Wind-rushes on Pipes, not included in the Musical Quality of Tone, which refers to the uniformly continuous musical sound, 65

Limitation of the conception of Musical Quality of Tone, 68

Investigation of the Upper Partial Tones which are present in different Musical Qualities of Tone, 69

1. ]\iusical Tones without Upper Partials, 69

2. Musical Tones with Inharmonic Upper Partials, 70

3. Musical Tones of Strings, 74

Strings excited by Striking, 74

Theoretical Intensity of the Partial Tones of Strings, 79

4. ]\Iusical Tones of Bowed Instruments, 80

5. :Musical Tones of Flute or Flue Pipes, 88

6. ilusical Tones of Reed Pipes, 95

7. Vowel Qualities of Tone, 103

Results for the Character of Musical Tones in general, 118

CHAPTEE YI. On the Apprehension of Qualities of Tone, pp. 119-151.

Does Quality of Tone depend on Difference of Phase ? 119

Electro-magnetic Apparatus for answering this question, 121

Artificial Vowels produced by Tuning-forks, 123

How to produce Difference of Phase, 125

Musical Quality of Tone independent of Difference of Phase, 126

Artificial Vowels produced by Organ Pipes, 128

The Hypothesis that a Series of S}-mpathetical Vibrators exist in the ear, explains its peculiar apprehension of Qualities of Tone, 129

Description of the parts of the internal ear which are capable of vibrating sympa- thetically, 129

Damping of Vibrations in the Ear, 142

Supposed Function of the Cochlea, 145

CONTENTS.

PAKT 11. (pp. 152-283.)

ON THE INTERRUPTIONS OF HARMONY.

Gonihinatiomil Tones and Beats, Consonance and Dissonance.

CHAPTEE VII. Combinational Tones, pp. 152-159.

Combinational Tones arise when Vibrations which are not of infinitesimal magnitude are

combined, 152 Description of Combinational Tones, 153 Law determining their Pitch Numbers, 254 Combinational Tones of different orders, 155

Difference of the strength of Combinational Tones on different instruments, 157 Occasional Generation of Combinational Tones in the ear itself, 158

CHAPTEE VIII. On the Beats of Simple Tones, pp. 159-173.

Interference of Two Simple Tones of the same pitch, 160

Description of the Polyphonic Siren, for experiments on Interference, 161

Eeinforcement or Enfeeblement of Sound, due to difference of Phase, 163

Interference gives rise to Beats when the Pitch of the two Tones is slightly different, 164

Law for the Number of Beats, 165

Visible Beats on Bodies vibrating sympathetically, 166

Limits of Kapidity of Audible Beats, 1G7

CHAPTEE IX. Deep and Deepest Tones, pp. 174-179.

Former Investigations were insufficient, because there was a possibility of the ear being deceived by Upper Partial Tones, as is shewn by the number of Beats on the Siren, 174

Tones of less than thirty Vibrations in a second fall into a Drone, of which it is nearly or quite impossible to determine the Pitch, 175

Beats of the Higher Upper Partials of one and the same Deep Compound Tone, 178

CHAPTEE X. Beats of the Uppee Partial Tones, pp. 179-197.

Any two Partial Tones of any two Compound Tones may beat if they are sufficiently near in pitch, but if they are of the same pitch there will be consonance, 179

Series of the different Consonances, in order of 'the Distinctness of their Delimitation, 183

Number of Beats which arise from Mistuning Consonances, and their effect in producing Roughness, 184

Disturbance of any Consonance by the adjacent Consonances, 186

Order of Consonances in respect to Harmoniousness, 188

CHAPTEE XL Beats due to Combinational Tones, pp. 197-211.

The Differential Tones of the first order generated by two Partial Tones are capable of producing very distinct beats, 197

Differential Tones of higher orders produce weaker beats, even in the case of simple gene- rating tones, 199

Influence of Quality of Tone on the Harshness of Dissonances and the Harmoniousness of Consonances, 205

CHAPTEE XII. Chords, pp. 211-233.

Consonant Triads, 211

Major and Minor Triads distinguished by their Combinational Tones, 214 Relative Harmoniousness of Chords in different Inversions and Positions, 218 Retrospect on Preceding Investigations, 226

CONTENTS.

PAKT III. (pp. 234-371.)

THE RELATIONSHIP OF MUSICAL TONES.

Scales and Tonality.

CHAPTEE XIII. General View of the Different Principles OF Musical Style in the Development of Music, pp. 234-249.

Difference between the Physical and the Esthetical Method, 234

Scales, Keys, and Harmonic Tissues depend upon esthetic Principles of Style as well as

Physical Causes, 235 Illustration from the Styles of Architecture, 235 Three periods of Music have to be distinguished, 236

1. Homophonic Music, 237

2. Polyphonic ]\Iusic, 244

3. Harmonic Music, 246

CHAPTEE XIV. The Tonality of Homophonic Music, pp. 250-290.

Esthetical Reason for Progression by Intervals, 250

Tonal Relationship in IMelody depends on the identity of two partial tones, 253

The Octave. Fifth, and Fourth were thus first discovered, 253

Variations in Thirds and Sixths, 255

Scales of Five Tones, used by Chinese and Gaels, 258

The Chromatic and Enharmonic Scales of the Greeks, 262

The Pythagorean Scales of Seven tones, 266

The Greek and Ecclesiastical Tonal ]\Iodes, 267

Early Ecclesiastical Modes, 272

The Rational Construction of the Diatonic Scales by the principle of Tonal Relationship in

the first and second degrees gives the five Ancient ^Melodic Scales, 272 Introduction of a more Accurate Notation for Pitch, 276

Peculiar discovery of natural Thirds in the Arabic and Persian Tonal Systems, 280 The meaning of the Leading Note and consequent alterations in the Modern Scales, 285

CHAPTEE XV. The Consonant Chords of the Tonal Modes, pp. 290-309.

Chords as the Representatives of compound Musical Tones with peculiar qualities, 290 Reduction of aU Tones to the closest relationship in the popular harmonies of the Manor Mode, 292 ^ i- i- . j

Ambiguity of Iifinor Chords, 294

The Tonic Chord as the centre of the Sequence of Chords, 296

Relationship of Chords of the Scale, 297

The ]\Iajor and INIinor IModes are best suited for Harmonisation of all the Ancient Modes,

298 Modern Remnants of the old Tonal IModes, 306

CHAPTEE XVI. The System of Keys, pp. 310-330.

Relative and Absolute Character of the different Keys, 310 Modulation leads to Tempering the Intonation of the Intervals, 312

Hauptmann's System admits of a Simplification vfhich makes its Realisation more Practi- cable, 315 Description of an Harmonium with Just Intonation, 316 Disadvantages of Tempered Intonation, 322 Modulation for Just Intonation, 327

CONTENTS. CHAPTEE XVII. Of Discords, pp. 330-350.

Envuneration of the Dissonant Intervals in the Scale, 331

Dissonant Triads, 338

Chords of the Seventh, 341

Conception of the Dissonant Note in a Discord, 346

Discords as representatives of compound tones, 347

CHAPTEE XVIII. Laws of Progression of Parts, pp. 350-362.

Tlie IMusical Connection of the Notes in a INIelody, 350

Consequent Rules for the Progression of Dissonant Notes, 353

Resolution of Discords, 354

Choral Sequences and Resolution of Chords of the Seventh, 355

Prohibition of Consecutive Fifths and Octaves, 369

Hidden Fifths and Octaves, 3G1

False Relations, 361

CHAPTEE XIX. EsTHETicAL Eelations, pp. 362-371.

Review of Results obtained, 362

Law of Unconscious Order in Works of Art, 366

The Law of IMelodic Succession depends on Sensation, not on Consciousness, 368

And similarly for Consonance and Dissonance, 869

Conclusion, 371

APPENDICES, pp. 327-556.

I. On an Electro-Magnetic Driving IMachine for the Siren, 372

II. On the Size and Construction of Resonators, 372

III. On the Motion of Plucked Strings, 374

IV. On the Production of Simple Tones by Resonance, 377 V. On tlie Vibrational Forms of Pianoforte Strings, 380

VI. Analysis of the ]\Iotion of Violin Strings, 384 VII. On the Theory of Pipes, 388

A. Influence of Resonance on Reed Pipes, 388

B. Theory of the Blowing of Pipes, 390

I. The Blowing of Reed Pipes, 390

II. The Blowing of Flue Pipes, 394 [Additions by Translator, 396]

VIII. Practical Directions for Performing the Experiments on the Composition of Vowels, 398 IX. On the Phases of Waves caused by Resonance, 400 X. Relation between the Strength of Sympathetic Resonance and the Length of Time

required for the Tone to die away, 405 XL Vibrations of the Wembrana Basilaris in the Cochlea, 406 XII. Theory of Combinational Tones, 411

XIII. Description of the Mechanism employed for opening the several Series of Holes in

the Polyphonic Siren, 413

XIV. Variation in the Pitch of Simple Tones that Beat, 414

XV. Calculation of the Intensity of the Beats of Different Intervals, 415 XVI. On Beats of Combinational Tones, and on Combinational Tones in the Siren and

Harmonium, 418 XVII. Plan for Justly-Toned Instruments with a Single Manual, 421 XVIII. Just Intonation in Singing, 422 XIX. Plan of Mr. Bosanquet's Manual, 429 [XX. Additions by the Translator, 430-556

*»* See separate Tables of Contents prefixed to each Section. [Sect. A. On Temperament, 430 [Sect. B. On the Determination of Pitch Numbers, 441

CONTENTS.

[App. XX. Additions by the Tra.nsla,toi-—coniinued.

*^* See separate Tables of Contents prefixed to each Section. [Sect. C. On the Calculation of Cents from Interval Ratios, 446 [Sect. D. Musical Intervals, not exceeding an Octave, arranged in order of Width 451 '

[Sect. E. On Musical Duodenes, or the Development of Just Intonation for

Harmony, 457 [Sect. P. Experimental Instruments for exhibiting the effects of Just Intonation 466 '

[Sect. G. On Tuning and Intonation, 483 [Sect. H. The History of Musical Pitch in Europe, 493 [Sect. K. Non-Harmonic Scales, 514

[Sect. L. Recent Work on Beats and Combinational Tones, 527 [Sect. M. Analysis and Synthesis of Vowel Sounds, 538 [Sect. N. Miscellaneous Notes, 544

[INDEX, 557-576]

LIST OF FIGURES

1. Seebeck's Siren, lie

2, 3, 4. Cagniard de la Tour's Siren, 12b

5. Tuning-fork tracing its Curve, 206

6. Curve traced in Phonautograph, 20d

7. Curve of Simple Vibration, 216

8. Curve of ]\Iotion of Hammer moved by

Water-wheel, 2lc

9. Curve of :Motion of Ball struck up on

its descent, 21c

10. Reproduction of fig. 7, 2M

11. Curve shewing the Composition of a

simple Note and its Octave in two different phases, 306, c

12. Curve shewing the Composition of a

simple note and its Twelfth in two different phases, 326

13. Tuning-fork on Resonance Box, 40a

14. Forms of Vibration of a Circular Mem-

brane, 40f, d

15. Pendulum excited by a membrane

covering a bottle, 42«

16. a. Spherical Resonator, 436 b. Cylindi-ical Resonator, 43f

17. Forms of Vibration of Strings, 46«, 6

18. Forms of Vibration of a String de-

flected by a Point, 54«, 6

19. Action of such a String on a Sounding-

board, 54c

20. Bottle and Blow-tube for producing a

simple Tone, 60c

21. Sand figures on circular elastic plates,

71c

22. The Vibration Microscope, 816

23. Vibrations as seen in the Vibration

Microscope, 826 •4. Vibrational Forms for the middle of a Viohn String, 836

25. Crumples on the vibrational form of a

violin string, 846

26. Gradual development of Octave on a

violin string bowed near the bridge, 856

27. An open wooden and stopped metal

organ flue-pipe, 88

28. Free reed or Harmonium vibrator, 956

29. Free and striking reed on an organ

pipe partly in section, 96rt, 6

30. IMembranous double reed, 97a

31. Reproduction of fig. 12, 120^, b

32. Fork with electro-magnetic exciter, and

sliding resonance box with a lid (aa-tificial vowels), 1216

35

33. Fork with electro-magnet to serve as

interrupter of the current (artificial vowels), 1226

34. Appearance of figiires seen through the

vibration microscope by two forks when the phase changes but the tuning is correct, I26d The same when the tuning is slightly altered, 127ft

56. Construction of the ear, general view,

with meatus auditorius, labyrinth, cochlea, and Eustachian tube, 129c

57. The three auditory ossicles, hammer,

anvil, and stirrup, in their relative positions, 130c

38. Two views of tlie hammer of the ear,

1316

39. Left temporal bone of a newly-born

child with the auditory ossicles in situ, 131c

40. Right drumskin with hammer seen

from the inside, 131c

41. Two views of the right anvil, 133«

42. Three views of the right stirrup, 134a

43. A, left labyrinth from without. B,

right labyrinth from within. C, left labyrinth from above, 1366, c

44. Utriculus and membranous semicircular

canals (left side) seen from without, 137«

45. Bony cochlea (right side) opened in

front, 1.37c, d Transverse section of a spire of a

cochlea which has been softened

in hydrochloric acid, ISSa, b Max Schultze's hairs on the internal

surface of the epithsnum in the

am^ndkc, 138c, d

48. Expansion of the cochlean nerve, 139c

49. Corti's membrane, 140rt, 6, c

50. Corti's rods or arches separate, 140(/

51. Corti's rods or arches in situ, 1416, c

52. Diagram of the law of decrease of sym-

pathetic resonance, 144c, d Interference of similarly disposed

waves, 1606 Interference of dissimilarly disposed

waves, 160c

55. Lines of silence of a tuning-fork,

161c

56. The Polyphonic Siren, 162

57. Diagram of origin of beats, 165ff

46.

47.

53.

54.

XVI

LIST OF PASSAGES IN MUSICAL NOTES.

58. Phouautographic representation of

beats, 166rt

59. Identical with fig. 52 but now taken to

shew the intensity of beats excited by tones making different intervals, 172c

60. A and B. Diagram of the comparative

roughness of intervals in the first and second octaves, 193b, c

61. Diagram of the roughness of dissonant

intervals, 333«

62. Reproduction of fig. 24 A, p. 385&

63. Diagram of the motion of a violin

string, 387c

14. Diagram of the arrangements for the

experiments on the composition of

vowels, 399b, c <5. Mechanism for opening the several

series of holes in the Polyphonic

Siren, 414rt, 16. Section, Elevation, and Plan of Mr.

Bosanquet's Manual, 429

Th Additions by Translator. 57. Perspective view of Mr. Colin Brown's

Fingerboard, 47 Id 18. Perspective view, 69 plan, 70 section

of Mr. H. W. Poole's Keyboard, 475

LIST OF PASSAGES IN MUSICAL NOTES.

The small octave, 15fZ

The once and twice accented octave, 16a, b

The great octave, 166

The first 16 Upper Partials of C'66, 22c

The first 8 Upper Partials of 6132, 50«-

Prof. Helmholtz's Vowel Resonances, UOb

First differential tones of the usual har- monic interval, 1546

Differential tones of different orders of the usual harmonic intervals, 1556, c

Summational tones of the usual harmonic intervals, 156ft

Examples of beating partials, 180c

Coincident partials of the principal con- sonant intervals, 183(Z

Coincident converted into beating partials by altering pitch of upper tone, 186c

Examples of intervals in which a pair of partials beat 33 times in a second, 1 92«

Major Triads with their Combinational Tones, 215a

Minor Triads with their Combinational Tones, 2156

Consonant Intervals and their Combina- tional Tones, 218c

The most Perfect Positions of the Major Triads, 219c

The less Perfect Positions of the Major Triads, 220c

The most Perfect Positions of the Minor Triads, 2216

The less Perfect Positions of the INIinor Triads, 221c

The most Perfect Positions of Major

Tetrads within the Compass of Two

Octaves, 223c Best Positions of Minor Tetrads with their

false Combinational Tones, 224« Ich bin spatziercn gegangen, 2386 Sic canta comma, 2396 Palestrina's Stabat Mater, first 4 bars,

247c Chinese air after Barrow, 260« Cockle Shells, older form, 2606 Blythe, blythc, and merry are vx, 261ffl Chinese temple hymn after Bitschurin, 2616 Braes of Bulqtihidder, 261c Five Forms of Closing Chords, 291c Two complete closes, 293c IMode of the Fourth, three forms of com- plete cadence, 302(7 Concluding bars of S. Bach's Chorale, Was

viein Gott ivill, das gescheJi' allzeit, 3046 End of S. Bach's Hymn, Veni redemptor

gentium, 305a Doric cadence from And with His stripes

we are healed, in Handel's Messiah, 307a Doric cadence from Hear, Jacob's God, in

Handel's Samson, 3076 Examples of False Minor Triad, 340a Examples of Hidden Fifths, 361c? Example of Duodenals, 465c Mr. H. W. Poole's method of fingering and

treatment of the harmonic Seventh, 477a Mr, H. W. Poole's Double Diatonic or Di-

chordal Scale in Ci' with accidentals, 478a

LIST OF TABLES.

Pitch Numbers of Notes in Just JMajor Scale, 17«

[Scale of Haimonical, '17c, d]

[Analogies of notes of the piano and colours of the Spectrum, IBd']

Pitch of the different forms of vibration of a circular membrane, 41c

Relative Pitch Numbers of the prime and proper tones of a red free at both ends, 56«

Proper Tones of circular elastic plates, 72a

Proper Tones of Bells, 72c

Proper Tones of Stretched Membranes, 7Sb

Theoretical Intensity of the Partial Tones of Strings, 7i'c

[Velocity in Soimd in tubes of different diameters — Blaikley, 9Qd]

[Partials of £\) Clarinet— Blaikley, 99c]

[Harmonics of £\^ horn, 99d]

[Compass of Eegisters of male and female voices — Behnke, ] 01 d]

Vowel trigram — Du Bois Raymond, senior, 106&

Vowel Resonances according to Helmholtz and Donders, l(i9b

[Vowel Resonances according to (1) Reyher, [i) Hellwag, (3) Florcke, (4) Donders after Helmholtz, (5) Dondeis after Merkel, (6) Helmholtz, (7) Merkel, (8) Koenig.. (9) Trautmann, 109rf]

Willis's Vowel Resonances, 117c

[Relative force of the partials for producing

different vowels, j24f/] Relation of Strength of Resonance to Alterations of Phase, 12oa

Difference of pitch, &c., necessary to reduce sympathetic vibration to J^ of that pro- duced by perfect unisonance, 143a

Numbers from wh ich fig. 52 was constructed, 145a

Measurements of the basilar membrane in a new-born child, 145c

Alteration of size of Corti's rods as they approach the vertex of the cochlea, 145ci

[Preyer's distinguishable and undistin- guishable intervals, 147f/]

First differential tones of the usual har- monic intervals, L'.4«

[Differential tones of different orders of the usual harmonic intervals, 155(/]

Different intervals which would give 33 beats of their primes, 172a

[Pitch numbers of Appunn's bass reeds,

1776] [Experiments on audibility of very deep

tones, 177c] Coincident partials for the principal con- sonances, 183a Pitch numbers of the primes which make

consonant iaitei-vals with a tone of 300

vib., 184c Beating partials of the notes in the last

table with a note of 301 vib., and number

of beats, 184c^ Disturbance of a consonance by altering

one of its tones by a Semitone, 185c Influence of different consonances on each

other, 187b [Upper partials of a just Fifth, 188d] [Upper partials of an altered Fifth, 189c] [Comparison of the upper partials of a

Fourth and Eleventh, major Sixth and

major Thirteenth, minor Sixth and

minor Thirteenth, lS9c?and 1906, c] [Comparison of the upper partials of a

major and a minor Third, 190c?] [Comparison of the uj^per partials of aU

the usual consonances, pointing out

those which beat, 1916, c] [Comparison of the upper partials of

septimal consonances, involving the

seventh partial, and pointing out which

beat, 195c, d] [General Table of the first 16 harmonics of

C'66, shewing how they affect each other

in any combination, 197c, d] Table of partials of 200 and 301, shewing

their differential tones, 198c Table of possible triads, shewing consonant,

dissonant, and septimal intervals, 2126, c Table of consonant triads, 214a [The first 16 harmonics of C, 2Ud] [Calculation of the Combinational Tones of

the Major Triads, 214rf] [Most of the first 40 harmonics oiA^,\f, 215c] [Calculation of the Combinational Tones of

the Minor Triads, 21:>d] [Calculation of the Differential Tones of

the Major Triads in their most Perfect

Positions, 2l9d] [Calculation of the Combinational Tones

of the Major Triads in the less Perfect

Positions, 220d] [Calculation of the Combinational Tones of

LIST OF TABLES.

the Minor Triads in the most and less

Perfect Positions of the Minor Triads,

221d, d'] [Calculation of the false Combinational

Tones of Minor Tetrads in their best

positions, 224f?] Ecclesiastical Modes, 245c, d Partial Tones of the Tonic, 257a [Pentatonic Scales, 259c, d] [Tetrachords 1 to 8, with intervals in

cents, 263d'] Greek Diatonic Scales, 267c [Greek Diatonic Scales with the intervals

in cents, 268c] [Greek Diatonic Scales reduced to begin- ning with c, with the intervals in cents,

268f^'] Greek modes with the Greek Ecclesiastical

and Helmholtzian names, 269a Later Greek Scale, 270a Tonal Keys, 270c Ecclesiastical Scales of Ambrose of Milan,

2716, c The Five Melodic Tonal Modes, 272b [The Seven Ascending and Descending

Scales, compared with Greek, with inter- vals in cents, 274e, d] [The different scales formed by a dif- ferent choice of the intercalary tones,

277c', rf'] The Five Modes with variable intercalary

tones, 278a, b [J. Curwen's characters of the tones in

the major scale, 279&, c] [Arabic Scale in relation to the major

Thirds, 281rf'] Arabic Scales, 2826-283c [Prof. Land's account of the 12 Arabic

Scales, 284 note] Five Modes as formed from three chords

each, 293c?, 294a The same with double intercalary tones,

297c, d The same, final form, 2986, c Trichordal Eelations of the Tonal Modes,

:309rf [Thirds and Sixths in Just, Equal, and

Pythagorean Intonation compared, 313c] [Combinational Tones of Just, Equal, and

Pythagorean Intonation compared, 314(i] The Chordal System of Prof. Helmholtz's

Just Harmonium, 316c [Duodenary statement of the tones on Prof.

Helmholtz's Just Harmonium, 317c, d] The Chordal System of the minor keys

on Prof. Helmholtz's Just Harmonium,

318a, b, d [Table of the relation of the Cycle of 58 to

Just Intonation, 3296, c] [Tabular Expression of the Diagram, fig.

61, 332] [Table of Roughness, 3ZM]

Measurements of Glass Resonators, 373c

Measurements of resonance tubes men- tioned on p. 55a, Z77d

Table of tones of a conical pipe of zinc, calculated from formula 393c [with sub- sidiary tables, 393c? and 394c]

Table of Mayer's observations on numbers of beats, 418a

Table of four stops for a single manual justly intoned instrument, 421c

Table of five stops for the same, 422a

In the Additions by Translator. Table of Pythagorean Intonation, 4336, c Table of Meantone Intonation, 4346 Table of Equal Intonation, 437c, d Synonymity of Equal Temperament, 4386 Synonymity of Mr. Bosanquet's Notes in

Fifths, 439a Notes of Mr. Bosanquet's Cycle of 53 in

order of Pitch, 4396, c, d Expression of Just Intonation in the Cycle

of 1200, p. 440 Principal Table for calculation of cents,

450a, Auxiliary Tables, 451a Table of Intervals not exceeding one Octave,

4536 Unevenly numbered Harmonics up to the

D3rd, 457a Number of any Interval not exceeding a

Tritone, contained in an Octave, 457c Harmonic Duodene or Unit of Modulation,

461a The Duodenarium, 463a Fingerboard of the Harmonical, first four

Octaves, with scheme, 4676, fifth Octave,

468rf Just Harmonium scheme, 470a Just English Concertina scheme, 4706 Mr. Colin Brown's Voice Harmonium

Fingerboard and scheme, 471a Rev. Henry Liston's Organ and scheme,

4736 Gen. Perronet Thompson's Organ scheme,

A7Zd Mr. H. Ward Poole's 100 tones, 474c Mr. H. W. Poole's scheme for keys of F,

C, G, 476a Mv. Bosanquet's Generalised Keyboard,

480 Expression of the degrees of the 53 divi- sion by multiples of 2, 5 and 7, p. 481c Typographical Plan of Mr. J. Paul White's

Fingerboard, 4826 Specimens of tuning in Meantone Tem- perament, 484c Specimens of tuning in Equal Tempera- ment, 4856 Pianoforte Tuning — Fourths and Fifths,

485d Cornu and Mercadier's observation on

Violin Intonation, 486c to 4876

LIST OF TABLES.

Scheme for tuning in Equal Temperament, 4895

Proof of rule for tuning in Equal Tempera- ment, 490e, d

Proof of rule for Tuning in Meantoue Tem- perament, 492«

Historical Pitches in order from Lowest to Highest, 49 5« to 504rt

Classified Index to the last Table, ri04& to

Effects of the length of the foot in differ- ent countries on the pitch of organs, 512a

Non-harmonic scales, 514c to 519c

Vowel sound ' Oh ! ' Analysis at various

pitches by Messrs. Jenkin & Ewing, 539d

to 5416 Vowel sounds ' oo,' 'awe,' 'ah,' analysis

at various pitches by Messrs. Jenkin

& Ewing, p. 541c, d Mean and actual Compass of the Human

Voice, 545«, b, e True Tritonic, False Tritonic, Zarlino's,

Meantone and Equal Temperaments,

compared, 548a

Presumed Characters of Major and Minor

Keys, 551-«, h

INTEODUCTION.

In the present work an attempt will be made to connect the boundaries of two sciences, which, although drawn towards each other by many natural atiinities, have hitherto remained practically distinct — I mean the boundaries of physical and physiological acoustics on the one side, and of musical science and esthetics on the other. The class of readers addressed will, consequently, have had very different cultivation, and will be affected by very different interests. It will therefore not be superfluous for the author at the outset distinctly to state his intention in undertaking the work, and the aim he has sought to attain. The horizons of physics, philosophy, and art have of late been too widely separated, and, as a consequence, the language, the methods, and the aims of any one of these studies present a certain amount of difficulty for the student of any other H of them ; and possibly this is the principal cause why the problem here undertaken has not been long ago more thoroughly considered and advanced towards its solution.

It is true that acoustics constantly employs conceptions and names borrowed from the theory of harmony, and speaks of the 'scale,' 'intervals,' ' consonances,' and so forth ; and similarly, manuals of Thorough Bass generally begin with a physical chapter which speaks of ' the numbers of vibrations,' and fixes their 'ratios' for the different intervals; but, up to the present time, this apparent connection of acoustics and music has been wholly external, and may be regarded rather as an expression given to the feelmg that such a connection must exist, than as its actual formulation. Physical knowledge may indeed have been useful for musical instrument makers, but for the development and foundation of the theory of harmony H It has hitherto been totally barren. And yet the essential facts within the field here to be explained and turned to account, have been known from the earliest times. Even Pythagoras (fl. circa B.C. 540-510) knew that when strings of different lengths but of the same make, and subjected to the same tension, were used to give the perfect consonances of the Octave, Fifth, or Fourth, their lengths must be in the ratios of 1 to 2, 2 to '6, or 3 to 4 respectively, and if, as is probable, his knowledge was partly derived from the Egyptian priests, it is impossible to conjecture in what remote antiquity this law was first known. Later physics has extended the law of Pythagoras by passing from the lengths of strings to the number of vibra- tions, and thus making it applicable to the tones of all musical instruments,

and the numerical relations 4 to 5 and 5 to «i have been added to the above

u

- PLAN OF THE WORK. introd.

for the less perfect consonances of the major and minor Thirds, but I am not aware that any real step was ever inade towards answering the ques- tion : What have musical consonances to do ivith the ratios of the first six numbers ! Musicians, as well as philosophers and physicists, have generally contented themselves with saying in effect that human minds were in some unknown manner so constituted as to discover the numerical relations of musical vibrations, and to have a peculiar pleasure in contemplating simple ratios which are readily comprehensible.

Meanwhile musical esthetics has made unmistakable advances in those points which depend for their solution rather on psychological feeling than on the action of the senses, by introducing the conception of movement in

IT the examination of musical works of art. E. Hanslick, in his book On the Beautiful in Music {Ueher das musihalisch Schone), triumphantly attacked the false standpoint of exaggerated sentimentality, from which it was fashionable to theorise on music, and referred the critic to the simple elements of melodic movement. The esthetic relations for the structure of musical compositions, and the characteristic differences of individual forms of composition are explained more fully in Vischer's Esthetics (Aesthetik). In the inorganic world the kind of motion we see, reveals the kind of moving force in action, and in the last resort the only method of recognising and measuring the elementary powers of nature consists in determining the motions they generate, and this is also the case for the motions of bodies or of voices which take place under the influence of human feelings. Hence

^the properties of musical movements which possess a graceful, dallying, or a heavy, forced, a dull, or a powerful, a quiet, or excited character, and so on, evidently chiefly depend on psychological action. In the same way questions relating to the equilibrium of the separate parts of a musical composition, to their development from one another and their connection as one clearly intelligible whole, bear a close analogy to similar questions in architecture. But all such investigations, however fertile they may have been, cannot have been otherwise than imperfect and uncertain, so long as they were without their proper origin and foundation, that is, so long as there was no scientific foundation for their elementary rules relating to the construction of scales, chords, keys and modes, in short, to all that is usually contained in works on ' Thorough Bass '. In this elementary region

U we have to deal not merely with unfettered artistic inventions, but with the natural power of immediate sensation. Music stands in a much closer connection with pure sensation than any of the other arts. The latter rather deal with what the senses apprehend, that is with the images of outward objects, collected by psychical processes from immediate sensation. Poetry aims most distinctly of all at merely exciting the formation of images, by addressing itself especially to iinagination and memory, and it is only by subordinate auxiliaries of a more musical kind, such as rhythm, and imitations of sounds, that it appeals to the immediate sensation of hearing. Hence its efltects depend mainly on psychical action. The plastic arts, although they make use of the sensation of sight, address the eye almost in the same way as poetry addresses the ear. Their main purpose is to excite in us the image of an external object of determinate form and colour. The spectator is essentially intended to interest himself in this

iN-TROD. PLAN OF THE WORK. 3

image, and enjoy its beauty ; not to dwell upon the means by which it was created. It must at least be allowed that the pleasure of a connoisseur or virtuoso in the constructive art shown in a statue or a picture, is not an essential element of artistic enjoyment.

It is only in painting that we find colour as an element which is directly appreciated by sensation, without any intervening act of the intellect. On the contrary, in music, the sensations of tone are the material of the art. So far as these sensations are excited in music, we do not create out of them any images of external objects or actions. Again, when in hearing a concert we recognise one tone as due to a violin and another to a clarinet, our artistic enjoyment does not depend upon our conception of a violin or clarinet, but solely on our hearing of the tones they produce, whereas the ^ artistic enjoyment resulting from viewing a marble statue does not depend on the white light which it reflects into the eye, but upon the mental image of the beautiful human form which it calls up. In this sense it is clear that music has a more immediate connection with pure sensation than any other of the fine arts, and, consequentl}^, that the theory of the sensations of hearing is destined to play a much more important part in musical esthetics, than, for example, the theory of chiaroscuro or of perspective in painting. Those theories are certainly useful to the artist, as means for attaining the most perfect representation of nature, but they have no part in the artistic effect of his work. In music, on the other hand, no such perfect representation of nature is aimed at ; tones and the sensations of tone exist for themselves alone, and produce their effects independently "^ of anything behind them.

This theory of the sensations of hearing belongs to natural science, and comes in the first place under ^A?/sio/o^/<;«/ rtco/^s^/c.s\^ Hitherto it is the physical part of the theory of sound that has been almost exclusively treated at length, that is, the investigations refer exclusively to the motions produced by solid, liquid, or gaseous bodies when they occasion the sounds which the ear appreciates. This physical acoustics is essentially nothing but a section of the theory of the motions of elastic bodies. It is physically indifferent whether observations are made on stretched strings, by means of spirals of brass wire (which vibrate so slowly that the eye can easily follow their motions, and, consequently, do not excite any sensation of sound), or by means of a violin string (where the eye can scarcely perceive the vibrations ^i which the ear readily appreciates). The laws of vibratory motion are pre- cisely the same in both cases ; its rapidity or slowness does not affect the laws themselves in the slightest degree, although it compels the observer to apply different methods of observation, the eye for one and the ear for the other. In physical acoustics, therefore, the phenomena of hearing are taken into consideration solely because the ear is the most convenient and handy means of observing the more rapid elastic vibrations, and the physicist is compelled to study the peculiarities of the natural instrument which he is employing, in order to control the correctness of its indications. In this way, although physical acoustics as hitherto pursued, has, undoubtedly, collected many observations and much knowledge concerning the action of the ear, which, therefore, belong to physiolocjical aconstics, these results were not the principal object of its investigations ; they were merely secondary

B 2

4 PLAN OF THE WORK. introd.

and isolated facts. The only justification for devoting a separate chapter to acoustics in the theory of the motions of elastic bodies, to which it essentially belongs, is, that the application of the ear as an instrument of research influenced the nature of the experiments and the methods of observation.

But in addition to a physical there is a physiological theory of acousticSy the aim of v^hich is to investigate the processes that take place within the ear itself. The section of this science which treats of the conduction of the motions to which sound is due, from the entrance of the external ear to the expansions of the nerves in the labyrinth of the inner ear, has received much attention, especially in Germany, since ground was broken by 11 Johannes Mueller. At the same time it must be confessed that not many results have as yet been established with certainty. But these attempts attacked only a portion of the problem, and left the rest untouched. Investigations into the processes of each of our organs of sense, have in general three different parts. First we have to discover how the agent reaches the nerves to be excited, as light for the eye and sound for the ear. This may be called the physical part of the corresponding physiological investigation. Secondly we have to investigate the various modes in which the nerves themselves are excited, giving rise to their various sensations, and finally the laws according to which these sensations result in mental images of determinate external objects, that is, in perceptions. Hence we have secondly a specially physiological investigation for sensations, and 11 thirdly, a specially psychological investigation for perceptions. Now whilst the physical side of the theory of hearing has been already frequently attacked, the results obtained for its physiological and psychological sections are few, imperfect, and accidental. Yet it is precisely the physio- logical part in especial — the theory of the sensations of hearing — to which the theory of music has to look for the foundation of its structure.

In the present work, then, I have endeavoured in the first place to collect and arrange such materials for the theory of the sensations of hearing as already existed, or as I was able to add from my own personal investigations. Of course such a first attempt must necessarily be somewhat imperfect, and be limited to the elements and the most interesting divisions of the subject discussed. It is in this light that I wish these studies to be regarded. 11 Although in the propositions thus collected there is little of entn-ely new discoveries, and although even such apparently new facts and observations as they contain are, for the most part, more properly speaking the imme- diate consequences of my having more completely carried out known theories and methods of investigation to their legitimate consequences, and of my having more thoroughly exhausted their results than had hare- tofore been attempted, yet I cannot but think that the facts frequently receive new importance and new illumination, by being regarded from a fresh point of view and in a fresh connection.

The First Part of the following investigation is essentially physical and physiological. It contains a general investigation of the phenomenon of harmonic uppier partial tones. The nature of this phenomenon is established, and its relation to qnality of tone is proved. A series of qualities of tone are analysed in respect to their harmonic upper partial tones, and it results

iNTROD. PLAN OF THE WORK. 5

that these upper partial tones are not, as was hitherto thought, isolated phenomena of small importance, but that, with very few exceptions, they determine the qualities of tone of almost all instruments, and are of the greatest importance for those qualities of tone which are best adapted for musical purposes. The question of how the ear is able to perceive these harmonic upper partial tones then leads to an hypothesis respecting the mode in which the auditory nerves are excited, which is well fitted to reduce all the facts and laws in this department to a relatively simple mechanical conception.

The Second Part treats of the disturbances produced by the simultaneous production of two tones, namely the comhimitional tones and heat:. The physiologico-physical investigation shows that two tones can besimul-^ taneously heard by the ear without mutual disturbance, when and only when they stand to each other in the perfectly determinate and well-known relations of intervals which form musical consonance. We are thus imme- diately introduced into the field of music proper, and are led to discover the physiological reason for that enigmatical numerical relation announced by Pythagoras. The magnitude of the consonant intervals is independent of the quality of tone, but the harmoniousness of the consonances, and the distinctness of their separation from dissonances, depend on the quality of tone. The conclusions of physiological theory here agree precisely with the musical rules for the formation of chords ; they even go more into par- ticulars than it was possible for the latter to do, and have, as I believe, the authority of the best composers in their favour. ^

In these first two Parts of the book, no attention is paid to esthetic considerations. Natural phenomena obeying a blind necessity, are alone treated. The Third Part treats of the construction of musical scales and )wtes. Here we come at once upon esthetic ground, and the differences of national and individual tastes begin to appear. Modern music has especially developed the principle of tonality, which connects all the tones in a piece of music by their relationship to one chief tone, called the tonic. On admitting this principle, the results of the preceding investigations furnish a method of constructing our modern musical scales and modes, from which all arbitrary assumption is excluded.

I was unwilling to separate the physiological investigation from its musical consequences, because the correctness of these consequences must H be to the physiologist a verification of the correctness of the physical and physiological views advanced, and the reader, who takes up my book for its musical conclusions alone, cannot form a perfectly clear view of the meaning and bearing of these consequences, unless he has endeavoured to get at least some conception of their foundations in natural science. But in order to facihtate the use of the book by readers w^ho have no special knowledge of physics and mathematics, I have transferred to appendices, at the end of the book, all special instructions for performing the more comphcated experiments, and also all mathematical investigations. These appendices are therefore especially intended for the physicist, and contain the proofs of my assertions.* In this way I hope to have consulted the interests of both classes of readers.

* [The additional Appendix XX. bj' the Translator is intended especially for the use of musical students. — Translator.]

6 PLAN OF THE WORK. ixtrod.

It is of course impossible for any one to understand the investigations thoroughly, who does not take the trouble of becoming acquainted by per- sonal observation with at least the fundamental phenomena mentioned. Fortunately with the assistance of common musical instruments it is easy for any one to become acquainted with harmonic upper partial tones, com- binational tones, beats, and the like.* Personal observation is better than the exactest description, especially when, as here, the subject of investiga- tion is an analysis of sensations themselves, which are always extremely difficult to describe to those who have not experienced them.

In my somewhat unusual attempt to pass from natural philosophy into the theory of the arts, I hope that I have kept the regions of physiology

H and esthetics sufficiently distinct. But I can scarcely disguise from myself, that although my researches are confined to the' lowest grade of musical grammar, they may probably appear too mechanical and unworthy of the dignity of art, to those theoreticians who are accustomed to summon the enthusiastic feelings called forth by the highest works of art to the scientific investigation of its basis. To these I would simply remark in conclusion, that the following investigation really deals only with the analysis of actually existing sensations — that the physical methods of observation employed are almost solely meant to facilitate and assure the work of this analysis and check its completeness — and that this analysis of the sensations would suffice to furnish all the results required for musical theory, even independently of my physiological hypothesis concerning the mechanism of

^ hearing, already mentioned (p. oa), but that I was unwilling to omit that hypothesis because it is so well suited to furnish an extremely simple con- nection between all the very various and very complicated phenomena which present themselves in the course of this investigation. t

* [But the use of the H(trmonical, described London, ]Macmillan, 1873. Such readers will

in App. XX. sect. F. No. 1, and invented for also find a clear exposition of the physical

the purpose of illustrating the theories of this relations of sound in J. Tyndall, On Souiul,

work, is recommended as greatly superior for a course of eight lectures, London, 1867, (the

students and teachers to any other instrument. last or fourth edition 188.3) Longmans, Green,

— Transhitor.'] & Co. A German translation of this work,

t Readers unaccustomed to mathematical entitled Der Schall, edited by H. Helmholtz

and physical considerations will find an and G. Wiedemann, was published at Bruns-

abridged account of the essential contents of wick in 1874. this Ijook in Sedley Taylor, Sound and Musk,

*^* [The marks ^ in the outer margin of each page, separate the page into 4 sections, referred to as a, >>, c, d, placed after the number of the page. If any section is in doul)le columns, the letter of the second column is accented, as p. i:3.r.]

PART I.

ON THE COMPOSri'ION OF YIBHATIONS,

uppp:r partial tones, a^td qualttib:s of toxe.

CHAPTER I.

ox THE SENSATION OF SOUND IN GENERAL.

Sensations result from the action of an external stimulus on the sensitive apparatus of oiir nerves. Sensations differ in kind, partly with the organ of sense excited, and partly with the nature of the stimulus employed. Each organ of sense pro- duces peculiar sensations, which cannot be excited by means of any other; the eye gives sensations of light, the ear sensations of sound, the skin sensations of touch. Even when the same sunbeams which excite in the eye sensations of light, impinge on the skin and excite its nerves, they are felt only as heat, not as light, wt In the same way the vibration of elastic bodies heard by the ear, can also be felt by the skin, but in that case produce only a whirring fluttering sensation, not sound. The sensation of sound is therefore a species of reaction against external stimulus, peculiar to the ear, and excitable in no other organ of the body, and is completely distinct from the sensation of any other sense.

As our problem is to study the laws of the sensation of hearing, our fir:jt business will be to examine how many kinds of sensation the ear can generate, and what differences in the external means of excitement or sound, correspond to these differences of sensation.

The first and principal difference between various sounds experienced by our ear, is that between 7ioises and musical tom^s. The soughing, howling, and whistling of the wind, the splashing of water, the rolling and rumbling of carriages, are examples of the first kind, and the tones of all musical instruments of the second. Noises and musical tones may certainly intermingle in very various degrees, and *t pass insensibly into one another, but their extremes are widely separated.

The nature of the difference between musical tones and noises, can generally be determined by attentive aural observation without artificial assistance. We perceive that generally, a noise is accompanied by a rapid alternation of different kinds of sensations of sound. Think, for example, of the rattling of a carriage over granite paving stones, the splashing or seething of a waterfall or of the waves of the sea, the rustling of leaves in a wood. In all these cases we have rapid, irregular, but distinctly perceptible alternations of vr.rious kinds of sounds, which crop up fitfully. When the wind howls the alternation is slow, the sound slowly and gradually rises and then falls again. It is also more or less possible to separate restlessly alternating sounds in case of the greater number of other noises. We shall hereafter become acquainted with an instrument, called a resonator, which will materially assist the ear in making this separation. On the other hand, a musical tone strikes the ear as a perfectly luidisturbed, luiiform sound which

8 NOISE AND MUSICAL TONE. tart i.

remains unaltered as long as it exists, and it presents no alternation of various kinds of constituents. To this then corresponds a simple, regular kind of sensation, whereas in a noise many various sensations of musical tone are irregularly mixed up and as it were tumbled about in confusion. We can easily compound noises out of musical tones, as, for example, by simultaneously striking all the keys con- tained in one or two octaves of a pianoforte. This shows us that musical tones are the simpler and more regular elements of the sensations of hearing, and that we have consequently first to study the laws and peculiarities of this class of sensations.

Then comes the further question : On what difterence in the external means of excitement does the difference between noise and musical tone depend 1 The normal and usual means of excitement for the human ear is atmospheric vibration. ^ Tiie irregularly alternating sensation of the ear in the case of noises leads us to conclude that for these the vibi-ation of the air must also change irregularl}'. For musical tones on the other hand we anticipate a regular motion of the air, con- tiniiing uniformly, and in its turn excited by an equally regular motion of the sonorous body, whose impulses were conducted to the ear by the air.

Those regular motions which produce musical tones have been exactly investi- gated by physicists. They are oscillations, vibrations, or swings, that is, up and down, or to and fro motions of sonorous bodies, and it is necessary that these oscillations should be regularly perioilic. By a periodic motion we mean one which constantly returns to the same condition after exactly equal intervals of time. The length of the equal intervals of time between one state of the motion and its next exact repetition, we call the length of the oscillation, vibration, or swing, or the period of the motion. In what manner the moving body actually moves during one period, is perfectly inditterent. As illustrations of periodical motion, take the ^motion of a clock pendulum, of a stone attached to a string and whirled round in a circle with uniform velocity, of a hammer made to rise and fall uniformly by its connection with a water wheel. All these motions, however different be their form, are periodic in the sense here explained. The length of their periods, which in the cases adduced is generally from one to several seconds, is relatively long in comparison with the much shorter periods of the vibrations producing nuisical tones, the lowest or deepest of which makes at least 30 in a second, while in other cases their number may increase to several thousand in a second.

Our definition of periodic motion then enables us to answer the question pro- posed as follows : — The sensation of a musical tone is due to a rapid periodic inotion of the sonorous body ; the sensation of a noise to non-periodic motions.

The musical vibrations of solid bodies are often visible. Although they may be too rapid for the eye to follow them singly, wc easily recognise that a sounding- string, or tuning-fork, or the tongue of a reed-pipe, is rapidly vibrating between two ^ fixed limits, and the regulai", apparently immovable image that we see, notwith- standing the real motion of the body, leads us to conclude that the backward and forward motions are quite regular. In other cases we can feel the swinging motions of sonorous solids. Thus, the player feels the trembling of the reed in the mouth- piece of a clarinet, oboe, or bassoon, or of his own lips in the mouthpieces of trumpets and trombones.

The motions proceeding from the sounding bodies are usually conducted to our ear by means of the atmosphere. The particles of air must also execute periodi- cally recurrent vibrations, in order to excite the sensation of a musical tone in our ear. This is actually the case, although in daily experience sound at first seems to be some agent, which is constantly advancing through the air, and projjagating itself further and further. We must, however, here distinguish between the motion of the individual particles of air — which takes place periodically backwards and forwards within very narrow limits — and the propagation of the sonorous tremor. The latter is constantly advancing by the constant attraction of fresh particles into its sphere of tremor.

CHAP. I. PROPAGATION OF SOUND. 9

This is a peculiarity of all so-called ii)i<hd(tto)-r/ motions. Suppose a stone to be thrown into a piece of calm water. Round the spot struck there forms a little ring of wave, which, advancing equally in all directions, expands to a constantly increasing circle. Corresponding to this ring of wave, sound also proceeds in the air from the excited point and advances in all directions as far as the limits of the mass of air extend. The process in the air is essentially identical with that on the surface of the water. The principal difference consists in the spherical propagation of sound in all directions through the atmosphere which fills all surrounding space, whereas the waves of the water can only advance in rings or circles on its surface. The crests of the waves of water correspond in the waves of sound to spherical shells where the air is condensed, and the troughs to shells where it is rarefied. On the free surface of the water, the mass when compressed can slip upwards and so form ridges, but in the interior of the sea of air, the mass must be condensed, as there is no unoccupied spot for its escape. «[I

The waves of water, therefore, continually advance without returning. But we nuist not suppose that the particles of water of which the waves are composed advance in a similar manner to the waves themselves. The motion of the particles of water on the surface can easily be rendered visible b}^ floating a chip of wood upon it. This will exactly share the motion of the adjacent particles. Now, such a chip is not carried on by the rings of wave. It only bobs up and down and finally rests on its original spot. The adjacent particles of water move in the same manner. When the ring of wave reaches them they are set bobbing ; when it has passed over them they are still in their old place, and remain there at rest, while the ring of wave continues to advance towards fresh spots on the surface of the water, and sets new particles of water in motion. Hence the waves which pass over the surface of the water are constantly built up of fresh particles of water. What really advances as a wave is only the tremor, the altered form of the surface, while the individual particles of water themselves merely move up and down ^ transiently, and never depart far from their original position.

The same relation is seen still more clearly in the waves of a rope or chain. Take a flexible string of several feet in length, or a thin metal chain, hold it at one end and let the other hang down, stretched by its own weight alone. Now, move the hand by which you hold it quickly to one side and back again. The excursion which we have caused in the upper end of the string by moving the hand, will run down it as a kind of wave, so that constantly lower parts of the string will make a sidewards excursion while the upper return again into the straight position of rest. But it is evident that while the wave runs down, each individual particle of the string can have only moved horizontally backwards and forwards, and can have taken no share at all in the advance of the wave.

The experiment succeeds still better with a long elastic line, such as a thick piece of india-rubber tubing, or a brass-wire spiral spring, from eight to twelve feet in length, fastened at one end, and slightly stretched by being held with the hand ^ at the other. The hand is then easily able to excite waves wliich will run very regularly to the other end of the line, be there reflected and return. In this case it is also evident that it can be no part of the line itself which runs backwards and forwards, but that the advancing wave is composed of continually fresh particles of the line. By these examples the reader will be able to form a mental image of the kind of motion to which sound belongs, where the material particles of the body merely make periodical oscillations, while the tremor itself is constantly propagated forwards.

Now let us return to the surface of the water. We have supposed that one of its points has been struck by a stone and set in motion. This motion has spread out in the form of a ring of wave over the surface of the water, and having reached the chip of wood has set it bobbing iip and down. Hence by means of the wave, the motion which the stone first excited in one point of the surface of the water has been communicated to the chip which was at another point of the same surface.

10 FORCE, PITCH, AND QUALITY. part i.

The process which goes on in the atmospheric ocean about us, is of a precisely similar nature. For the stone substitute a sounding body, which shakes the air ; for the chip of wood substitute the human ear, on which impinge the waves of air excited by the shock, setting its movable parts in vibration. The waves of air proceeding from a sounding body, transport the tremor to the human ear exactly in the same way as the water transports the tremor produced by the stone to the floating chip.

In this way also it is easy to see how a body which itself makes periodical oscillations, will necessarily set the particles of air in periodical motion. A falling stone gives the surface of the water a single shock. Now replace the stone by a regular series of drops falling from a vessel with a small orifice. Every separate drop will excite a ring of wave, each ring of wave will advance over the surface of the water precisely like its predecessor, and will be in the same way followed by

^ its successors. In this manner a regular series of concentric rings will be formed and propagated over the surface of the water. The number of drops which fall into the water in a second will be the number of waves which reach our floating chip in a second, and the number of times that this chip will therefore bob up and down in a second, thus executing a periodical motion, the period of which is equal to the interval of time between the falling of consecutive drops. In the same way for the atuiosphere, a periodically oscillating sonorous body produces a similar periodical motion, first in the mass of air, and then in the drumskin of our ear, and the period of these vibrations must be the same as that of the vibration in the sonorous body.

Having thus spoken of the principal division of sound into Noise and Musical Tones, and then described the general motion of the air for these tones, we pass on to the peculiarities which distinguish such tones one from the othei*. We are acquainted with three points of difference in musical tones, confining oiu' attention

m in the first place to such tones as are isolatedly produced by our usual musical instruments, and excluding the sinudtaneous sounding of the tones of different instruments. Musical tones are distinguished : —

1. By their force,

2. By their 2jifc/i,

3. By their qtialiti/.

It is unnecessary to explain what we mean by the force and pitch of a tone. By the (piality of a tone we mean that peculiarity which distinguishes the musical tone of a violin from that of a flute or that of a clarinet, or that of the hiunan voice, when all these instruments produce the same note at the same pitch.

We have now to explain what peculiarities of the motion of sound correspond to these three principal diftcrences between musical tones.

First, We easily recognise that the force of a musical tone increases and dimi- nishes with the extent or so-called amplitude of the oscillations of the particles of ■T the sounding body. When we strike a string, its vibrations are at first sufficiently large for us to see them, and its corresponding tone is loudest. The visible vibrations become smaller and smaller, and at the same time the loudness diminishes. The same observation can be made on strings excited by a violin bow, and on the reeds of reed-pipes, and on many other sonorous bodies. The same conclusion results from the diminution of the loudness of a tone when we increase our distance from the sounding body in the open air, although the pitch and quality remain unaltered ; for it is only the amplitude of the oscillations of the particles of air which diminishes as their distance from the sounding body increases. Hence loudness must depend on this amplitude, and none other of the properties of sound do so.*

* Mechanically the force of the oscillations no measure can be found for the intensity of

for tones of different pitch is measured by the sensation of sound, that is, for the loudness

their vis viva, that is, by the square of the of sound which will hold all pitches. [See

greatest velocity attained by the oscillating the addition to a footnote on p. 75f/, referring

particles. But the ear has different degrees of especially to this passage. — Translator.] sensibility for tones of different pitch, so that

1

PITCH AND THE SIREN.

The second essential difference between difterent musical tones consists in their J) if c/i. Daily experience shows iis that mnsical tones of the same pitch can be prodnced upon most diverse instruments by means of most diverse mechanical contrivances, and with most diverse degrees of loudness. All the motions of tlie air thus excited must be periodic, because they would not otlierwise excite in us the sensation of a nmsical tone. But the sort of motion within each single period may be any whatever, and yet if the length of the periodic time of two musical tones is the same, they have the same pitch. Hence : Fitrk (Ujwnih solely on the length of time in which each single vibration is executed, or, which comes to the same thing, on the number of vibrations completed in a given time. We are accustomed to take a second as the unit of time, and shall consequently mean by the pitch number [or frequoici/] of a tone, the number of vibrations which the particles of a sounding body perform in one second of time.* It is self-evident that we find the periodic time or vibrational period, that is length of time which H is occupied in performing a single vibration backwards and forwards, by dividing- one second of time by the pitch number.

Musical tones are said to be higher, the greater their pi frit numbers, f/i<(t is, the shorter their vibrational periods.

The exact determination of the pitch niuuber for such elastic bodies as produce audible tones, presents considerable difficulty, and physicists had to contrive many comparatively complicated processes in order to solve this problem for each particular case. Mathematical theory and numerous experiments had to render mutual assistance. t It is consequently very convenient for the demonstration of the fundamental facts in this department of knowledge, to be able to apply a peculiar instrument for producing musical tones — the so-called siren — which is constructed in such a manner as to determine the pitch number of tlie tone produced, by a direct observation. The principal parts of the simplest form of the siren are shown in fig. 1, after Seebeck. ^

A is a thin disc of cardboard or tinplate, which can be set in rapid rotation about its axle b by means of a string f f, which passes over a larger wheel. On the margin of the disc there is punched a set of holes at equal intervals : of these

there are twelve in the figure ; one or more similar series of holes at equal distances are introduced on concentric circles (there is one such of eight holes in the figure), c is a pipe which is directed over one of the holes. Now, on setting the disc in rotation and blow- ing through the pipe c, the air will pass freely whenever one of the holes comes under the end of the pipe, but will be checked whenever an unpierced portion ^ of the disc comes under it. Each hole of the disc, then, that passes the end of the pipe lets a single puft" of air escape. Supposing the disc to make a single revolution and the pipe to be directed to the

* [The pitch number was called the ' vibra- tional number' in the first edition of this trans- lation. The pitch n umber of a note is commonly called the pitch of the note. By a convenient abbreviation we often write a' 440, meaning the note a' having the i^itch number 440 ; or say that the pitch of a' is 440 vib. that is, 440 double vibrations in a second. The second texxn. frequency, which I have introduced into the text, as it is much used by acousticians, properly represents Ihc number of times that any periodically recurrinq event happens in one second of time, and, applied to double vibrations, it means the same as pitch number.

The pitch of a musical instrument is the pitch of the note by wliich it is tuned. But as i^itch is properly a sensation, it is necessary here to distinguish from this sensation the pitch number or frcq^icncy of vibration by which it is measured. The larger the pitch number, the higher or sharper the pitch is said to be. The lower the pitch nmnber the deeper or flatter the pitch. These are all metaphorical expressions which must be taken strictly in this sense. — Translator.']

t [An account of the more exact modern methods is given in App. XX. sect. B. — Ti-anslator.']

12

PITCH AND THE SIREN.

outer circle of holes, we have twelve puffs corresponding to the twelve holes; but if the pipe is directed to the inner circle we have only eight puffs. If the disc is made to revolve ten times in one second, the outer circle will produce 120 puffs, in one second, which would give rise to a v/eak and deep musical tone, and the inner circle eighty puff's. Generally, if we know the number of revolutions which the disc makes in a second, and the number of holes in the series to which the tube is directed, the product of these two numbers evidently gives the number of puff's in a second. This number is consequently far easier to determine exactly than in any other musical instrument, and sirens are accordingly extremely well adapted for studying all changes .in musical tones resulting from the alterations and ratios of the pitch numbers.

The form of siren here desci-ibed gives only a weak tone. I have placed it first because its action can be most readily understood,- and, by chana'inu' the disc, it

can be easily applied to experiments of very different descriptions. A stronger tone is produced in the siren of Cagniard de la Tour, shown in figures 2, 3, and 4, above. Here s s is the rotating disc, of which the upper surface is shown in fig. 3, and the side is seen in figs. 2 and 4. It is placed over a windchest A A, which is connected with a bellows by the pipe B B. The cover of the windchest A A, which lies immediately under the rotating disc s s, is pierced with precisely the same number of holes as the disc, and the direction of the holes pierced in the cover of the chest is oblique to that of the holes in the disc, as shown in fig. 4, which is a vertical section of the instrument through the line n n in fig. 3. This position of the holes enables the wind escaping from A A to set the disc s s in rotation, and by increasing the pressure of the bellows, as much as 50 or 60 rotations in a second can be produced. Since all the holes of one circle are blown through at the same time in this siren, a much more powerful tone is produced than in Seebeck's, fig. 1 (p. lie). To record the revolutions, a counter z z is

CHAP. I. PITCH AND INTERVAL. 13

introducsd, C3iiaeste:l with a toothei wheel which works in the screw t, ;uid advances one tooth for each revohitioii of the disc s s. By the handle h this counter niav be moved slightly to one side, so that the wheelwork and screw may be connected or disconnected at pleasure. If they are connected at the beginning of one second, and disconnected at the beginning of another, the hand of the counter shows how many revolutions of the disc have been mide in the corre- sponding number of seconds.*

Dove+ introduced into this siren several rows of holes through which the wind might be directed, or from which it might be cut oflf, at pleasure. A polyphonic siren of this description with other peculiar arrangements will be figured and described in Chapter VIII., fig. 56.

It is clear that when the pierced disc of one of these sirens is made to revolve with a uniform velocity, and the air escapes through the holes in puffs, the motion of the air thus produced must be x)eriod.k in the sense already explained. TheH holes stand at equal intervals of space, and hence on rotation follow each other at equal intervals of time. Through every hole there is poured, as it were, a drop of air into the external atmospheric ocean, exciting waves in it, which succeed each other at uniform intervals of time, just as was the case when regulaidy falling drops impinged upon a surface of water (p. lOa). Within each separate period, each individual pufF will have considerable variations of form in sirens of difterent construction, depending on the different diameters of the holes, their distance from each other, and the shape of the extremity of the pipe which conveys the air ; but in every case, as long as the velocity of rotation and the position of the pipe remain unaltered, a regulaidy periodic motion of the air must result, and consequently the sensation of a musical tone must be excited in the ear, and this is actually the case.

It results immediately from experiments with the siren that two series of the same number of holes revolving with the same velocity, give musical tones of the ^ same pitch, quite independently of the size and form of the holes, or of the pipe. We even obtain a musical tone of the same pitch if we allow a metal point to strike in the holes as they revolve instead of blowing. Hence it follows firstly that the pitch of a tone depends only on the Huinb<n- of puffs or swings, and not on their form, force, or method of production. Further it is very easily seen with this instrument that on increasing the velocity of rotation and consequently the number of puffs produced in a second, the pitch becomes sharper or higher. The same result ensues if, maintaining a uniform velocity of rotation, we first blow into a series with a smaller and then into a series with a greater niimber of holes. The latter gives the sharper or higher pitch.

With the same instrument we also very easily find the remarkable relation which the pitch numbers of two musical tones must possess in order to form a consonant interval. Take a series of 8 and another of 16 holes in a disc, and blow into both sets while the disc is kept at uniform velocity of rotation. TwoH tones will be heard which stand to one another in the exact relation of an Octave. Increase the velocity of rotation ; both tones will become sharper, but both will continue at the new pitch to form the interval of an Octave. J Hence we conclude that a musical tone which is an Octave higher than another, inaki^s exaetli/ twice as manij vibrations in a given time as the latter.

* See Appendix I. names of all the intervals usually distinguished

t [Pronounce Doh-reh, in two syllables. — are also given in App. XX. sect. I)., with the

Transtntor.] corresponding ratios and cents. These names

\ [When two notes have different pitch were in the first place derived from the ordinal

numbers, there is said to be an interval number of the note in the scales, or succes-

between them. This gives rise to a sensa- sions of continually sharper notes. The Octave

tion, very differently appreciated by different is the eighth note in the major scale. An octave

individuals, but in all cases the interval is is a set of notes lying within an Octave. Ob-

measured by the ratio of the pitch mimbers, serve that in this translation all names of in-

and, for some purposes, more conveniently by tervals commence with a capital letter, to

other numbers called cents, derived from these prevent ambiguity, as almost all such v/ords

ratios, as explained in App. XX. sect. C. The are also used in other senses.— Translator.]

U PITCH AND INTERVAL. part i.

The disc shown m fig. 1, p. 11 r, has two circles of 8 and 12 holes respectively. Each, blown successiveh', gives two tones which form with each other a perfect Fifth, independently of the velocity of rotation of the disc. Hence, two musical tones stand in the relation of a so-called Fifth irJien the highe)- tone makes three vibrations in the same time as the lower makes two.

If we obtain a musical tone by blowing into a circle of 8 holes, we require a circle of 16 holes for its Octave, and 12 for its Fifth. Hence the ratio of the pitch numbers of the Fifth and the Octave is 12 : 16 or 3 : 4. But the interval between the Fifth and the Octave is the Fourth, so that we see that when two musical tones form a Fourth, the higher makes four vibrations irhile the lower makes three.

The polyphonic siren of Dove has ixsually four circles of 8, 10, 12 and 16 holes respectively. The series of 16 holes gives the Octave of the series of 8 holes, and U the Fourth of the series of 1 2 holes. The series of 1 2 holes gives the Fifth of the series of 8 holes, and the minor Third of the series of 10 holes. While the series of 10 holes gives the major Third of the series of 8 holes. The four series con- sequently give the constituent musical tones of a major chord.

By these and similar experiments we find the following relations of the pitch numbers : —

1 : 2 Octave

2 : 3 Fifth

3 : 4 Fourth

4 : 5 major Third

5 : 6 minor Third

When the fundamental tone of a given interval is taken an Octave higher, the interval is said to be inverted. Thus a Fourth is an inverted Fifth, a minor Sixth ^ an inverted major Third, and a major Sixth an inverted minor Third. The corre- sponding ratios of the pitch numbers are consequently obtained by doubling the smaller number in the original interval.

From 2 : 3 the Fifth, we thus have 3 : 4 the Fourth „ 4:5 the major Third ... 5:8 the minor Sixth „ 5:6 the minor Third, 6 : 10= 3 : 5 the major Sixth.

These are all the consonant intervals which lie within the compass of an Octave. With the exception of the minor Sixth, which is really the most imperfect of the above consonances, the ratios of their vibrational numbers are all expressed by means of the whole numbers, 1, 2, 3, 4, 5, 6.

Comparatively simple and easy experiments with the siren, therefore, corrobo- rate that remarkable law mentioned in the Introduction (p. Id), according to which the pitch numbers of consonant musical tones bear to each other ratios expressible H by small whole numbers. In the course of our investigation we shall employ the same instrument to verify more com})letely the strictness and exactness of this law.

Long before anything was known of pitch numbers, or the means of counting them, Pythagoras had discovered that if a string be divided into two parts by a bridge, in such a way as to give two consonant musical tones when struck, the lengths of these parts must be in the ratio of these whole numbers. If the bridge is so placed that f of the string lie to the right, and ~ on the left, so that the two lengths are in the ratio of 2 : 1, they produce the interval of an Octave, the greater length giving the deeper tone. Placing the bridge so that f of the string lie on the right and f on the left, the ratio of the two lengths is 3 : 2, and the interval is a Fifth.

These measurements had been executed with great precision by the Creek musicians, and had given rise to a system of tones, contrived with considerable art. For these measurements they used a peculiar instnunent, the monochord.

CHAP. I. PITCH NUMBERS IN JUST MAJOR SCALE. 15

consisting of a sonnding board and box on which a single string was stretched with a scale below, so as to set the bridge correctly.*

It was not till ninch later that, through the investigations of CJalileo (1638), Newton, Euler (1729), and Daniel Bernouilli (1771), the law governing the motions of strings became known, and it was thus found that the simple ratios of the lengths of the strings existed also for the pitch numbers of the tones they pro- duced, and that they consequently belonged to the musical intervals of the tones of all instruments, and were not confined to the lengths of strings through which the law had been first discovered.

This relation of whole numbers to musical consonances was from all time looked upon as a wonderful mystery of deep significance. Tlie Pythagoreans themselves made use of it in their speculations on the harmony of the spheres. From that time it remained partly the goal and partly the starting point of the strangest and most venturesome, fantastic or philosophic combinations, till in ^ modern times the majority of investigators adopted the notion accepted by Eider himself, that the human mind had a peculiar pleasure in simple ratios, because it could better understand them and comprehend their bearings. But it remained uninvestigated how the mind of a listener not versed in physics, who perhaps was not even aware that musical tones depended on periodical vibrations, contrived to recognise and compare these ratios of the pitch numbers. To show what pro- cesses taking place in the ear, render sensible the diflference between consonance and dissonance, will be one of the principal problems in the second part of this work .

Calculatiox of the Pitch Numbers for all the Tones of the Musical Scale.

By means of the ratios of the pitch numbers already assigned for the consonant intervals, it is easy, by pursuing these intervals throughout, to calculate the ratios ^ for the whole extent of the musical scale.

The major triad or chord of three tones, consists of a major Third and a Fifth. Hence its ratios are :

C : E : G

1 : A :^

or 4:5: 6

If we associate with this triad that of its dominant G : B : D, and that of its sub-dominant F : A : C, each of which has one tone in common with the triad of the tonic C : E : G, we obtain the complete series of tones for the major scale of C, with the following ratio of the pitch numbers :

C : D : E \ F : G : A \ B : v

[or 24 : 27 : 30 : 32 : 36 : 40 : 45 : 48] ^I

In order to extend the calculation to other octaves, we shall adopt the following notation of musical tones, marking the higher octaves by accents, as is usual in (ilermany,t as follows :

1. The unaccented or small octave (the 4-foot octave on the organj) : —

c d e f (J a h

* [As the monochord is very liable to error, below the letters, which are typographically

these results were happy generalisations from inconvenient. Hence the German notation is

necessarily imperfect experiments. — Tmns- retained. — Translator.]

lator.] + [The note C in the small octave was

t [English works use strokes above and once emitted by an organ pipe 4 feet in length :

16 PITCH NUMBERS IN JUST MAJOR SCALE. pari

2. IVie once-accented octave (2-foot) : —

^._ ^ ^

c' d' e' f

3. The twice-accented octave (1-foot) : —

*^ c" d" e" f" <j" a" h"

And so on for higher octaves. Below the small octave lies the great octave, written with unaccented capital letters ; its C requires an organ pipe of eight feet H in length, and hence it is called the 8-foot octave.

4. Great or 8-foot octave : —

"O"

-&-

-^ &-

C D E F G A B

Below this follows the KS-foot or contra-octave ; the lowest on the pianoforte and most organs, the tones of which may be represented by C ^ D ^ E ^ F ^ G ^ A^ B ., with an inverted accent. On great organs there is a still deeper, 32-foot octave, the tones of which may be written C ^^ D ^^ E^^ F^^ G^^ A ^^ B^, with two inverted accents, but they scarcely retain the character of musical tones. (See Chap. JX.)

Since the pitch numbers of any octave are always twice as great as those for 11 the next deeper, we find the pitch numbers of the higher tones by multiplying those of the small or unaccented octave as many times by 2 as its symbol has upper accents. And on the contrary the pitch numbei's for the deeper octaves are found by dividing those of the great octave, as often as its symbol has lower accents.

Thus (•" = 2x2xc=2x2x2C C. = i X i X <^' = i X * X i c.

For the pitch of the musical scale German physicists have generally adopted that proposed by Scheibler, and adopted siibseqiiently by the German Association of Natural Philosophers {die deutsche Naturforscherversammhing) in 1834. Tliis makes the once-accented a execute 440 vibrations in a second.* Hence results the

thus Bedos {L'Art du Fadcur d'Orgues, 1766) backward and forward niotiois it would be 51 made it 4 old French feet, which gave a indifferent by which method we counted, but note a full Semitone flatter than a pipe of for non-symmetrical musical vibrations which 4 English feet. But in modern organs not even are of constant occurrence, the French method so much as 4 English feet are used. Organ of counting is very inconvenient. The number builders, however, in all countries retain the 440 gives fewer fractions for the first [just] nanres of the octaves as here given, which major scale of C, than «' = 435. The difference must be considered merely to determine the of pitcli is less than a comma. [The practical place on the staff, as noted in the text, inde- settlement of pitch has no relation to such pendently of the precise X3itch. — Travslatnr.'\ arithmetical considerations as are here sug- * The Paris Academy has lately fixed the gested, but depends on the compass of the pitch number of the same note at 435. This human voice and the music written for it at is called 870 by the Academy, because French different times. An Abstract of my History physicists have adoj)ted the inconvenient of Musical Fitch is given in Appendix XX. habit of counting the forward motion of a sect. H. Scheibler's proposal, named in the swinging body as one vibration, and the back- text, was chosen, as he tells us {Der Tunmcsscr, ward as another, so that the whole vibra- 1884, p. 53), as being the mean between the tion is counted as two. This method of limits of pitch within which Viennese piano- counting has been taken from the seconds fortes at that time rose and fell by heat and pendulum, which ticks once in going forward cold, which he reckons at J vibration either and once again on returning. For symmetrical way. That this proposal had no reference to the

I

CHAP. I. PITCH NUMBERS IN JUST :\IAJOR SCALE. 17

following table for the scale of C major, which will serve to determine the ])itch of all tones that are defined by their pitch numbers in the following work.

Contra Octave

Notes.

C toB,

16 foot

C

33

D

37-125

E

41-25

F

44

G

49-5

A

55

B

61-875

Great Octave Cto JS

8 foot

60

74-25

82-5

88

99 110 123-75

Unaccented Octave c to 6 4 foot

132

148-5

165

170

198

220

247-5

Once- accented Octave

264 297 330 352 396 440 495

Twice- accented Octave c" to h" Ifoot

528 594 660 704 792

Thrice- accented

Octave c" to b-

ifoot

1056

1188 1320 1408 1584 1760 1980

Fovir-times

accented

Octave

c'" to b'"'

J foot

2112 2376 2640 2816 3168 3520 3960*

The lowest tone on orchestral instruments is the E^ of the double bass, making Modern pianofortes and organs usually go down to C, ^

expression of the Just major scale in whole numbers, is shown by the fact that he proposed it for an equally tempered scale, for wbicli he calculated the pitch numbers to four places of decimals, and for which, of course, none but the octaves of a' are ex- pressible by whole numbers. — Translator.']

* [As itis important that students should be able to hear the exact intervals and pitches spoken of throughout this book, and as it is quite impossible to do so on any ordinary in- strument, I have contrived a specially-tuned harmonium, called an Harmonical, fully de- scribed in App. XX. sect. F. No. 1, which Messrs. Moore & Moore, 104 Bishopsgate Street, will, in the interests of science, supply to order, for the moderate sum of 165i-. The follow- ing are the pitch numbers of the first four octaves, the tuning of the fifth octave will be

explained in App. XX. sect. F. The names of the notes are in tlie notation of the latter part of Chap. XIV. below. Read the sign i*, as '/) one,' E^\) as 'one E flat,' and 'B\ji as ' seven B flat '. In playing observe that Z*, is on the ordinary D\f or G'| digital, and that '/)'[j is on the ordinary G\) or i*^ digital, and that the only keys in which chords can be played are U major and 0 minor, with the minor chovA D^F A ^ and the natural chord of the Ninth CE\G''B\jB. The mode of measuring intervals by ratios and cents is fully explained hereafter, and the results are added for con- venience of reference. The pitches of c" 528, a' 440, ft'i[j 422-4 and 'h'\y 462, were taken from forks very carefully tuned by myself to these numbers of vibrations, by means of my unique series of forks described in App. XX., at the H end of sect. B.

Scale

OF THE HaRMOMICAL.

Pitch Numbers,

Ratios.

Cents.

Notes.

Sfoot

4 foot

2 foot

Ifoot

Note to Note

C to Note

Note to Note

C to Note

6'

66

132

264

528

9: 10

1 : 1

182

0

^1

731

1461

2931

5861

80 : 81

9 : 10

22

182

D

m

148*

297

594

15 :16

8:9

112

204

E'\>

m

1584

316|

633f

24 : 25

5 : 6

70

316

E,

82 .i

165

3.30

660

15 : 16

4 : 5

112

386

F

88

176

352

704

8 :9

3:4

204

498

G

99

198

396

792

15: 16

2 : 3

112

702

A'\y

105j

211i

422|

8444

24 : 25

5: 8

70

814

^1

110

220

440

880

20 : 21

3:5

85

884

-B'r,

11.5^

231

462

924

35 : 36

4 :7

49

969

B^b

1184

237f

475i

950|

24 : 25

5:9

70

1018

B\

123=

247*

495

990

8: 15

—

1088

15 : 16

112

c

132 1 264

528

1056

1 : 2

—

1200

Translator.']

t [The following account of the actual tones 3d is adapted from my History of Musical

Fitch. G , commencement of the 32-foot oc- tave, the 'lowest tone of verj' large organs, two

C

18

COMPASS OF INSTRUMENTS.

with 33 viljivations, and the latest grand pianos even down to A^^ with 27^ vibra- tions. On larger organs, as already mentioned, there is also a deeper Octave reach- ing to C„ with 16i vibrations. But the musical character of all these tones below F^ is imperfect, because we are here near to the limit of the power of the ear to combine vibrations into musical tones. These lower tones cannot therefore be vised musically except in connection with their higher octaves to which they impart a character of o-reater depth without rendering the conception of the pitch indeterminate.

Upwards, pianofortes generally reach a"" with b520, or even c" with 4224 vibra- tions. The highest tone in the orchestra is probably the five-times accented J" of the piccolo flute with 4752 vibrations. Appunn and W. Preyer by means of small tunino--forks excited by a violin bow have even reached the eight times accented «="" with 40,960 vibrations in a second. These high tones were very painfully unplea- sant, and the pitch of those which exceed the boundaries of the musical scale was ^ very imperfectly discriminated by musical observers."* More on this in Chap. IX.

The musical tones which can be used with advantage, and have clearly dis- tinguishable pitch, have therefore between 40 and 4000 vibrations in a second, extendino- over 7 octaves. Those which are audible at all have from 20 to 40,000 vibrations, extending over about 11 octaves. This shows what a great variety of different pitch numbers can be perceived and distinguished by the ear. In this respect the ear is far superior to the eye, which likewise distinguishes light of dif- ferent periods of vibration by the sensation of different colours, for the compass of the vibrations of light distinguishable by the eye but slightly exceeds an Octave.t

Fo7re and 2}itch were the two first differences which we found between musical tones ; the third was quality of tone, which we have now to investigate. When

of Tone,' [ilbcr die Grcnzen dcr Tonv:ahrnc]i- muncj, 1876, p. 20), are in the South Kensing- ton Museum, Scientific Collection. I have several times tried them. I did not myself find the tones painful or cutting, probably because there was no beating of inharmonic upper ^Dartials. It is best to sound them with two violin bows, one giving the octave of the other. The tones can be easily heard at a distance of more than 100 feet in the gallery of the Museum. — Translator.']

t [Assuming the undulatory theory, which attributes the sensation of light to the vibra- tions of a supposed luminous ' ether,' resem- bling air but more delicate and mobile, then the phenomena of ' interference ' enables us to calculate the lengths of waves of light in empty space, &c. , hence the numbers of vibra- tions"in a second, and consequently the ratios of these numbers, which will then clearly resemble the ratios of the pitch nimibers that measure musical intervals. Assuming, then, that the yellow of the spectriun answers to the tenor c in music, and Fraunhofer's ' line A ' corresponds to the G below it, Prof. Helm- holtz, in his Physiological Optics, {Hand- buch der physiologischen Optik, 1867, p. 237), gives the following analogies between the notes of the piano and the colours of the spectrmn :—

Octaves below the lowest tone of the Violon- cello. A,„ the lowest tone of the largest pianos. C\, commencement of the 16-foot octave, the lowest note assigned to the Double

U Bass in Beethoven's Pastoral Symphony. JS,, the lowest tone of the German four-stringed Double Bass, the lowest tone mentioned in the text. F„ the lowest tone of the English four-stringed Double Bass. G,, the lowest tone of the Italian three- stringed Double Bass. A„ the lowest tone of the English three-stringed Double Bass. C, conmiencement of the 8-foot octave, the lowest tone of the Violoncello, written on the second leger line below the bass stafi. G, the tone of the third open string of the Violoncello. c, commencement of the 4-foot octave ' tenor C,' the lowest tone of the Viola, written on the second space of the bass staff, d, the tone of the second open string of the Violoncello. /, the tone signified by the bass or i^-clef. ;/, the lowest tone of the Violin, a, the tone of the highest open string of the Violoncello, c', conmiencement of the

51 2-foot octave, ' middle 6',' written on the leger line between the bass and treble staves, the tone signified by the tenor or C-clef . d', the tone of the third open string of the Violin, g', the tone signified by the treble or G-clei. a', the tone of the second open string of the Violin, the 'tuning note ' for orchestras, t", commencement of the 1-foot octave, the usual ' tuning note ' for pianos. e", the tone of the first or highest open string of the Violin, c", commencement of the ^-foot octave, g'", the usual highest tone of the Flute. Civ, commencement of the ^-foot octave. €'", the highest tone on the Violin, being the double Octave harmonic of the tone of the highest open string, a}"", the usual highest tone of large pianos. tZ^', the highest tone of the piccolo flute. c^"i, the highest tone reached by Appunn's forks, see next note. — Translator.} * [Copies of these forks, described in Prof. Preyer's essay ' On the Limits of the Perception

F 1 end of the Red. G,Iied.

f i, Violet. ■(/, Ultra-violet.

G i Red. ^,*Red.

^t' "

((,„ ..

A 5, Orange-red. ^,X>range.

b, end of the solar

spectrum.

c. Yellow.

The scale there-

c it. Green.

fZ, Greenish-blue.

fore extends to

about a Fourth

d |, Cyanogen-blue. e, Indigo-blue.

beyond the oc-

tave. — 2'ransla-

/, Violet.

tor.]

CHAP. I. QUALITY OF TONE AND FORM OF VIBRATION. 19

we hear notes of the same force and same pitch sonnded snccessively on a piano- forte, a vioUn, clarinet, oboe, or trumpet, or by the liuman voice, the character of the musical tone of each of these instruments, notwithstanding the identity of force and pitch, is so different that by means of it we recognise witli the greatest ease which of these instruments was used. Varieties of quality of tone appear to be infinitely numerous. Not only do we know a long series of musical instruments which could each produce a note of the same pitch ; not only do diflerent individual instruments of the same species, and the voices of different individual singers show certain more delicate shades of quality of tone, which our ear is able to distinguish ; but notes of the same pitch can sometimes be sounded on the same instrument with several qualitative varieties. In this respect the ' bowed ' instruments (i.e. those of the violin kind) are distinguished above all other. But the human voice is still richer, and human speech employs these very qualitative varieties of tone, in order to distinguish different letters. The different vowels, namely, belong to the class H of sustained tones which can be used in music, while the character of consonants mainly depends upon brief and transient noises.

On inquiring to what external physical difference in the waves of sound the different qualities of tone correspond, we must remember that the amplitude of the vibration determines the force or loudness, and the period of vibration the pitch. Quality of tone can therefore depend upon neither of these. The only possible hypothesis, therefore, is that the quality of tone should depend upon the manner in which the motion is performed within the period of each single vibra- tion. For the generation of a musical tone we have only required that the motion should be periodic, that is, that in any one single period of vibration exactly the same state should occur, in the same order of occurrence as it presents itself in any other single period. As to the kind of motion that should take place within any single period, no hypothesis was made. In this respect then an endless variety of motions might be possibly for the production of sound. ^

Observe instances, taking first such periodic motions as are performed so slowly that -we can follow them with the eye. Take a pendulum, which we can at any time construct by attaching a weight to a thread and setting it in motion. The pendulum swings from right to left with a imiform motion, uninterrupted by jerks. Near to either end of its path it moves slowly, and in the middle fast. Among sonorous bodies, which move in the same way, only very much faster, we may . mention tuning-forks. When a tuning-fork is struck or is excited by a violin bow, and its motion is allowed to die away slowly, its two prongs oscillate backwards and forwards in the same way and after the same law as a pendulum, only they make many hundred swings for each single swing of the pendulum.

As another example of a periodic motion, take a hammer moved by a water- wheel. It is slowly raised by the millwork, then released, and falls down suddenly, is then again slowly raised, and so on. Here again we have a periodical backwards and forwards motion ; but it is manifest that this kind of motion is totally diflf'erent ^ from that of the pendulum. Among motions wdiich produce musical sounds, that of a violin string, excited by a bow, would most nearly correspond with the hammer's, as will be seen from the detailed description in Chap. V. The string clings for a time to the bow, and is carried along by it, then suddenly releases itself, like the hammer in the mill, and, like the latter, retreats somewhat with much greater velocity than it advanced, and is again caught by the bow and carried forward.

Again, imagine a ball thrown up vertically, and caught on its descent with a blow which sends it up again to the same height, and suppose this operation to be performed at equal intervals of time. Such a ball would occupy the same time in rising as in falling, but at the lowest point its motion would be suddenly interrupted, whereas at the top it wovdd pass through gradually diminishing speed of ascent into a gradually increasing speed of descent. This then would be a third kind of alternating periodic motion, and would take place in a manner essentially different from the other two.

c 2

20

FORM OF VIBRATION.

To render the law of such motions more comprehensible to the eye than is le by lengthy verbal descriptions, mathematicians and physicists are in the habit of applying a graphical method, which must be frequently employed in this work, and should therefore be well understood.

To render this method intelligible suppose a drawing point b, fig. 5, to be fastened to the prong A of a tuning-fork in such a manner as to mark a surface of pauer B B. Let the tuning-fork be moved with a uniform velocity in the direc- tion of the upper arrow, or else the paper be drawn under it in the opposite direction, as shown by the lower arrow. When the fork is not sounding, the point will describe the dotted straight line d c. But if the prongs have been first set in vibration, the point will describe the undulating line d c, for as the prong vibrates, the attached point b will constantly move backwards and forwards, and hence be

^

sometimes on the right and sometimes on the left of the dotted straight line d c, as is shown by the wavy line in the figure. This wavy line once drawn, remains as a permanent image of the kind of motion performed by the end of the fork during ^ its musical vibrations. As the point b is moved in the direction of the straight line d c with a constant velocity, equal sections of the straight line d c will corre- spond to equal sections of the time during which the motion lasts, and the distance of the wavy line on either side of the straight line will show how far the point b has moved from its mean position to one side or the other during those sections of time.

In actually performing such an experiment as this, it is best to wrap the paper over a cylinder which is made to rotate uniformly by clockwork. The paper is wetted, and then passed over a turpentine flame which coats it with lampblack, on which a fine and somewhat smooth steel point will easily trace delicate lines.

Fig. 6 is the copy of a drawing actually made in this way on the rotating cylinder of Messrs. Scott and Koenig's Phonmttograph.

Fig. 7 shows a portion of this curve on a larger scale. It is easy to see the meaning of such a curve. The drawing point has passed with a uniform velocity in the direction e h. Suppose that it has described the section e g in -^^ of a second. Divide e g into 12 equal parts, as in the figure, then the point has been y^o^ of a second in describing the length of any such section horizontally, and the curve shows us on what side and at what distance from the position of rest the vibrating point will be at the end of ■^~-^, yf^, and so on, of a second, or, generally, at any given short interval of time since it left the point e. We see, in the figure, that after yi^ of a second it had reached the height 1, and that it rose gradually till the end of yf ^ of a second ; then, however, it began to descend gradually till, at the end of Tfo = oV seconds, it had reached its mean

FORM OF VUUIATION.

21

position f, and then it continued descending on the (j{)posite side till the end of y^ of a second and so on. We can also easily determine where the vibrating point was to be found at the end of any fraction of this hundred-and-twentieth of a second. A drawing of this kind consequently shows immediately at what point of its path a vibrating particle is to be found at any given instant, and hence gives a complete image of its motion. If the reader wishes to reproduce the motion of the vibrating point, he has only to cut a narrow vertical slit in a piece of paper, and place it over fig. 6 or fig. 7, so as to show a vei-y small portion of the curve through the vertical slit, and draw the book slowly but uniformly under the slit, from right to left ; the white or black point in the slit will then appear to move backwards and forwards in precisely the same manner as the original drawing point attached to the fork, only of course much more slowly.

We are not yet able to make all vibrating bodies describe their vibrations

U

directly on paper, although nmch methods required for this purpose.

has recently been made in the are able ourselves to draw such

progress But we curves for all sounding bodies, when the law of their motion is known, that is, when we know how far the vibrating point will be from its mean position at a,ny given moment of time. We then set off on a horizontal line, such as e f, fig. 7, lengths corresponding to the interval of time, and let fall perpendiculars to it on^ either side, making their lengths equal or proportional to the distance of the vibrat- ing point from its mean position, and then by joining the extremities of these per- pendiculars we obtain a curve such as the vibrating body would have drawn if it had been possible to make it do so.

Thus fig. 8 represents the motion of the hammer raised by a water-wheel, or of a point in a string excited by a violin bow. For the first 9 intervals it rises slowly and xuiiformly, and during the 10th it falls suddenly down.

LiA.

V

Fig. 9 represents the motion of the ball which is struck up again as soon as it^ "comes down. Ascent and descent are performed with equal rapidity, whereas in fig. 8 the ascent takes much longer time. But at the lowest point the blow suddenly changes the kind of motion.

Physicists, then, having in their mind such curvilinear forms, representing the law of the motion of sounding bodies, speak briefly of the form of vibratum of a sounding body, and assert that the (juality of tone depends on the form of vibration. This assertion, which has hitherto been based simply on the fact of our knowing that the quality of the tone could not possibly depenfl on the periodic time of a vibration, or on its amplitude (p. 10c), will be strictly examined hereafter. It will be shown to be in so far correct that every different quality of tone recpiires a difterent form of vibration, but on the other hand it will also appear that different forms of vibration may correspond to the same quality of tone.

On exactly and carefully examining the effect produced on the ear by difterent forms of vibration, as for example that in fig. 8, corresponding nearly to a violin

22 COMPOUND AND PARTIAL TONES. part i.

string, we meet with a strange and imexpected phenomenon, long known indeed to individual musicians and ph^'sicists, but commonly regarded as a mere curiosity, its generality and its great significance for all matters relating to musical tones not having been recognised. The ear when its attention has been properly' directed to the effect of the vibrations which strike it, does not hear merely that one musical tone whose pitch is determined by the period of the vibrations in the manner already explained, but in addition to this it becomes aware of a whole series of higher musical tones, which we will call the harmonic upper partial tones, and sometimes simply the iipper jMrtials of the whole musical tone or note, in contra- distinction to the fundamental or prime x><irtial tone or simply the prime, as it may be called, which is the lowest and generally the loudest of all the partial tones and by the pitch of which we judge of the pitch of the whole compound musical tone itself. The series of these upper partial tones is precisely the same for all com- H pound musical tones which correspond to a uniformly periodical motion of the air. It is as follows : —

The first upper partial tone [or second partial tone] is the upper Octave of the prime tone, and makes double the number of vibrations in the same time. If we call the prime 6', this upper Octave will be c.

The .second upper partial tone [or third partial tone] is the Fifth of this Octave, or g, making three times as mtxny vibrations in the same time as the prime.

The third upper partial tone [or fourth partial tone] is the second higher Octave or c', making four times as many vibrations as the prime in the same time.

The fourth upper partial tone [or fifth partial tone] is the major Third of this second higher Octave, or e , with five times as many vibrations as the prime in the same time.

The fifth upper partial tone [or sixth partial tone] is the Fifth of the second higher Octave, or (f , making six times as many vibrations as the prime in the ^ same time.

And thus they go on, becoming continually fainter, to tones making 7, 8, 9, ifcc, times as many vibrations in the same time, -.s the prime tone. Or in musical notation

i^^§=^=bEL=l?

fj

fj "/>'[? c" d' e" ^y fi" ^^'a" ~h"\) h" c"

Ordinal mmher of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Pitch mfmber 66 132 198 264 380 396 462 528 594 660 726 792 858 924 990 1054*

where the figures [in the first line] beneath show how many times the corresponding

pitch number is greater than that of the prime tone [and, taking the lowest note

to have 66 vibrations, those in the second line give the pitch numbers of all the

*\ other notes].

The whole sensation excited in the ear by a periodic vibration of the air we

* [This diagram has been slightly altered to This slightly flattens each note, and slow beats introduce all the first 16 harmonic partials can be produced in ever_v case (except, of of C 66 (which, excepting 11 and 13, are course, 11 and 13, which are not on the given on the Harmouical as harmonic notes), instrument) up to 16. It should also be ob- aiid to show the notation, symbolising, both in served that the pitch of the beat is very nearly letters and on the staff, the 7th, 11th, and that of the upper {not the lower) note in each 13th harmonic partials, which are not used in case. The whole of these 16 harmonics of C 66 general music. It is easy to show on the (except the 11th and 13th) can Ije played Harmonical that its lowest note, C of this at once on the Harmonical by means of the series, contains all these partials, after the harmonical bar, first without and then with theory of the beats of a disturbed unison the 7th and 14th. The whole series will be has been explained in Chap. VIII. Keep found to sound like a single fine note, and the down the note C, and touch in succession the 7th and 14th to materially increase its rich- notes c, g, c', c', cj', &c., but in touching the latter ness. The relations of the partials in this case press the fmger-key such a little way down may be studied from the tables in the footnotes that the tone of the note is only just audible. to Chap. X. — Translator.']

DEFINITION OF TERMS EMPLOYED.

23

have called a mHsica/ tone. We now find that this is ronijjoujtd, c()ntainin<>- a series of ditt'erent tones, which we distinguish as the constituents or 2M)-tial tones of the compound. The first of these constitnents is the pritne j^artial tone of the compoiuid, and the rest its harmonic upper partial tones. The number which shows the order of any partial tone in the series shows how many times its vibrational number exceeds that of the prime tone.* Thus, the second partial tone makes twice as many, the third three times as many vibrations in the same time as the prime tone, and so on.

G. S. Ohm was the first to declare that there is only one form of vibration which will give rise to no harmonic upper partial tones, and which will therefore consist solely of the prime tone. This is the form of vibration which we have described above as pecidiar to the pendulum and tmiing-forks, and drawn in figs. G and 7 (p. 10). We will call these j^^ndular vibrations, or, since they caiuiot be analysed into a compound of diflferent tones, simple vibrations. In what sense not H merely other musical tones, but all other forms of vibration, may be considered as compowid, will be shown hereafter (Chap. IV.). The terms simple or pendular vibration, f will therefore be used as synonymous. We have hitherto used the expression tone and musical tone indifferently. It is absolutely necessary to dis- tinguish in acoustics first, a musical tone, that is, the impression made by an// periodical vibration of the air ; secondW, a simple tone, that is, the impression l)roduced by a simpde or pendular vibration of the air ; and thirdly a comjwund tone, that is, the impression produced by the simultaneous action of several simple tones with certain definite ratios of pitch as already explained. A musical tone may be either simjde or comptound. For the sake of brevity, tone will be used in

* [The ordinal number of a partial tone in general, must be distinguished from the ordinal number of an upper partial tone in particular. For the same tone the former number is always greater by unity than the latter, because the partials in general include the prime, which is reckoned as the first, and the upper partials exclude the prime, which being the loudest partial is of course not an upper partial at all. Thus the partials gene- rally numbered 2 3 4 5 6 7 8 9 are the same as the upper partials numbered 12 3 4 5 6 7 8 respectively. As even the Author has occasionally failed to carry out this distinction in the original German text, and other writers have constantly neglected it, too much weight cannot be here laid upon it. The presence or absence of the word wppcr before the word partml must always be care-

fully observed. It is safer never to speak of an vipper partial by its ordinal number, but to call the pfth upixr partial the sixth partial, omitting the word uyper and increasing the 51 ordinal number by one place. And so in other cases. — Translator.']

t The law of these vibrations may be popularly explained by means of the constritc- tion in fig. 10. Suppose a point to describe the circle of which c is the centre with a uniform velocity, and that an observer stands at a considerable distance in the prolongation of the line e h, so that he does not see the surface of the circle but only its edge, in which case the point will appear merely to move up and down along its diameter a b. This up and down motion would take place exactl}- according to the law of pendular vibration. To represent this motion graphi-

cally by means of a curve, divide the length e g, supposed to correspond to the time of a single period, into as many (here 12) equal parts as the circumference of the circle, and draw the perpendiculars 1, 2, 3, &c., on the dividing points of the line e g, in order, equal in length to and in the same direction with, those drawn in the circle from the correspond- ing points 1, 2, 3, &c. In this way we obtain the curve drawn in fig. 10, which agrees in

form witli that drawn by the tuning-fork, fig. 6, p. 206, but is of a larger size. Mathe- matically expressed, the distance of the vibrat- ing point from its mean position at any time is equal to the sine of an arc proportional to the corresponding time, and hence the form of simple vibrations are also called the sivc- vibrations [and the above curve is also known as the curve of sines'].

24

DEFINITION OF TERMS EMPLOYED.

the general sense of a musical tone, leaving the context or a prefixed tjualitication to determine whether it is simple or compound. A compound tone will often bo briefly called a note, and a simple tone will also be frequently called a imrtial, when used in connection with a compound tone : otherwise, the full expression simple tone will be employed. A note has, properly speaking, no single pitch, as it is made up of various partials each of which has its own pitch. By the 2'''^^'^^ of a note or compo^ind tone then we shall therefore mean the ^)?YcA of its lowest pjartial or prime tone. By a chord or combination of tones we mean several musical tones (whether simple or compound) produced by difl^erent instruments or diff'erent parts of the same instrument so as to be heard at the same time. • The facts here adduced show us then that every musical tone in which harmonic upper partial tones can be distinguished, although produced by a single instrument, may really be con- sidered as in itself a chord or combination of various simple tones.*

H

* [The above paragraph relating to the English terms used in this translation, neces- sarily differs in many respects from the original, in which a justification is given of the use made by the Author of certain German ex- pressions. It has been my object to employ terms which should be thoroughly English, and should not in any way recall the German words. The word tone in English is extremely ambiguous. Prof. Tj'ndall {Lectures on Sound, 2nd ed. 1869, p. 117) has ventured to define a tone as a sini/ile lone, in agreement with Prof. Helmholtz, who in the present passage limits the German word Ton in the same way. But I felt that an English reader could not be safely trusted to keep this very peculiar and important class of musical tones, which he has very rarely or never heard separately, invariably distinct from those musical tones

•fl with which he is familiar, unless the word tone were uniformly qualified by the epithet simple. The only exception I could make was in the case of a partial tone, which is received at once as a new conception. Even Prof. Helmholtz himself has not succeeded in using his word Ton consistently for a simple tone only, and this was an additional warning to me. English musicians have been also in the habit of using tone to signify a certain musical interval, and semitone for half of that interval, on the equally tempered scale. In this case I write Tone and Semitone with capital initials, a practice which, as already explained (note, p. 13c?'), I have found con- venient for the names of all intervals, as Thirds, Fifths, &c. Prof. Helmholtz uses the word Klang for a musical tone, which gene-

^ rally, but not always, means a compound tone. Prof. Tyndall (ibid.) therefore proposes to use the English word clang in the same sense. But clang has already a meaning in English, thus defined by Webster : ' a sharp shrill sound, made by striking together metallic substances, or sonorous bodies, as the clang of arms, or any like sound, as the claiig of trumpets. This word implies a degree of harshness in the sound, or more harshness than clink.' Interpreted scientifically, then, clang according to this definition, is either noise or one of those musical tones until in- harmonic upper partials, which will be sub- sequently explained. It is therefore totally unadapted to represent a musical tone in general, for which the simple word tone seems eminently suited, being of course originally the tone produced by a stretched string. The coiomon word note, properly the mark by

which a musical tone is written, will also, in accordance with the general practice of musi- cians, be used for a musical tone, which is generally compound, without necessarily im- plying that it is one of the few recognised tones in our musical scale. Of course, if clang could not be used. Prof. Tyndall's suggestion to translate Prof. Helmholtz's Klangfarbc by clangtint (ibid.) fell to the ground. I can find no valid reason for sup- planting the time-honoured expression qualitg of tone. Prof. Tyndall [ibid.) quotes Dr. Young to the effect that ' this quality of sound is sometimes called its register, colour, or timbre'. Register has a distinct meaning in vocal music which must not be disturbed. Timbre, properly a kettledrum, then a helmet, then the coat of arms surmounted with a helmet, then the official stamp bearing that coat of arms (now used in France for a postage label), and then the mark which declared a thing to be what it pretends to be, Burns's 'guinea's stamp,' is a foreign word, often odiously mispronounced, and not worth preserving. Colour I have never met with as applied to music, except at most as a passing metaphorical expression. But the difference of tones in qualitg is familiar to our language. Then as to the Partial Tones, Prof. Helmholtz uses Theiltone and Particd- tone, which are aptly Englished by partial simple tones. The words simple and tone, however, may be omitted when partials is employed, as partials are necessarily both tones and simple. The constituent tones of a chord may be either simple or compound. The Grundion or fundamental tone of a compound tone then becomes its prime tone, or briefly its prime. The Grundton or root of a chord will be further explained hereafter. Upper partial (simple) tones, that is, the partials exclusive of the prime, even when harmonic (that is, for the most part, belong- ing to the first six partial tones), must be distinguished from the sounds usually called harmonics when produced on a violin or harp for instance, for such harmonics are not neces- sarily simple tones, but are more generally compounds of some of the complete series of j)artial tones belonging to the musical tone of the whole string, selected by damping the remainder. The fading harmonics heard in listening to the sound of a pianoforte string, struck and undamped, as the sound dies away, are also compound and not simple partial tones, but as they have the successive partials for their successive primes, they have the

CHAPS. I. 11. COEXISTENCE OF DISTINCT WAVES OF SOUND. 25

Now, since quality of tone, as we have seen, depends on the form of vibration, which also determines the occurrence of upper partial tones, we have to inquire how far differences in quality of tone depend on different force or loudness of upper partials. This inq\iiry will be found to give a means of clearing- up our concep- tions of what has liitherto been a perfect enigma, — the nature of quality of tone. And we must then, of course, attempt to explain how the ear manages to analyse every musical tone into a series of partial tones, and what is the meaning of this analysis. These investigations will engage our attention in the following chapters.

CHAPTER II.

ON THE COMPOSITION OF VIBRATIONS.

A.T the end of the last chapter we came upon the remarkable fact that the human ear is capable, under certain conditions, of separating the musical tone produced by a single musical instrument, into a series of simple tones, namely, the prime partial tone, and the various upper partial tones, each of which produces its own separate sensation. That the ear is capable of distinguishing from each other tones proceeding from different sources, that is, which do not arise from one and the same sonorous body, we know from daily experience. There is no difficulty during a concert in following the melodic progression of each individual instru- ment or voice, if we direct our attention to it exclusively ; and, after some practice, most persons can succeed in following the simultaneous progression of several united parts. This is true, indeed, not merely for musical tones, but also for noises, and for mixtures of music and noise. When several persons are speaking at once, we can generally listen at pleasure to the words of any single one of them, H and even understand those words, provided that they are not too much overpowered by the mere loudness of the others. Hence it follows, first, that many different trains of waves of sound can be propagated at the same time through the same mass of ail-, without mutual disturbance ; and, secondly, that the human ear is capable of again analysing into its constituent elements that composite motion of the air which is produced by the simultaneous action of several musical instru- ments. We will first investigate the nature of the motion of the air when it is produced by several simultaneous musical tones, and how such a compound motion is distinguished from that due to a single musical tone. We shall see that the ear has no decisive test by which it can in all cases distinguish between the effect of a

pitch of those partials. But these fading meaning upper, but the English preposition harmonics are not regular compound tones of over is equivalent to the German preposition the kind described on p. 22«, because the lower iiher. Compare Obcrzolm, &n 'upper tooth,' ^ partials are absent one after another. Both i.e., a tooth in the upper jaw, with UeherzaJm, sets of harmonics serve to indicate the exist- an ' overtooth,' i.e., one grown over another, ence and place of the partials. But they are a projecting tooth. The continual recurrence no more those upper partial tones themselves, of such words as cJancj, clancjtint, overtone, than the original compound tone of the string would combine to give a strange un-English is its own prime. Great confusion of thought appearance to a translation from the German, having, to my own knowledge, arisen from On the contrary I have endeavoured to put it conionndiug such ha rmunics with tqjpcr parti(il into as straightforward EngHsh as possible. tones, I have generally avoided using the am- But for those acquainted with the original and biguous substantive Af/r/iwy/uV. Properly speak- with Prof. Tyndall's work, this explanation ing the harmonics of any compound tone are seemed necessary. Finally I would caution other compound tones of which the primes are the reader against using overtones for partial partials of the original compound tone of tones in general, as almost every one who which they are said to be harmonics. Prof. adopts Prof. Tyndall's word is in the habit of Helmholtz's term Oherfihie is merely a con- doing. Indeed I have in the course of this traction for Oberpartiattonc, but the casual translation observed, that even Prof. Helmholtz resemblance of the sounds of ober and over, has himself has been occasionally misled to em- led Prof. Tyndall to the erroneous translation ploy Obertone in the same loos^e manner. See overtones. The German ober is an adjective my remarks in note, p. 23f. — Translator.]

26 COMPOSITION OF WAVES. part i.

motion of the air caused by several different musical tones arising from different sources, and that caused by the musical tone of a single sounding body. Hence the ear has to analyse the composition of single musical tones, under proper con- ditions, by means of the same faculty which enabled it to analyse the composition of simultaneous musical tones. We shall thus obtain a clear concei^tion of v.-hat is meant by analysing a single musical tone into a series of partial simple tones, and we shall perceive that this phenomenon depends upon one of the most essential and fundamental properties of the human ear.

We begin by examining the motion of the air which corresponds to several simple tones acting at the same time on the same mass of air. To illustrate this kind of motion it will be again convenient to refer to the waves foi-med on a calm surface of water. We have seen (p. 9a) that if a point of the surface is agitated by a stone thrown upon it, the agitation is propagated in rings of waves over the surface

f to more and more distant points. Now, throw two stones at the same time on to different points of the surface, thus producing two centres of agitation. Each will give rise to a separate ring of waves, and the two rings gradually expanding, will finally meet. Where the waves thus come together, the water will be set in motion by both kinds of agitation at the same time, but this in no wise prevents botli series of waves from advancing further over the surface, just as if each were alone present and the other had no existence at all. As they proceed, those parts of both rings which had just coincided, again appear separate and mialtered in form. These little waves, caused by throwing in stones, may be accompanied by other kinds of waves, such as those due to the wind or a passing steamboat., Our circles of waves will spread out over the water tluis agitated, with the same quiet regularity as they did upon the calm surface. Neither will the greater waves be essentially disturbed by the less, nor the less by the greater, provided the waves never break ; if that happened, their regular course would certainly be impeded.

H Indeed it is seldom possible to survey a large surface of water from a high point of sight, without perceiving a great multitude of different systems of waves mutually overtopping and crossing each other. This is best seen on the surface of the sea, viewed from a lofty cliff, when there is a lull after a stiff breeze. We first see the great waves, advancing in far-stretching ranks from the blue distance, here and there more clearly marked oiit by their white foaming crests, and following one another at regular intervals towards the shore. From the shore they rebound, in different directions according to its sinuosities, and cut obliquely across the advancing waves. A passing steamboat forms its own wedge-shaped wake of waves, or a bird, dai'ting on a fish, excites a small circular system. The eye of the spectator is easily able to pursue each one of these diHerent trains of waves, great and small, wide and narrow, straight and curved, and observe how each passes over the surface, as undisturbedly as if the water over which it flits Avere not agitated at the same time by other motions and other forces. I must own that

II whenever I attentively observe this spectacle it awakens in me a peculiar kind of intellectual pleasure, because it bares to the bodily eye, what the mind's eye grasps only by the help of a long series of complicated conclusions for the waves of the invisible atmospheric ocean.

We have to imagine a perfectly similar spectacle proceeding in the interior of a ball-room, for instance. Hera we have a number of musical instruments in action, speaking men and women, rustling garments, gliding feet, clinking glasses, and so on. All these causes give rise to systems of waves, which dart through the mass of air in the room, are reflected from its walls, return, strike the opposite wall, are again reflected, and so on till they die out. We have to imagine that from the mouths of men and from the deeper musical instruments there proceed waves of from 8 to 12 feet in length [c to F], from the lips of the women waves of 2 to 4 feet in length [c" to c'], from the rustling of the dresses a fine small crumple of wave, and so on ; in short, a tumbled entanglement of the most difterent kinds of motion, complicated beyond conception.

CHAP. II. ALGEBRAICAL ADDITIOX OF WAVES. 27

And yet, ;is the ear is able to distinguish all the sei)arate constituent parts of this confused whole, we are forced to conclude that all these different systems of wave coexist in the mass of air, and leave one another mntually undisturbed. But how is it possible for them to coexist, since every individual train of waves has at any particular point in the mass of air its own particular degree of condensa- tion and rarefaction, which determines the velocity' of the particles of air to this side or that ? It is evident that at each point in the mass of air, at each instant of time, there can be only one single degree of condensation, and that the particles of air can be moving with only one single determinate kind of motion, having only one single determinate amount of velocity, and passing in only one single deter- minate direction.

What happens under such circumstances is seen directly by the eye in the waves of water. If where the water shows large waves we throw a stone in, tiie waves thus caused will, so to speak, cut into the larger moving surface, and thislj surface will be partly raised, and partlj- depressed, by the new waves, in such a way that the fresh crests of the rings will rise just as much above, and the troughs sink just as much below the curved surfaces of the previous larger waves, as they would have risen above or svink below the horizontal surface of calm water. Hence where a crest of the smaller system of rings of waves comes upon a crest of the greater system of waves, the surface of the water is raised by the sum of the two heights, and where a trough of the former coincides with a trough of the latter, the surface is depressed by the sum of the two depths. This may be expressed more briefly if we consider the heights of the crests above the level of the surface at rest, as positive magnitudes, and the depths of the troughs as negative magnitudes, and then form the so-called algebraical sum of these positive and negative magnitudes, in which case, as is well known, two positive magnitudes (heights of crests) must be added, and similarly for two negative magnitudes (depths of troughs) ; but when both negative and positive concur, one is to be subtracted U from the other. Performing the addition then in this algebraical sense, we can express our description of the surface of the water on which two systems of waves concur, in the following simple manner : The distance of the surface of the water at any point from its jjosition of rest is at any moment eqiial to the [alyeljraica/] sum of the distances at vjliich it ^vould have stood had each wave acted separately at the same jjlace and at the same time.

The eye most clearly and easily distinguishes the action in such a case as has been just adduced, where a smaller circular system of waves is produced on a large rectilinear system, because the two systems are then strongly distinguished from each other both by the height and shape of the waves. But with a little attention the eye recognises the same fact even when the two systems of waves have but slightly diff"erent forms, as when, for example, long rectilinear waves advancing towards the shore concur with those reflected from it in a slightly different direction. In this case we observe those well-known comb-backed waves where H the crest of one system of waves is heightened at some points by the crests of the other system, and at others depressed by its troughs. The multiplicity of forms is here extremely great, and any attempt to describe them would lead us too far. The attentive observer will readily comprehend the result by examining any disturbed surface of water, without further description. It will s\iffice for our purpose if the first example has given the reader a clear conception of what is meant by adding waves together/''

Hence although the surface of the water at any instant of time can assume only one single form, while each of two different systems of waves simultaneously attempts to impress its own shape upon it, we are able to suppose in the above

* Tho velocities and displacements of the addition of waves as is spoken of in the text,

particles of water are also to be added accord- is not perfectly correct, unless the heights of

ing to the law of the so-called parallelogram the waves are infinitely small in comparison

of forces. Strictly speaking, such a simple with their lengths.

28 ALGEBRAICAL ADDITION OF WAVES. part i.

sense that the two systems coexist and are superimposed, by considering the actual elevations and depressions of the surface to be suitably separated into two parts, each of which belongs to one of the systems alone.

In the same sense, then, there is also a superimposition of different systems of sound in the air. By each train of waves of sound, tlie density of the air and the velocity and position of the particles of air, are temporarily altered. There are places in the wave of sound comparable with the crests of the waves of water, in which the quantity of the air is increased, and the air, not having free space to escape, is condensed ; and other places in the mass of air, comparable to the troughs of the waves of water, having a diminished quantity of air, and hence diminished density. It is true that two different degrees of density, produced by two different systems of waves, cannot coexist in the same place at the same time ; nevertheless the condensations and rarefactions of the air can be (algebraically)

H added, exactly as the elevations and depressions of the surface of the water in the former case. Where two condensations are added we obtain increased condensation, where two rarefactions are added we have increased rarefaction ; while a concur- rence of condensation and rarefaction mutually, in whole or in part, destroy or neutralise each other.

The displacements of the particles of air are compounded in a similar manner. If the displacements of two different systems of waves are not in the same direc- tion, they are compounded diagonally ; for example, if one system would drive a particle of air upwards, and another to the right, its real path will be obliquely upwards towards the right. For our present purpose there is no occasion to enter more particularly into such compositions of motion in different directions. We are only interested in the effect of the mass of air upon the ear, and for this we are only concerned with the motion of the air in the passages of the ear. Now the passages of our ear are so narrow in comparison with the length of the waves of

^ sound, that we need only consider such motions of the air as are parallel to the axis of the passages, and hence have only to distinguish displacements of the particles of air outwards and inwards, that is towards the outer air and towards the interior of the ear. For the magnitude of these displacements as well as for their velocities with which the particles of air move outwards and inwards, the same (algebraical) addition holds good as for the crests and troughs of waves of water.

Hence, vjhen several sonorous bodies in the surroxmding atmosphere, simnl- taneously excite different systems of waves of sound, the changes of density of the air, and the disj)lacements and velocities of the ^)a^*^?'c^6's of the air ivithin the passages of the ear, are each equal to the [algebraical) sum of the corresponding changes of density, disjolacements, and- velocities, inhich each system of waves would have sejmrately produced, if it had acted independently ; * and in this sense we can say that all the separate vibrations which separate waves of sound would

H have produced, coexist undisturbed at the same time within the passages of our ear. After having thus in answer to the first question explained in what sense it is possible for several different systems of waves to coexist on the same surface of water or within the same mass of air, we proceed to determine the means possessed by our organs of sense, for analysing this composite whole into its original consti- tuents.

I have already observed that an eye which surveys an extensive and disturbed surface of water, easily distinguishes the separate systems of waves from each other and follows their motions. The eye has a great advantage over the ear in being able to survey a large extent of surface at the same moment. Hence the eye readily sees whether the individual waves of water are rectilinear or curved, and whether they have the same centre of curvature, and in what direction they

* The same is true for the whole mass of according to the law of the parallelogram of external air, if only the addition of the dis- forces, placements in different dii'ections is made

CHAP. II. EYE AND EAR C'ONTRASTED. 29

are advancin"'. All these observations assist it in determining whotlier two systems of waves are connected or not, and hence in discovering their corresponding parts. Moreover, («i the snrface of the water, waves of unequal length advance with unecpial velocities, so that if they coincide at one moment to such a degree as to be difficult to distinguish, at the next instant one train pushes on and the other lags behind, so that they become again separately visible. In this way, then, the observer is greatly assisted in referring each system to its point of departure, and in keeping it distinctly visible during its further course. For the eye, then, two systems of waves having difterent points of departure can never coalesce ; for example, such as arise from two stones thrown into the water at different points. If in any one place the rings of wave coincide so closely as not to be easily separable, they always remain separate during the greater part of their extent. Hence the eye could not be easily brought to confuse a compound with a simple undulatory motion. Yet this is precisely what the ear does under similar circum-H stances when it separates the musical tone which has proceeded from a single source of sound, into a series of simple partial tones.

But the ear is much more unfavourably situated in relation to a system of waves of sound, than the eye for a system of waves of water. The ear is affected only by the motion of that mass of air which happens to be in the immediate neigh- bourhood of its tympanum within the aural passage. Since a transverse section of the aural passage is comparatively small in comparison with the length of waves of sound (which for serviceable musical tones varies from 6 inches to .32 feet),* it corresponds to a single point of the mass of air in motion. It is so small that distinctly different degrees of density or velocity could scarcely occur upon it, because the positions of greatest and least density, of greatest positive and nega- tive velocity, are always separated by half the length of a wave. The ear is therefore in nearly the same condition as the eye would be if it looked at one point of the surface of the water, through a long narrow tube, which would permit of ^ seeing its rising and falling, and were then required to undertake an analysis of the compound waves. It is easily seen that the eye would, in most cases, completely fail in the solution of such a problem. The ear is not in a condition to discover how the air is moving at distant spots, Avhether the waves which strike it are spherical or plane, whether they interlock in one or more circles, or in what direction they are advancing. The cii'cumstances on which the eye chiefly depends for foi'ming a judgment, are all absent for the ear.

If, then, notwithstanding all these difficulties, the ear is capable of distin- guishing musical tones arising from different sources — and it really shows a marvellous readiness in so doing— it must employ means and possess properties altogether difterent from those employed or possessed by the eye. But whatever these means may be — and we shall endeavour to determine them hereafter — it is clear that the analysis of a composite mass of musical tones must in the first place be closely connected with some determinate properties of the motion of the ^ air, capable of impressing theniselves even on such a very minute mass of air as that contained in the aural passage. If the motions of the particles of air in this passage are the same on two different occasions, the ear will receive the same sensation, whatever be the origin of those motions, whether they spring from one or several sources.

We have already explained that the mass of air which sets the tympanic membrane of the ear in motion, so far as the magnitudes here considered are concerned, must be looked upon as a single point in the surrounding atmosphere. Are there, then, any peculiarities in the motion of a single particle of air which would differ for a single musical tone, and for a combination of musical tones ? We have seen that for each single musical tone there is a corresponding periodical

* [These are of course rather more than flue organ pipes. See Chap. Y. sect. 5, and twice the length of the corresponding open compare p. 26rf. — Trmislator.]

30

COMPOSITION OF SIMPLE WAVES.

motion of the air, and that its pitch is determined by the length of the periodic time, but that the kind of motion during any one single period is perfectly arbitrary, and may indeed be infinitely various. If then the motion of the air lying in the aural passage is not periodic, or if at least its periodic time is not as short as that of an audible musical tone, this fact will distinguish it from any motion which belongs to a musical tone ; it must belong either to noises or to several simultaneous musicll tones. Of this kind are really the greater number of cases where the dif- ferent musical tones have been only accidentally combined, and are therefore not designedly framed into musical chords; nay, even where orchestral music is per- fornied, the method of tempered tuning which at present prevails, prevents an accurate fulfilment of the conditions under which alone the resulting motion of the air can be exactly periodic. Hence in the greater number of cases a want of periodicity in the motion might furnish a mark for distinguishing the presence ^ of a composite mass of musical tones.

But a composite mass of musical tones may also give rise to a jmrely periodic motion of the air, namely, token all the musical tones which intermingle, have pitch numbers which are all multiples of one and the same old mimher, or which

Fio. 11.

comes to the same thing, when all these musical tones, so far as their pjitch is concerned, may he regarded as the upper partial tones of the same prime tone. It was mentioned in Chapter I. (p. 22a, h) that the pitch numbers of the upper partial tones are multiples of the pitch number of the prime tone. The meaning of this rule will be clear from a particular example. The curve A, fig. II, represents a pendular motion in the manner explained in Chap. I. (p. 21/^), as produced in the air of the aural passage by a tuning-fork in action. The horizontal lengths in the curves of fig. 11, consequently represent the passing time, and the vertical heights the corresponding displacements of the particles of air in the aural passage. Now suppose that \\ith the first simple tone to which the curve A corresponds, there is sounded a second simple tone, represented by the curve B, an Octave higher than the first. This condition requires that two vibrations of the curve B should be made in the same time as one vibration of the curve A. In A, the sections of the curve d„8 and 8 8i are perfectly equal and similar. The curve B is also divided into equal and similar sections e e and c ej by the points e, c, €,. We could cer- tainly halve each of the sections e e and c c„ and thus obtain equal and similar sections, each of which would then correspond to a single period of B. But by

CHAP. II. COMPOSITION OF .SIMPLE WAVES. 31

taking sections consisting of two periods of B, we divide B into larger sections, each of which is of the same horizontal length, and hence corres])onds to the same duration of time, as the sections of A.

If, then, both simple tones are heard at once, and the times of the points e and dj, € and 8, e, and S, coincide, the heights of the portions of the section of curve e e have to be [algebraically] added to heights of the section of curve <i„8, and similarly for the sections e c, and 8 S,. The result of this addition is shown in the curve C. The dotted line is a duplicate of the section d^S in the curve A. Its object is to make the composition of the two sections immediately evident to the eye. It is easily seen that the curve C in every place rises as much above or sinks as nnich below the curve A, as the curve B respectively rises above or sinks beneath the horizontal line. The heights of the curve C are consequently, in ac- cordance with the rule for compounding vibrations, equal to the [algebraical] sum of the corresponding heights of A and B. Thus the perpendicular Ci in C is the H sum of the perpendiculars a, and bi in A and B ; the lower part of this perpen- dicular Ci, from the straight line up to the dotted curve, is equal to the perpen- dicular ai, and the upper part, from the dotted to the continuous curve, is equal to the perpendicular bj. On the other hand, the height of the perpendicular Cj is equal to the height a^ diminished by the depth of the fall bo. And in the same way all other points in the curve C are found.*

It is evident that the motion represented by the curve C is also periodic, and that its periods have the same duration as those of A. Thus the addition of the section d„S of A and e e of B, must give the same result as the addition of the perfectly equal and similar sections 8 8, and e e„ and, if we supposed both curves to be continued, the same would be the case for all the sections into which they would be divided. It is also evident that equal sections of both curves could not continually coincide in this way after completing the addition, unless the ciu'ves thus added could be also separated into exactly equal and similar sections of the same II length, as is the case in fig. 11, whei-e two periods of B last as long or have the same horizontal length as one of A. Now the horizontal lengths of our figure represent time, and if we pass from the curves to the real motions, it results that the motion of air caused by the composition of the two simple tones, A and B, is also periodic, just because one of these simple tones makes exactly twice as many vibrations as the other in the same time.

It is easily seen by this example that the peculiar form of the two curves A and B has nothing to do with the fact that their sum C is also a periodic curve. Whatever be the form of A and B, provided that each can be separated into equal and similar sections which have the same horizontal lengths as the equal and similar sections of the other — no matter whether these sections correspond to one or two, or three periods of the individual curves — then any one section of the curve A compounded with any one section of the curve B, will always give a section of the curve C, which will have the same length, and will be precisely equal and •fl similar to any other section of the curve C obtained by compounding any other section of A with any other section of B.

When such a section embraces several periods of the corresponding curve (as in fig. 11, the sections e e and e e, each consist of two periods of the simple tone B), then the pitch of this second tone B, is that of an upper partial tone of a prime (as the simple tone A in fig. 11), whose period has the length of that principal section, in accordance with the rule above cited.

In order to give a slight conception of the multiplicity of forms producible by comparatively simple compositions, I may remark that the compound curve would

* [Readers not used to geometrical con- spending perpendiculars in A and B in proper

structions are strongly recommended to trace directions, and joining the extremities of the

the two curves A and B, and to construct the lengths thus found by a curved line. In this

curve C from them, by drawing a number of way only can a clear conceiDtion of the cona-

perpendiculars to a straight line, and then position of vibrations be rendcnnl sufficiently

setting oS. upon them the lengths of the corre- familiar for subsequent use. — Translatvr.'^

32

DIFFERENCE OF PHASE.

receive another form if the curves B, fig. 11, were displaced a little with respect to the curve A before the addition were commenced-. Let B be displaced by being slid to the right until the point e falls under dj in A, and the composition will then give the curve D with narrow crests and broad troughs, both sides of the crest being, however, equally steep ; whereas in the curve C one side is steeper than the other.' If we displace the curve B still more by sliding it to the right till e falls under do, the compound curve would resemble the reflection of C in a mirror : that is, it would have the same form as C reversed as to right and left ; the steeper inclination which in C lies to the left would now lie to the right. Again, if we displace B till e falls under dj we obtain a curve similar to D, fig. 11, but reversed as to up and down, as may be seen by holding the book upside-down, the crests being broad and the troughs narrow.

Fig. 12.

^

All these curves with their various transitional forms are periodic curves. Other composite periodic curves are shown at C, D, fig. 12 above, where they are compounded of the two curves A and B, having their periods in the ratio of 1 to 3. The dotted curves are as before copies of the first complete vibration or period of the curve A, in order that the reader may see at a glance that the compound curve is always as much Iiigher or lower than A, as B is higher or lower than the horizontal line. In C, the curves A and B are added as they stand, but for D the curve B has been first slid half a wave's length to the right, and then the addition •j has been effected. Both forms differ from each other and from all preceding ones. C has broad crests and broad troughs, D narrow crests and narrow troughs.

In these and similar cases we have seen that the compound motion is perfectly and regiilarly periodic, that is, it is exactly of the same kind as if it proceeded from a single musical tone. The curves compounded in these examples correspond to the motions of single simple tones. Thus, the motions shown in fig. 11 (on p. SOb, c) might have been produced by two tuning-forks, of which one soimded an Octave higher than the other. But we shall hereafter see that a flute by itself when gently blown is sufticient to create a motion of the air corresponding to that shown in C or D of fig. 11. The motions of fig. 12 might be produced by two tuning-forks of which one sounded the twelfth of the other. Also a single closed organ pipe of the narrower kind (the stop called Quintnten*) would give nearly the same motion as that of C or D in fig. 12.

* [The names of the stops on German organs do not always agree with those on

English organs. I find it best, therefore, not to translate them, but to give their explana-

CHAP. n. ANALYSIS INTO SIMPLE VIBIUTIONS. 33

Here, then, the motion of the air in the aural passage has no property by whicli tlie composite* musical tone can be distinguished from the single nuisical tone. If the ear is not assisted by other accidental circumstances, as by one tuning-fork beginning to sound before the other, so that we hear them struck, or, in the other case, the rustling of the wind against the mouthpiece of the flute or lip of the organ pipe, it has no means of deciding whether the musical tone is sim])le or composite.

Now, in what relation does the ear stand to such a motion of the air 1 Does it analyse it or does it not ? F.xperience shows us that when two ti;ning-forks, an Octave or a Twelfth apart in pitch, are sounded together, the ear is quite able to distinguish their simple tones, although the distinction is a little more difficult with these than with other intervals. But if the ear is able to analyse a compo- site musical tone produced by two tuning-forks, it cannot but be in a condition to carry out a similar analysis, when the same motion of the air is produced by a H single flute or organ pipe. And this is really the case. The single musical tone of such instruments, proceeding from a single source, is, as we have already men- tioned, analysed into partial simple tones, consisting in each case of a prime tone, and one upper partial tone, the latter being different in the two cases.

The analysis of a single musical tone into a series of partial tones depends, then, \ipon the same property of the car as that which enables it to distinguish different musical tones from each other, and it miist necessaril}'^ effect both analyses by a rule which is independent of the fact that the waves of sound are produced by one or by several musical instruments.

The rule by which the ear proceeds in its anal^'sis was first laid down as generally true by G. S. Ohm. Part of this rule has been already enunciated in the last chapter (p. 2'3a), where it was stated that only that particular motion of the air which we have denominated a simple vibration, for whicli the vibrating particles swing backwards and forwards according to the law of pendular motion, H is capable of exciting in the ear the sensation of a single simple tone. Every motion of the air, then, which corresp07ids to a compiosite mass of nntsical tones, is, according to Ohtn^s laio, capable of being analysed into a sum of simple pen- dular vibrations, and to each such single simple vibration corresponds a simjile tone, sensible to the ear, and having a pntch detertnined, by the periodic time of the correspjonding motion of the air.

The proofs of the correctness of this law, the reasons why, of all vibrational forms, only that one which we have called a simple vibration plays such an important part, must be left for Chapters IV. and VI. Our present business is only to gain a clear conception of what the rule means.

The simple vibrational form is inalterable and always the same. It is only its amplitude and its periodic time which are subject to change. But we have seen in figs. 11 and 12 (p. 306 and p. 326) what varied forms the composition of only two simple vibrations can produce. The number of these forms might be greatly in- %. creased, even without introducing fresh simple vibrations of different periodic times, by merely changing the proportions wdiich the heights of the two simple

tions from E. J. Hopkins's The Organ, its iu other cases, ' a pipe for sounding the Twelfth

History and Construction, 1870, pp. 444-448. in addition to the fundamental tone'. It seems

In this case Mr. Hopkins, following other to be properly the English stop ' Tii:elfUi,

authorities, prints the word ' quintato;;,' and Octave Quin', Ihioi/fciinu,' No. 611, p. 141 of

defines it, in IG feet tone, as ' double stopped Hopkins. — Tnni^hitur.]

diapason, of rather small scale, producing the * [The reader must distinguish between

Twelfth of the fundamental sound, as well as single and simple musical tones. A single tone

the ground-tone itself, that is, sounding the may be a conqknuul tone inasmuch as it may

IG and 5| ft. tones ' which means sounding the be compounded of several simple musical tones,

notes beginning with C'„ simultaneously with but it is si)igle because it is produced by one

tlie notes beginning with G, which is called the sounding body. A composite _ musical tone is

5^ foot tone, because according to the organ- necessarily compound, but it is called coinjiosite

makers' theory ^not practice) the length of the because it is made up of tones (simple orcom-

G pipe is \ of the length of the 0 pipe, and ^ of pound) produced by several sounding bodies.—

16 is 5J. [See p. IM, noteJ.J And similarly, Translator.]

D

34 ANALYSIS INTO SIMPLE VIBRATIONS. part i.

vibrational curves A and B bear to eacb other, or displacing the curve B by other distances to the right or left, than those ah-eadj selected in the figures. By these simplest possible examples of such compositions, the reader will be able to form some idea of the enormous variety of forms which would result from using more than two simple forms of vibration, each form representing an upper partial tone of the same prime, and hence, on addition, always producing fresh periodic curves. We should be able to make the heights of each single simple vibrational curve greater or smailer at pleasure, and displace each one separately by any amount in respect to the prime, — or, in physical language, we should be able to alter their amplitudes and the difterence of their phases ; and each such alteration of ampli- tude and difference of phase in each one of the simple vibrations would produce a fresli change in the resulting composite vibrational form. [See App. XX. sect. M. No. 2.1

The multiplicity of vibrational forms which can be thus produced by the com- 11 position of simple pendular vibrations is not merely extraordinarily great : it is so great that it cannot be greater. The French mathematician Fourier has proved the correctness of a mathematical law, which in reference to our present subject may be thus enunciated : Any given regular periodic form of vibration can always be produced by the addition of simple vibrations, having 2^itch numbers which are once, twice, thrice, four times, dx., as great as the pitch nwnbers of the given motion.

The amplit%ides of the elementary simple vibrations to which the height of our wave-curves corresponds, and the difference of phase, that is, the relative amount of horizontal displacement of the wave-curves, can always be found in every given case, as Fourier has shown, by peculiar methods of calculation (which, however, do not admit of any popular explanation), so that any given regularly periodic motion can always be exhibited in one single way, and in no other way whatever, as the sum of a certain number of j^endidar vibrations. tI Since, according to the results already obtained, any regularly periodic motion corresponds to some musical tone, and any simple pendular vibration to a simple musical tone, these propositions of Fourier ma\ be thus expressed in acoustical terms :

Any vibrational motion of the air in the entrance to the ear, corresjwnding to a musical tone, may be cdtvays, and for each case only in one single way, exhibited as the sum of a number of simjde vibrational motions, corresponding to the partials of this musical tone.

Since, according to these propositions, any form of vibration, no matter what shape it maj^ take, can be expressed as the sum of simple vibrations, its analysis into such a sum is quite independent of the power of the eye to perceive, by looking at its representative curve, whether it contains simple vibrations or not, and if it does, what they are. I am obliged to lay stress upon this point, because I have by no means unfrequently found even physicists start on the false hypothesis, that the H vibrational form must exhibit little waves corresponding to the several audible upper partial tones. A mere inspection of the figs. II and 12 (p. 30b and p. 32b) will suffice to show that although the composition can be easily traced in the parts where the curve of the prime tone is dotted in, this is quite impossible in those parts of the curves C and D in each figure, where no such assistance has been provided. Or, if we suppose that an observer who had rendered himself thoroughly familiar with the curves of simple vibrations imagined that he could trace the com- position in these easy cases, he would certainly utterly fail on attempting to dis- cover by his eye alone the composition of such curves as are shown in figs. 8 and 9 (p. 21c). In these will be found straight lines and acute angles. Perha]js it will be asked how it is possible by compounding such smooth and uniformly rounded curves as those of our simple vibrational forms A and B in figs. 1 1 and 12, to generate at one time straight lines, and at another acute angles. The answer is, that an infinite number of simple vibrations are required to generate curves with such discontinuities as are there shown. But when a great many

CHAP. II. ANALYSIS INTO SIMPLE VIBRATIONS. 3o

such curves are combined, and are so chosen that in certain places thev all bend in the same direction, and in oth^rs'in opposite directions, the curvatures mutually strengthen each other in the first case, finally producing an infinitely great curva- ture, that is, an acute angle, and in the second case they mutually weaken each other, so that ultimately a straight line results. Hence we can generally lay it down as a rule that the force or loudness of the upper partial tones is the gi'eater, the sharper the discontinuities of the atmospheric motion. When the motion alters uniformly and gradually, answering to a vibrational curve j)roceeding in smoothly curved forms, only the deeper partial tones, which lie nearest to the prime tone, have any perceptible intensity. But where the motion alters by jumps, and hence the vibrational curves show angles or sudden changes of curvature, the upper partial tones will also have sensible force, although in all these cases the amplitudes decrease as the pitch of the upper partial tones becomes higher.*

We shall become acquainted with examples of the analysis of given vibrational H forms into separate partial tones in Chapter V.

The theorem of F'ourier here adduced shows first that it is mathematicallv possible to consider a musical tone as a sum of simple tones, in the meaning we have attached to the words, and mathematicians have indeed always found it convenient to base their acoustic investigations on this mode of analysing vibrations. But it by no means follows that we are obliged to consider the matter in this way. We have rather to inquire, do these partial constituents of a musical tone, such as the mathemathical theory distinguishes and the ear perceives, really exist in the mass of air external to the ear? Is this means of analysing forms of vibration which Fourier's theorem prescribes and renders possible, not merely a mathematical fiction, permissible for facilitating calculation, but not necessarily having any corresponding actual meaning in things themselves? What makes us hit upon pendidar vibrations, and none other, as the simplest element of all motions pro- ducing sound ? We can conceive a whole to be split into parts in very different and arbitrary ways. Thus we may find it convenient for a certain calcxdation to ^ consider the number 12 as the sum 8 + 4, because the 8 may have to be cancelled, but it does not follow that 12 must always and necessarily be considered as merely the sum of 8 and 4. In another case it might be more convenient to consider 12 as the sum of 7 and 5. Just as little does the mathematical possibility, proved by Fourier, of compoimding all pei'iodic vibrations out of simple vibrations, justifv us in concluding that this is the only permissible form of analysis, if we cannot in addition establish that this analysis has also an essential meaning in nature. That this is indeed the case, that this analysis has a meaning in nature independently of theory, is rendered probable by the fact that the ear really effects the same analysis, and also by the circumstance already named, that this kind of analysis has been found so much more advantageous in mathematical investigations than any other. Those modes of regarding phenomena that correspond to the most intimate constitution of the matter under investigation are, of course, also always those which lead to the most suitable and evident theoretical treatment. But it ^ would not be advisable to begin the investigation with the functions of the ear, because these are very intricate, and in themselves require much explanation. In the next chapter, therefore, we shall inquire whether the analysis of compound into simple vibrations has an actually sensible meaning in the external world, independently of the action of the ear, and we shall really be in a condition to show that certain mechanical effects depend upon whether a certain partial tone

* Supposing n to be the number of the a sudden jump, .and hence the curve lias an order of a partial tone, and n to be very large, 1

then the amplitude of the upper partial tones acute angle ; (3) as ^-^-^, when the curvature

decreases : (1) as -, when tbe amplitude of the f^^f' suddenly ; (4) when none of the differen-

w '■ tial quotients are discontniuous, they must

vibrations themselves makes a sudden jump ; , j. i j. ^ i. -«-

"1^ J i ' decrease at least as fast as c .

(21as — , when their differential quotient makes

36 MECHANICS OF SYMPATHETIC RESONANCE. paut i.

is or is not contained in a composite mass of musical tones. Tlie existence of partial tones will thus acquire a meaning in nature, and our knowledge of their mechanical effects will in turn shed a new light on their relations to the human ear.

CHAITEH III.

ANALYSIS OF MUSICAL TONES CY SYMrATHETIC RESONANCE.

AVE proceed to show that the simple partial tones contained in a composite mass of musical tones, produce peculiar mechanical effects in nature, altogether inde- pendent of the human ear and its sensations, and also altogether independent of

^ merely theoretical considerations. These effects consequently give a peculiar objec- tive significance to this peculiar method of analysing vibrational forms.

Such an effect occurs in the phenomenon of sympathetic resonance. This phenomenon is always found in those bodies which when once set in motion by any impiilse, continue to perform a long series of vibrtitions before they come to rest. When these bodies are struck gently, but periodically, although each blow may be separately quite insufficient to produce a sensible motion in the vibratory body, yet, provided the periodic time of the gentle blows is precisely the same as the periodic time of the body's own vibrations, very large and powerful oscilla- tions may result. But if the periodic time of the regular blows is different from the periodic time of the oscillations, the resulting motion will be weak or quite insensible.

Periodic impulses of this kind generally proceed from another body which is already vibrating regularly, and in this case the swings of the latter in the course

H of a little time, call into action the swings of the former. Under these circum- stances we have the pi'ocess called syrnjmthetic oscillation or sympathetic resonance. The essence of the mechanical effect is independent of the rate of motion, which may be fast enough to excite the sensation of sound, or slow enough not to produce anything of the kind. Musicians are well acquainted with symjmthetic resonance. When, for example, the strings of two violins are in exact unison, and one string is bowed, the other will begin to vibrate. But the nature of the process is best seen in instances where the vibrations are slow enough for the eye to follow the whole of their successive phases.

Thus, for example, it is known that the largest clnu-ch-bells may be set in motion by a man, or even a boy, who pulls the ropes attached to them at proper aiid regular intervals, even when their weight of metal is so great that the strongest man could scarcely move them sensibly, if he did not apply his strength in determinate periodical intervals. When such a l»ell is once set in motion, it continues, like a

H struck pendulum, to oscillate for some time, until it gradually returns to rest, even if it is left quite by itself, and no force is employed to arrest its motion. The motion diminishes gradually, as we know, because the friction on the axis and the resistance of the air at every swing destroy a portion of the existing moving force.

As the bell swings backwards and forwards, the lever and vo\)e fixed to its axis rise and fall. If when the lever falls a boy clings to the lower end of the bell-rope, his weight will act so as to increase the rapidity of the existing motion. This increase of velocity may be very small, and yet it will produce a coi-responding increase in the extent of the bell's swings, which again will continue for a while, until destroyed by the friction and resistance of the air. But if the boy clung to the bell-rope at a wrong time, while it Avas ascending, for instance, the weight of his body would act in opposition to the motion of the bell, and the extent of swing Avould decrease. Now, if the boy continued to cling to the rope at each swing so long as it was falling, and then let it ascend freely, at every swing the motion of the bell would be only increased in speed, and its swings would gradually become

CHAP. III. MKCHANICS OF SYMPATHETIC RESONANCE. 37

greater and greater, \iiitil by their increase the motion imparted on every osciUation of the bell to the walls of the belfry, and the external air would become so great as exactly to be covered by the power exerted by the boy at each swing.

The success of this process depends, therefore, essentially on the boy's applying his force only at those moments when it will increase the motion of the bell. That is, he must employ his strength periodically, and the ])criodic time must be equal to that of the bell's swing, or he will not be successful. He would just as easily bring the swinging bell to rest, if he clung to the rope only during its ascent, and thus let his weight be raised by the bell.

A similar experiment which can be tried at any instant is the following. Con- struct a pendulum by hanging a heavy body (such as a ring) to the lower end of a thread, holding the upper end in the hand. On setting the ring into gentle pen- dular vibration, it will be found that this motion can be gradually and considerably increased by watching the moment when the pendulum has reached its greatest H departure from the vertical, and then giving the hand a very small motion in the opposite direction. Thus, when the pendulum is furthest to the right, move the hand very slightly to the left ; and when the pendulum is furthest to the left, move the hand to the right. The pendulum may be also set in motion from a state of rest by giving the h^md similar very slight motions having the same periodic time as the pendulum's own swings. The displacements of the hand may be so small under these circumstances, that they can scarcely be perceived with the closest attention, a circumstance to which is due the superstitious application of this little apparatus as a divining rod. If namely the observer, without thinking of his hand, follows the swings of the pendulum with his eye, the hand readily follows the eye, and involuntarily moves a little backwards or forwards, precisely in the same time as the pendulum, after this has accidentally begun to move. These involuntary motions of the hand are usnally overlooked, at least when the observer is not accustomed to exact observations on such unobtrusive influences. By this "^ nieans any existing vibration of the pendulum is increased and kept up, and any accidental motion of the ring is readily converted into pendular vibrations, which seem to arise spontaneously without any co-operation of the observer, and are hence attributed to the influence of hidden metals, running streams, and so on.

If on the other hand the motion of the hand is intentionally made in the con- trary direction, the pendiilum soon comes to rest.

The explanation of the process is very simple. When the iipper end of the thread is fastened to an immovable support, the pendulum, once struck, continues to swhig for a long time, and the extent of its swings diminishes very slowly. We can suppose the extent of the swings to be measured by the angle which the thread makes with the vertical on its greatest deflection from it. If the attached body at the point of greatest deflection lies to the right, and we move the hand to the left, we manifestly increase the angle between the string and the vertical, and con- 1 sequently also augment the extent of the swing. By moving the upper end of the string in the opposite direction we should decrease the extent of the swmg.

In this case there is no necessity for moving the hand in the same periodic time as the pendulum swings. We miglit move the hand backwards and forwards only at every third or fifth or other swing of the pendulum, and we should still produce large swings. Thus, when the pendulum is to the right, move the hand to the left, and keep it still, till the pendulum has swung to the left, then again to the right, and then once more to the left, and then return the hand to its first position, afterwards wait till the pendidum has swung to the right, then to the left, and again to the right, and then recommence the first motion of the hand. In this way three complete vibrations, or double excursions of the pendulum, will corre- spond to one left and right motion of the hand. In the same way one left and right motion of the hand may be made to correspond with seven or more swings of the pendulum. The meaning of this i)rocess is always that the motion of the

38 MECHANICS OF SYMPATHETIC RESONANCE. part i.

liand must in each case be made at such a time and in such a direction as to be opposed to the deflection of the penduhun and consequently to inci'ease it.

By a sHght alteration of the process we can easily make two, four, si.x, etc., swings of the pendulum correspond to one left and right motion of the hand ; for a sudden motion of the hand at the instant of the pendulum's passage through the vertical has no influence on the size of the swings. Hence when the pendulum lies to the right move the hand to the left, and so increase its velocity, let it swing to the left, watch for the moment of its passing the vertical line, and at that instant return the hand to its original position, allow it to reach the right, and then again the left and once more the right extremity of its arc, and then recommence the first motion of the hand.

We are able then to communicate violent motion to the pendulum by very small periodical vibrations of the hand, having their periodic time exactly as great, Hor else two, three, four, &c., times as great as that of the peudular oscillation. We have here considered that the motion of the hand is backwards. This is not necessary. It may take place continuously in any other way we please. When it moves continuously there Avill be generally portions of time during which it will increase the pendulum's motion, and others perhaps in which it will diminish the same. In order to create strong vibrations in the pendulum, then, it will be necessary that the increments of motion should l)e permanently predominant, and should not be neutralised by the sum of the decrements.

Now if a determinate periodic motion were assigned to the hand, and we wished to discover wliether it would produce considerable vibrations in the pendulum, we could not alwaj's predict the result without calculation. Theoretical mechanics would, however, prescribe the following })i-ocess to be pursued : Analyse the lieriocUc motion of the hand into a sum of simple pendular vihrations of the Aa?ifZ— exactly in the same way as was laid down in the last chapter for the periodic motions of 51 the particles of air, — then, if the periodk time of one of these vibrations is eqwil to the periodic time of the 2'Ctiduliim's oi"n oscillations, the j/enduliim roill be set into violent motion, but not otherwise. We might compound small pendular motions of the hand out of viljrations of other periodic times, as much as we liked, but we should fail to produce any lasting strong swings of the pendulum. Hence the analysis of the motion of the hand into pendular swings has a real meaning in nature, producing determinate mechanical effects, and for the present purpose no other analysis of the motion of the hand into any other partial motions can be substituted for it.

In the above examples tlie pendulum could lie set into sympathetic vibration, when the hand moved periodically at the same rate as the pendulum ; in this case the longest partial vibration of the hand, corresponding to the prime tone of a resonant vibration, was, so to sjjeak, in unison with the pendulum. When three swings of the pendulum went to one backwards and forwards motion of the hand, H it was the third partial swing of the hand, answering as it were to the Twelfth of its prime tone, which set the pendulum in motion. And so on.

The same process that we have thus become acquainted with for swings of long periodic time, holds precisely for swings of so short a period as sonorous vibrations. Any elastic body which is so fastened as to admit of continuing its vibrations for some length of time when once set in motion, can also be made to vibrate sym- p.itheticall}', when it receives periodic agitations of comparatively small amounts, having a periodic time corresponding to that of its own tone.

Gently touch one of the keys of a pianoforte without striking the string, so as to raise the damper only, and then sing a note of the corresponding pitch forcibly directing the voice against the strings of the instrument. On ceasing to sing, the note will be echoed back from the piano. It is easy to discover that this echo is caused by the string which is in unison with the note, for directly the hand is removed from the key, and the damper is allowed to fall, tlie echo ceases. The sympathetic vibration of the sti-ing is still better shown by putting little paper

CHAP. III. DIFFERENT EXTENT OF SYMPATHETIC RESONANCE. 39

riders iipon it, which are jerked oft" as soon as the string vibrates. The more exactly the singer hits the pitch of the string, the more strongly it vibrates. A very little deviation from the exact pitch fails in exciting sympathetic vibration.

In this experiment the sounding board of the instrument is first struck by the vibrations of the air excited by the human voice. The sounding board is well known to consist of a broad flexible wooden plate, which, owing to its exten- sive siirface, is better adapted to convey the agitation of the strings to the air, and of the air to the strings, than the small surface over which string and air arc themselves directly in contact. The sounding board first commiuiicates the agita- tions which it receives from the air excited by the singer, to the points where the string is fastened. The magnitude of an}- single such agitation is of course infini- tesimally small. A very large number of such effects must necessarily be aggre- gated, before any sensible motion of the string can be caused. And such a con- tinuous addition of eff"ects really takes place, if, as in the preceding experiments with ^ the bell and the pendulum, the periodic time of the small agitations which are com- municated to the extremities of the string by the air, through the intervention of the sounding board, exactl}- corresponds to the periodic time of the string's own vibra- tions. When this is the case, a long series of such vibrations will really set the string into motion which is very violent in comparison with the exciting cause.

In place of the human voice we might of course use any other musical instru- ment. Provided only that it can produce the tone of the pianoforte string accu- rately and sustain it powerfully, it will bring the latter into sympathetic vibration. In place of a pianoforte, again, we can employ any other stringed instrument having a sounding board, as a violin, guitar, harp, etc., and also stretched mem- branes, bells, elastic tongues or plates, ifec, provided only that the latter are so fastened as to admit of their giving a tone of sensible duration when once made to sound. ^

Wlien the pitch of the original sounding body is not exactly that of the sym- |)athising body, or that which is meant to vibrate in sympathy with it, the latter will nevertheless often make sensible sympathetic vibrations, which will diminish in amplitude as the difference of pitch increases. But in this respect difterent sounding bodies show great difterences, according to the length of time for which they continue to soiuid after having been set in action before comnnuiicating their whole motion to the air.

Bodies of small mass, which readily communicate their motion to the air, and quickly cease to sound, as, for example, stretched membranes, or violin strings, are readily set in sympathetic vibration, because the motion of the air is conversely readily transferred to them, and they are also sensibly moved by sufficiently strong agitations of the air, even when the latter have not precisely the same periodic time as the natural tone of the sympathising bodies. The limits of pitch capable of exciting sympathetic vibration are consequently a little wider in this case. By the comparatively greater influence of the motion of the air upon light elastic H bodies of this kind which offer but little resistance, their natural periodic time can be slightly altered, and adapted to that of the exciting tone. Massive elastic bodies, on the other hand, which are not readily movable, and are slow in com- municating their sonorous vibrations to the air, such as bells and ])lates, and con- tinue to sound for a long time, are also more difficult to move by the air. A much longer addition of effects is required for this purpose, and consequently it is also necessary to hit the pitch of their own tone with much greater nicety, in order to make them vibrate sympathetically. Still it is well known that bell-shaped glasses can be put into violent motion by singing their proper tone into them ; indeed it is i-elated that singers with very powerful and pure voices, have sometimes been able to crack them by the agitation thus caused. The principal difficulty in this experi- ment is in hitting the pitch with sufficient precision, and retaining the tone at that exact pitch for a sufficient length of time.

Tuning-forks are the most difficult bodies to set in sympathetic \ibration. To

40 INFLUENCE OF PAUTIALS ON SYMl'ATHETIC RESONANCE, part i.

effect this they may be fastened on sounding boxes which have been exactly tuned to

their tone, as shown in fig. 13. If we have two such forks of exactly the same

pitch, and excite one by a violin bow,

the other will begin to vibrate in sym- pathy, even if placed at the further

end of the same room, and it will con- tinue to sound, after the first has been

damped. The astonishing nature of

such a case of sympathetic vibration

will appear, if we merely compare the

heavy and powerful mass of steel set

in motion, with the light yielding mass

of air which produces the effect by such U small motive powers that they could

not stir the lightest spring which was

not in tune with the fork. With such

forks the time required to set them

in full swing by sympathetic action,

is also of sensible duration, and the

slightest disagreement in pitch is sufficient to produce a sensible diminution in

the sympathetic effect. By sticking a piece of wax to one prong of the second

fork, sufficient to make it vibrate once in a second less than the first — a difference

of pitch scarcely sensible to the finest car — the sympathetic vibration will be

wholly destroyed.

After having thus described the phenon\enon of sympathetic vibration in

general, we proceed to investigate the influence exerted in sympathetic resonance

l)y the different forms of wave of a musical tone. ^ First, it must be observed that most elastic bodies which have been set into

sustained vibration by a gentle force acting periodically, are (with a few exceptions

to be considered hereafter) always made to swing in pendular vibrations. But they are in general capable of executing several kinds of such vibration with different periodic times and with a different distribution over the various i)arts of the vibrating body. Hence to the different lengths of the periodic times correspond different simple tones producible on such an elastic body. These are its so-called proper tones. It is, however, only exceptionally, as in strings and the narrower kinds of organ pipes, that these proper tones correspond in pitch with the har-

III. INFLUENCE OF PARTIALS ON SYMPATHETIC RESONANCE. 41

uionic upper i)artial tones of a musical tone already mentioned. They are for tl>e most part inharmonic in relation to the prime tone.

In many cases the vibrations and their mode of distribution over the vibrating bodies can be rendered visible by strewing a little fine sand over the latter. Take, for ■example, a menibrane (as a bladder or piece of thin india-rubber) stretched over a circular ring. In fig. 14 are shown the various forms which a membrane can jissume when it vibrates. The diameters and circles on the surface of the mem- brane mark those points which remain at rest during the vibration, and are known <is nodal linen. By these the surface is divided into a number of compartments which bend altei-nately up and down, in such a way that while those marked ( + ) rise, those marked (-) fall. Over the figures a, b, c, are shown the forms of a .section of the membrane during vibration. Only those forms of motion are drawn which correspond with the deepest and most easily producible tones of the mem- l)rane. The number of circles and diameters can be increased at pleasure by 51 taking a sufficiently thin membrane, and stretching it with sufiicient regularity, and in this case the tones would continually sharpen in pitch. By strewing sand on the membrane the figures are easily rendered visible, for as soon as it begins to vibrate the particles of sand'collect on the nodal lines.

In the same way it is possible to render visible the nodal lines and forms of vibration of oval and square membranes, and of differently-shaped plane elastic" plates, bars, and so on. These form a series of very interesting phenomena dis- covered by Chladni, but to pursue them would lead us too far from our proper .subject. It will suflice to give a few details respecting the simplest case, that of a circular memlirane.

In the time required by the membrane to execute 100 vibrations of the form a, fig. 14 (p. 40c), the number of vibrations executed by the other forms is as follows : — H

Form of \'ibration

a without uodal lines .

b with one circle ....

c with two circles

d with one diameter

e with one diameter and one circle

f with two diameters .

Pitch Number

Cents *

Notes nearly

100

0

(.

229-6

1439

rf'-f-

859-9

2217

h'h +

159

805

«[?

292

1858

'j\r

214

1317

4+

The prime tone has been here arbitrarily assumed as c, in order to note the inter- vals of the higher tones. Those simple tones produced by the membrane which are slightly higher than those of the note written, are marked ( -1-); those lower, by (-). In this case there is no commensurable ratio between the prime t(me and the other tones, that is, none expressible in whole numbers.

Strew a very thin membrane of this kind with sand, and somid its prime tone strongly in its neighbourhood ; the sand will be driven by the vibrations towards IT the edge, where it collects. On producing another of the tones of the membrane, the sand collects in the corresponding nodal lines, and we are thus easily able to determine to which of its tones the membrane has responded. A singer wdio knows how to hit the tones of the membrane correctly, can thus easily make the

spoken of in the text (as in this table), they must be considered as additions by the transla- tor. In the present case, they give the inter- vals exactly, and not roughly as in the column of notes. Thus, 1439 cents is sharper than 14 Semitones above c, that is, sharper than d' by 39 hundredths of a Semitone, or about ^ of a Semitone, and 1858 is flatter than 19 Semitones above c, that is flatter than g' by 42 hun- dredths of a Semitone, or nearly i a Semitone. — Translator.']

* [Cents are hundredths of iiu equal Semi- tone, and are exceedingly valuable as measures of any, especially unusual, musical intervals. They are fully exf)lained, and the method of calculating them from the Interval Ratios is given in App. XX. sect. C. Here it need only be said that the number of hundreds of cents is the number of equal, that is, pianoforte Semitones in the interval, and these may be counted on the keys of any piano, while the units and tens show the number of hundredths of a Semitone in excess. Wherever cents are

42 INFLUENCE OF PART I ALS ON SYMPATHETIC RESONANCE, i'aht i.

sand {irranged itself at pleasure in one order or the other, by singing the correspond- ing tones powerfully at a distance. But in general the simpler figures of the deeper tones are more easily generated than the complicated figures of the upper tones. It is easiest of all to set the membrane in general motion by sounding its prime tone, and hence such memln-anes have been much nsed in aconstics to prove the existence of some determinate tone in some determinate spot of the siuTounding air. It is most suitable for this pui-j)ose to connect the membrane with an inclosed mass of air. A, fig. 15, is a glass bottle, Fig. is.

having an open month a, and in place of its bottom b, a stretched membrane, consisting of Avet pig's bladder, al- lowed to dry after it has been stretched and fastened. At c is attached a H single fibre of a silk cocoon, bearing a drop of sealing-wax, and hanging down like a pendulum against the membrane.

As soon as the membrane vibrates, the little pendulum is violently agitated. Such a pendulum is very convenient as long as we have no reason to apprehend any con- fusion of the prime tone of the membrane with any other of its proper tones. There is no scattering of sand, and the apparatus is therefore always in order. But to decide with certainty what tones are really agitating the membrane, we must after all place the bottle with its mouth downwards and strew sand on the membrane. However, when the bottle is of the right size, and the membrane uniformly stretched and fastened, it is only the prime tone of the membrane (slightly altered by that of the sympathetically vibrating mass of air in the bottle) which is easily excited. This prime tone can be made deeper by increasing the size of the mem- brane, or the volume of the bottle, or by diminishing the tension of the membrane " or size of the orifice of the bottle.

A stretched membrane of this kind, whether it is or is not attached to tie bot- tom of a bottle, will not only be set in vibration by nuisical tones of the same pitch as its own proper tone, but also by such musical tones as contain the proper tone of the membrane among its upper partial tones. Generally, given a number of interlacing waves, to discover whether the membrane will vibrate sympathetically, we must suppose the motion of the air at the given place to be mathematically analysed into a sum of pendular vibrations. If there is one such vibration among them, of which the periodic time is the same as that of any one of the proper tones of the membrane, the corresponding vibrational form of the membrane will be super- induced. But if there are none such, or none sutfieiently powerful, the membrane will remain at rest.

In this case, then, we also find that the analysis of the motion of the air into pendular vibrations, and the existence of certain vibrations of this kind, are deci- % sive for the sympathetic vibration of the membrane, and for this purpose no other similar analysis of the motion of the air can be substituted for its analysis into pendular vibrations. The pendular vibrations into wdiich the composite motion of the air can be analysed, here show themselves capable of producing mechanical eflfects in external nature, independently of the ear, and independently of mathe- matical theory. Hence the statement is confirmed, that the theoretical view which first led mathematicians to this method of analysing compound vibrations, is founded in the nature of the thing itself.

As an example take the following descri})tiou of a single experiment : — A bottle of the shape shown in fig. 15 above Avas covered with a thin vulcan- ised india-rubber membrane, of which the vibrating surface was -49 millimetres (1-93 inches)" in diameter, the bottle being UO millimetres (5-51 inches) high, and

* [As 10 inches are exactly 25J: millimetres the calculation of one set of measures from and 100 metres, that is, 100,000 millimetres are the other. Roughly we may assume 25 mm. 3937 inches, it is easy to form little tables for to be 1 inch. But wlieuever dimensions are

II

CHAP. HI. RESONATOllS. 43

having- an opening at the brass month of 13 milUmetres (-51 inches) in diameter. When blown it gave./'ji, and the sand heaped itself in a circle near the edge of the meml)rane. The same circle resnlted from my giving the some tone /'jjl on an harmonium, or its deeper Octave /|, or the deeper Twelfth B. Both Fk and D gave the same circle, but more weakly. Now the f^ of the membrane is the prime tone of the harmonium tone /"|, the second partial tone of f^, the third of B, the fourth of F^ and fifth of D* All these notes on being sounded set the membrane in the motion due to its deepest tone. A second smaller circle, 19 millimetres (•75 inches) in diameter was produced on the membrane by // and the same more faintly by A, and there was a trace of it for the deeper Twelfth e, that is, for simple tones of which vibrational numbers were h and i that of !>' .\

Stretched membranes of this kind are very convenient for these and similar experiments on the ])artials of compound tones. They have the great advantage

of being independent of the ear, but they H are not xevy sensitive for the fainter simple tones. Their sensitiveness is far inferior to that of the re^'o^uitors which I have intro- duced. These are hollow spheres of glass or metal, or tubes, with two openings as shown in figs. 16 a and 16 b. One opening (a) has sharp edges, the other (b) is funnel- shaped, and adapted for insertion into the ear. This smaller end I usually coat with melted sealing wax, and when the wax has cooled down enough i;ot to hurt the finger on being touchdl, but is still soft, I press the opening into the entrance of my ear. The sealing wax thus moulds itself to the shape of the inner surface of this opening, and when I subsequently use the resonator, it fits easily and is air-tight. H Such an instriunont is very like the resonance bottle already described, fig. 15

(p. 4.2a), for which the observer's Fig. ic. 1). VF ;>

own tympanic membrane has

been made to replace the for- mer artificial membrane.

The mass of air in a reso- nator, together with that in the aural passage, and with the tympanic membrane or drumskin itself, forms an elastic system which is capal)le of vibrating in a peculiar manner, and, in especial, the prime tone of the sphere, which is much deeper than any other of its proper tones, can be set into very powerful sympathetic vibration, and then the ear, which is in immediate connec- tion with the air inside the sphere, perceives this augmented tone by direct action. If we stop one ear (which is best done by a plug of sealing wax moulded into the ^ form of the entrance of the ear), J and apply a resonator to the other, most of the tones produced in the surrounding air will be considerably damped ; but if the proper tone of the resonator is sounded, it brays into the ear most powerfully.

given inthetext in mm. (that is, millimetves), + [For ordinary purposes this is quite

thev will be reduced to inches and decimals of enough, indeed it is generally unnecessary to

an inch..—rranslatur.] stop the other ear at all. But for such experi-

* [As the instrument was tempered, wo ments as Mr. Bosanquet had to make on beats

should have, approximatelv, for fjt the partials (see App. XX. section L. art. 4, b) he was

f% ft, &c. ; for B the partials 'K, h, f'%., &c. ; obliged to use a jar as the resonator, conduct

'for>Vthe partials i^i, ft, 4^/% &c^; and the sound from it through first a glass and

for Z* the partials IJ, ii, a, U',/^, &c. To then an elastic tube to a semicircular metal tube

prevent confusion I have reduced the upper which reached from ear to ear, to each end of

partials of the text to ordinary partials, as which a tube coated with india-rubber, could be

suggested in p. 23//, note.— Translator.] screwed into the ear. By this means, when

t [Here the partials of b are /), b', Sec, and proper care was taken, all sound but that

of c are c, c! , //, &c., so that both b and r. coming from the resonance jar was perfectly

contain b'.— Translator.'] cyicXwAeA.— Translator.']

44 RESONATORS. vmvv i.

Hence any one, even if he has no ear for music or is quite unpractised in detecting musical somids, is put in a condition to pick the required simple tone, even if com- paratively faint, from out of a great number of others. The proper tone of the resonator may even be sometimes heard cropping up in the whistling of the wind, the rattling of carnage Avheels, the splashing of water. For these purposes such resonators are incomparably more sensitive than tuned membranes. When the simple tone to be observed is faint in comparison with those Avhich accompany it, it is of advantage to alternately apply and withdraw the resonator. We thus easily feel whether the proper tone of the resonator begins to sound when the instrument is applied, whereas a unifoi-m continuous tone is not so readily perceived.

A properly tuned series of such resonators is therefore an important instrument for experiments in which individiial faint tones have to be distinctly heard, although accompanied by others which are strong, as in observations on the combinational

^ and Tipper partial tones, and a series of other phenomena to be hereafter described relating to chords. By their means such researches can be carried out even by ears quite untrained in musical observation, whereas it had been previously impossible to conduct them except by trained musical ears, and much strained attention properly assisted. These tones were consequently accessible to the observation of only a very few individuals ; and indeed a large number of physi- cists and even musicians had never succeeded in distinguishing them. And again even the trained ear is now able, with the assistance of resonators, to carry the analysis of a mass of miisical tones much further than before. Without their help, indeed, 1 shou.ld scarcely have succeeded in making the observations hereafter described, with so much precision and certainty, as I have been enabled to attain at present.*

It must be carefully noted that the ear does not hear the required tone with augmented force, unless that tone attains a considerable intensity within the mass

H of air enclosed in the resonator. Now the mathematical theory of the motion of the air shows that, so long as the amplitude of the vibrations is sufficiently small, the enclosed air will execute pendular oscillations of the same periodic time as those in the external air, and none other, and that only those pendular oscillations whose periodic time corresponds with that of the proper tone of the resonator, have any considerable strength ; the intensity of the rest diminishing as the difTer- ence of their pitch from that of the proper tone increases. All this is independent of the connection of the ear and resonator, except in so far as its tympanic mem- brane forms one of the inclosing walls of the mass of air. Theoretically this apparatus does not differ from the bottle with an elastic membrane, in fig. 15 (p. 42a), but its sensitiveness is amazingly increased by using the drumskin of the ear for the closing membrane of the bottle, and thus bringing it in direct connection with the auditory nerves themselves. Hence we cannot obtain a powerful tone in the resonator except when an analysis of the motion of the external air into

^ pendular vibrations, would show that one of them has the same periodic time as the proper tone of the resonator. Here again no other analysis but that into pendular vibrations woidd give a correct result.

It is easy, for an observer to convince himself of the above-named properties of resonators. Apply one to the ear, and let a piece of harmonised miisic, in which the proper tone of the resonator frequently occurs, be executed by any instruments. As often as this tone is struck, the ear to which the instrument is held, will hear it violently contrast with all the other tones of the chord.

This proper tone will also often be heard, but more weakly, when deeper musical tones occur, and on investigation we find that in such cases tones have been struck which include the proper tone of the resonator among their upper partial tones. Such deeper musical tones are called the harmonic nnder tones of the resonator. They are musical tones whose periodic time is exactly 2, 3, 4, 5, and so on, times as great as that of the resonator. Thus if the proper tone of

* See Appendix II. for the measures and different foi'ins of these Resonators.

CHAP. iir.

SYMPATHETIC RESONANCE OF STRINGS. 45

the resonator is c", it will be heard when a nmsical instruuieut sounds r', ./", c, A\y, F, D, C, and so on.* In this case the resonator is made to sound in sympathy with one of the harmonic xipper partial tones of the compound nmsical tone which is vibrating in the external air. It must, however, be noted that by no means all the harmonic upper partial tones occur in the compound tones of every instrument, and that they have very difterent degrees of intensity in different instruments. In tlie musical tones of violins, pianofortes, and harmoniums, the first five or six are generally very distinctly present. A more detailed account of the iipper partial tones of strings will be given in the next chapter. On the harmonium the un- evenly numbered partial tones (1, 3, 5, &c.) are generally stronger than the evenly numbered ones (2, 4, 6, &c.). In the same way, the upper partial tones are clearly heard by means of the resonators in the singing tones of the luunan voice, but differ in strength for the different vowels, as will be shown hereafter. 11

Among the bodies capable of strong sympathetic vibration must be reckoned stretched strings which are connected with a sounding board, as on the pianoforte.

The principal mark of distinction between strings and the other bodies which vibrate sympathetically, is that different vibrating forms of strings give simple tones corresponding to the htwmonic upper partial tones of the prime tone, whereas the secondary simple tones of membranes, bells, rods, &c., are wdiarmonic with the prime tone, and the masses of air in resonators have generally only very high upper partial tones, also chiefly «?iharmonic with the prime tone, and not capalde of being much reinforced by the resonator.

The vibrations of strings may be studied either on elastic chords loosely stretched, and not sonorous, but swinging so slowly that their motion may be followed with the hand and eye, or else on sonorous strings, as those of the piano- forte, guitar, monochord, or violin. Strings of the first kind are best made of thin H spirals of brass wire, six to ten feet in length. They should be gently stretched, and both ends should be fastened. A string of this construction is capable of making very large excursions with great regularity, which are easily seen by a large audience. The swings are excited by moving the string regularly backwards and forwards by the finger near to one of its extremities.

A string may be first made to vibrate as in fig. 17, a (p. 466), so that its appear- ance when displaced from its position of rest is always that of a simple half wave. The string in this case gives a single simple tone, the deepest it can produce, and no other harmonic secondary tones are audible.

But the string may also during its motion assume the forms fig. 17, b, c, d. In this case the form of the string is that of two, three, or four half waves of a simple wave-curve. In the vibrational form b the string produces only the upper Octave of its prime tone, in the form c the Twelfth, and in the form d the second Octave. The dotted lines show the position of the string at the end of half its 11 periodic time. In b the point ft remains at rest, in c two points yi and y. remain at rest, in d three points Sj, 8.,, &,. These points are called nodes. In a swinging spiral wire the nodes are readily seen, and for a resonant string they are shown by little paper riders, which are jerked off from the vibrating parts and remain sitting on the nodes. When, then, the string is divided by a node into two swinging sections, it produces a simple tone having a pitch number double that of the prime

* [The (.•" occurs as the 2nd, 3rd, 4th, the 7th being rather flat. The partials are 5th, Gth, 7th, 8th partials of these notes, in fact :—

c' c"

f f <-■"

c c' r c"

A\y fAJj c\) a'\) c"

F f c' f a' c"

I) d a d' ft a. c"

a c f c e' /' h'V, c".-- Translator.^

46

SYMPATHETIC RESOXANCE OF STRINGS.

tone. For three sections the pitch number is tripled, for foiu- sections iiu;i(h-ui)led, and so on.

To bring a spiral wire into these different forms of vibration, we move it periodically with the finger near one extremity, adopting the period of its slowest swings for a, twice that rate for b, three times for c, and four times for d. Or else we just gently touch one of the nodes nearest the extremity with the finger, and jjluck the string half-way between this node and the nearest end. Hence when yi in c, or 81 in d, is kept at rest by the finger, we pluck the string at c. Tlie other nodes then appear when the vibrati(jn commences.

For a sonorous string the vibrational forms of fig. 17 above arc most purely produced by applying to its sounding board the handle of a tuning-fork which has been struck and gives the simple tone corresponding to the form required. If only a determinate number of nodes are desired, and it is indifferent whether the indi- vidual points of the string do or do not execute simple vibrations, it is sufticient to touch the string very gently at one of the nodes and either pluck the string or rub it with a violin bow. By touching the string with the finger all those simple vibra- tions are damped which have no node at that point, and only those remain which allow the string to be at rest in that place.

The number of nodes in long thin strings may be considerable. They cease to be formed when the sections which lie between the nodes are too short and stiff to U be capable of sonorous vibration. Very fine strings consequently give a greater number of higher tones than thicker ones. On the violin and the lower pianoforte strings it is not very difficult to produce tones with 10 sections; but with extremelv fine wires tones with 16 or 20 sections can be made to sound. [Also compare p. 7Sd.]

The forms of vibration here spoken of are those in which each point of the string performs pendular oscillations. Hence these motions excite in the ear the sensation of only a single simple tone. In all other vibratiomxl forms of the strings, the oscillations are not simply pendular, but take place according to a differ- ent and more complicated law. This is always the case when the string is plucked in the usual way with the finger (as for guitar, harp, zither) or is struck with a hammer (as on the pianoforte), or is rubbed with a violin bow. The resulting motions may then be regarded as compounded of many simple vibrations, which, when taken separately, correspond to those in fig. 17. The multiplicity of such com- posite foi-ms of motion is infinitely great, the string may indeed be considered as capable of assuming any given form (provided we confine ourselves in all cases

CHAP. III. SYMPATHETIC KESOXANCK ()E .STRINGS. 4 7

to very small deviations from the position of rest), because, according- to wiiat was said in Chapter II., any given form of wave can he compounded ont of a number of simple waves such as those indicated in tig. 17, a, b, c, d. A plucked, struck, or bowed string therefore allows a great number of harmonic upper partial tones to l)e heard at the same time as the prime tone, and generally the number increases with the thinness of the string. The peculiar tinkling sound of very fine metallic .strings is clearly due to these very high secondary tones. It is easy to distinguish the upper simple tones up to the sixteenth by means of resonators. Beyond the sixteenth they are too close to each other to be distinctly separable by this mean.s.

Hence when a string is sympathetically excited by a musical tone in its neigh- ])ourhood, answering to the i)itch of the prime tone of the string, a whole series of difterent simple vibrational forms will generally be at the same time generated in the string. For when the prime of the musical tone corresponds to the pi'ime of the string all the harmonic upper partials of the first correspond to those of the H second, and are hence capable of exciting the corresponding vibrational forms in the string. Generally the string will be brought into as many forms of sympa- thetic vibration by the motion of the air, as the analysis of that motion shows that it possesses simple vibrational forms, having a periodic time equal to that of some vibrational form, that the string is capable of assuming. But as a general rule when there is one such simple vibrational form in the air, there are several such, and it will often be difficult to determine by which one, out of the many possible simple tones which would produce the effect, the string has been excited. Conse- (|uently the usual unweighted strings are not so convenient for the determination of the pitch of any simple tones which exist in a composite mass of air, as the membranes or the inclosed air of resonators.

To make experiments with the pianoforte on the sympathetic vibrations of strings, select a flat instrument, raise its lid so as to expose the strings, then press down the key of the string (for c' suppose) wdiich you wish to put into sympathetic vibration, but so slowly that the hammer does not strike, and place a little chip of ^ wood across this c string. You will find the chip put in motion, or even thrown otf, when certain other strings are struck. The motion of the chip is greatest when one of the under tones of c (p. iid) is struck, as c, F, C, A}), F, D^, or C ^. Some, but much less, motion also occurs when one of the upper partial tones of c is struck, as c", g" , or c'", but in this last case the chip wdll not move if it has been placed over one of the corresponding nodes of the string. Thus if it is laid across the middle of the string it will be still for c" and c", but will move for g" . Placed at one third the length of the string from its extremity, it will not stir for (/", but will move for c" or c" . Finally the string c will also be put in motion when an under tone of one of its upper partial tones is struck; for example, the note/, of which the third partial tone c" is identical with the second partial tone of c'. In this case also the chip remains at rest when put on to the middle of the string /, which is its node for c". In the same way the string c will move, with the formation of H two nodes, for g , g, or e\f, all which notes have (/" as an upper partial tone, which is also the third partial of c r'

Observe that on the pianoforte, when one end of the strings is commonly concealed, the position of the nodes is easily found by pressing the string gently on both sides and striking the key. If the finger is at a node the corresponding upper partial tone will be heard purely and distinctly, otherwise the tone of the string is dull and bad.

As long as only one upper partial tone of the string c' is excited, the corre- sponding nodes can be discovei-ed, and hence the particular form of its vibration determined. But this is no longer possible by the above mechanical method when

* [These experiments can of course not be struck and damped. And this sounding of c',

conducted on the usual upright cottage piano. although unstruck, is itself a very interesting

But the experimenter can at least hear the phenomenon. But of course, as it depends on

tone of f', if c, F, C, &c., are struck and the ear, it does not establish the results of the

immediately damped, or if c", ij" , c'" are text. — Translator.']

48 OBJECTIVE EXISTENCE OV PAIITLVLS. part i.

two upper partial tones are excited, such ;is r" and ;/", as would l)e the case if both these notes were struck at once on the ijianoforte, because the whole string of r' would then be in motion.

Although the relations for strings appear more complicated to the eye, their sympathetic vibration is subject to the same law as that which holds for resonators, membranes, and othei- elastic bodies. The sympathetic vibration is always deter- mined by the analysis of whatever sonorous motions exist, into simple pendidar vibrations. If the periodic time of one of these simple vibrations corresponds to the periodic time of one of the proper tones of the elastic body, that body, whether it be a string, a membrane, or a mass of air, will be put into strong sympathetic vibration.

These facts give a real objective value to the analysis of sonorous motion into simple pendular vibration, and no such vahie would attach to any other analysis. H Every individual single system of waves formed by pendular vibrations exists as an independent mechanical unit, expands, and sets in motion other elastic bodies having the corresponding proper tone, perfectly undisturbed by any other simple tones of other pitches which may be expanding at the same time, and Avhich may proceed either from the same or any other source of sound. Each single simple tone, then, can, as we have seen, be separated from the composite mass of tones, by mechanical means, namely by bodies which will vibrate sympathetically with it. Hence every individual partial tone exists in the compound musical tone produced by a single musical instrument, just as truly, and in the same sense, as the different colours of the rainbow exist in the white light proceeding from the sun or any other luminous body. Light is also oidy a vibrational motion of a peculiar elastic medium, the luminous ether, just as sound is a vibrational motion of the air. In a beam of Avhite light there is a species of motion which /nai/ be repre- sented as the sum of many oscillatory motions of various periodic times, each of H which corresponds to one particular colour of the solar spectrum. But of course each particle of ether at any particular moment has only one determinate velocity, and only one determinate departvire from its mean position, just like each particle of air in a space traversed by many systems of sonorous waves. The really exist- ing motion of any particle of ether is of course only one and indi^'idual ; and our theoretical treatment of it as compound, is in a certain sense arbitrary. But the imdulatory motion of light can also be analysed into the waves corresponding to the separate colours, by external mechanical means, such as by refraction in a prism, or by transmission through fine gratings, and each individual simple wave of light corresponding to a simple colour, exists mechanically by itself, indepen- dently of any other colour.

We must therefore not liold it to be an illusion of the ear, or to be mere imagination, when in the musical tone of a single note emanating from a musical instrument, we distinguish many partial tones, as I have found musicians inclined ^ to think, even when they have heard those partial tones quite distinctly with their own ears. If we admitted this, we should have also to look upon the colours of the spectrum which are separated from white light, as a mere illusion of the eye. The real outward existence of partial tones in nature can be established at any moment by a sympathetically vibrating membrane which casts up the sand strewn upon it.

Finally I would observe that, as respects the conditions of sympathetic vibra- ti(jn, I have been obliged to refer frequently to the mechanical theory of the motion of air. Since in the theory of soiuid we have to deal wuth well-known mechanical forces, as the pressure of the air, and Avith motions of material particles, and not with any hypothetical explanation, theoretical mechanics have an unassailable a\ithority in this department of science. Of course those readers who are unacquainted with mathematics, must accept the results on faith. An experimental way of examining the problems in question will be described in the next chapter, in which the laws of the analysis of musical tones by the ear have

CHAPS. III. IV. METHODS OF OBSERVING PARTIAL TONES. 49

to be established. The experimental proof there given for the ear, can also be carried out in precisely the same way for membranes and masses of air which vibrate sympathetically, and the identity of the laws in l)oth cases will result from those iiwestiffations.*

CHAPTER IV.

ON THE ANALYSIS OF MUSICAL TONES BY THE EAR.

It was frequently mentioned in the preceding chapter that musical tones could be resolved by the ear alone unassisted by any peculiar apparatus, into a series of partial tones corresponding to the simple pendular vibrations in a mass of air, that ^ is, into the same constituents as those into which the motion of the air is resolved b}^ the sympathetic vibration of elastic bodies. We proceed to show the correctness of this assertion.

Any one who endeavours for the first time to distinguish the upper partial tones of a musical tone, generally finds considerable difficulty in merely hearing them.

The analysis of our sensations when it cannot be attached to corresponding differences in external objects, meets with peculiar difficulties, the nature and significance of which will have to be considered hereafter. The attention of the observer has generally to be drawn to the phenomenon he has to observe, by peculiar aids properly selected, until he knows precisely what to look for ; after he has once succeeded, he will be able to throw aside such crutches. Similar diffi- culties meet us in the observation of the upper partials of a musical tone. I shall first give a description of such processes as will most easily put an untrained H observer into a position to recognise upper partial tones, and I will remark in passing that a musically trained ear will not necessarily hear tipper partial tones with greater ease and certainty than an untrained ear. Success depends rather upon a peculiar power of mental abstraction or a peculiar mastery over attention, than upon musical training. But a musically trained observer has an essential advantage over one not so trained in his power of figuring to himself how the simple tones sought for, ought to sound, whereas the untrained observer has con- tinually to hear these tones sounded by other means in order to keep their effect fresh in his mind.

First we must note, that the unevenly numbered partials, as the Fifths, Thirds, Sevenths, &c. of the prime tones, are usually easier to hear than the even ones, which are Octaves either of the prime tone or of some of the upper partials which lie near it, just as in a chord we more readily distinguish whether it contains Fifths and Thirds than whether it has Octaves. The second, fourth, and eighth H partials are higher Octaves of the prime, the sixth partial an Octave above the third partial, that is, the Twelfth of the prime ; and some practice is required for distinguishing these. Among the uneven partials which are more easily dis- tinguished, the first place must be assigned, from its usual loudness, to the third partial, the Twelfth of the prime, or the Fifth of its first higher Octave. Then follows the fifth partial as the major Third of the prime, and, generally verj faint, the seventh partial as the minor Seventh + of the second higher Octave of the prime, as will be seen by their following expression in musical notation, for the compound tone c.

* Optical means for rendering visible weak f [Or more correctly Awiv-miuor Seventh ;

sympathetic motions of sonorous masses of as the real minor Seventh, formed by taking

air, are described in App. II. These means two Fifths down and then two Octaves up, is

are valuable for demonstrating the facts to sharper by 27 cents, or in the ratio of 63 : 64.

hearers unaccustomed to the observing and — Translator. '\ distinguishing musical tones.

50 METHODS OF OBSERVING PARTIAL TONES.

1 in r^-

BtE^^EE:

:p=

-:&

I 2 '^ 3 4 5^ &^ _7^ «^

r (■' r/ r" ^" v" '//'[j c'"

[Cents. 0 1200 1902 2400 2786 3i02 3369 3600]*

In commencing to observe upper partial tones, it is advisable just before pro- ducing the musical tone itself which you wish to analyse, to sound the note you wish to distinguish in it, very gently, and if possible in the same quality of tone as the compound itself. The pianoforte and harmonivim are well adapted for these experiments, because they both have upper partial tones of considerable power.

H First genth' strike on a piano the note g', as marked above, and after letting the digital + rise so as to damp the string, strike the note c, of which g' is the third partial, with great force, and keep your attention directed to the pitch of the (/' which you had just heard, and you will hear it again in the compound tone of c. Similarly, first stroke the fifth partial e" gently, and then c strongly. These upper partial tones are often more distinct as the sound dies away, because they appear to lose force more slowly than the prime. The seventh and ninth partials h"\f and d" are mostly weak, or quite absent on modern pianos. If the same ex- periments are tried with an harmonium in one of its louder stops, the seventh partial will generally be well heard, and sometimes even the ninth.

To the objection which is sometimes made that the observer only imagines he hears the partial tone in the compound, because he had just heard it by itself, I need only remark at present that if e" is first heard as a partial tone of c on a good piano, tuned in equal temperament, and then /' is struck on the instrument

ff itself, it is quite easy to perceive that the latter is a little sharper. This follows from the method of tuning. But if there is a difference in pitch between the two tones, one is certainly not a continuation of the mental effect produced by the other. Other facts which completely refute the above conception, will be subse- quently adduced.

A still more suitable process than that just described for the piano, can be adopted on any stringed instrument, as the piano, monochord, or violin. It con- sists in first producing the tone we wish to hear, as an harmonic [p. 25(7, note] by touching the corresponding node of the string when it is struck or rubbed. The resembhmce of the tone first heard to the corresponding partial of the compound is then much greater, and the ear discovers it more readily. It is usual to place a divided scale bj^ the string of a monochord, to facilitate the discovery of the nodes. Those for the third partial, as shown in Chap. III. (p. idd), divide the string into three equal parts, those for the fifth into five, and so on. On the piano and violin

U the position of these points is easily found experimentally, by touching the string gently with the finger in the neighbourhood of the node, which has been approxi- matively detennined by the eye, then striking or bowing the sti'ing, and moving the finger about till the required harmonic comes out strongly and purely. By then sounding the string, at one time with the finger on the node, and at another without, we obtain the required upper partial at one time as an harmonic, and at another in the compound tone of the whole string, and thus learn to recognise the existence of the first as part of the second, with comparative ease. Using thin strings which have loud upper partials, I have thus been able to recognise the

* [The cents (see p. ild, note), reckoned piano or organ, are best called digitals or from the lowest note, are assigned on the finger-keys, on the analogy of pedals and foot- supposition that the harmonics are perfect, keys on the organ. The word key ha\-ing as on the Harmonical, not tempered as on another musical sense, namely, the scale in the pianoforte. See also diagram, p. 22t:. — which a piece of music is written, will without Translator.] prefix be confined to this meaning. — Trans-

t [The keys played by the fingers on a lalor.]

CHAP. IV. METHODS OF 0BSP:RVING PARTIAL TONP:s. 51

partials separately, up to the sixteenth. Those which He still higher are too near to each other in pitch for the ear to separate them readily.

In such experiments I recommend the following process. Touch the node of the string on the pianoforte or monochord with a camel's-hair pencil, strike the note, and immediately remove the pencil from the string. If the ])encil has been pressed tightly on the strisig, we either continue to hear the required partial as an harmonic, or else in addition hear the prime tone gently sounding with it. On repeating the excitement of the string, and continuing to press more and more lightly with the camel's-hair pencil, and at last removing the pencil entirely, the prime tone of the string will be heard more and more distinctly with the harmonic till we have finally the full natural miisical tone of the string. By this means we obtain a series of gradual transitional stages between the isolated partial and the compound tone, in which the first is readily retained by the ear. By applying this last process I have generally succeeded in making perfectly untrained ears H recognise the existence of upper partial tones.

It is at first more difticult to hear the upper partials on most wind instruments and in the human voice, than on stringed instruments, harmoniums, and the more penetrating stops of an organ, because it is then not so easy first to produce the upper partial softly in the same quality of tone. But still a little practice sufliices to lead the ear to the required partial tone, by previously touching it on the piano. The partial tones of the human voice are comparativel}' most difficult to distinguish for reasons which will be given svibsequently. Nevertheless they were distin- guished even by Rameau* without the assistance of any apparatus. The process is as follows : —

Get a powerful bass voice to sing e]^ to the vowel 0, in sore [more like m',> in sail' than o in so], gently touch b'\) on the piano, which is the Twelfth, or third partial tone of the note e\f, and let its sound die away while you ai-e listening to it attentively. The note l>'\f on the piano will appear really not to die away, H but to keep on sounding, even when the string is damped by removing the finger from the digital, because the ear unconsciously passes from the tone of the piano to the partial tone of the same pitch produced by the singer, and takes the latter for a continuation of the former. But when the finger is removed from the key, and the damper has fallen, it is of course impossible that the tone of the string should have continued sounding. To make the experiment for g" the fifth partial, or major Third of the second Octave above e\), the voice should sing to the vowel A in father.

The resonators described in the last chapter furnish an excellent means for this purpose, and can be used for the tones of any musical instrument. On apply- ing to the ear the resonator corresponding to any given upper partial of the com- pound c, such as g', this g' is rendered much more powerful when c is sounded. Now hearing and distinguishing g' in this case by no means proves that the ear alone and without this apparatus would hear g' as part of the compound c. But U the increase of the loudness of g' caused by the resonator may be used to direct the attention of the ear to the tone it is required to distinguish. On gradually removing the resonator from the ear, the force of g' will decrease. But the attention once directed to it by this means, remains more readily fixed upon it, and the observer continues to hear this tone in the natural and unchanged compound tone of the given note, even with his unassisted ear. The sole office of the resonators in this case is to direct the attention of the ear to the required tone.

By frequently instituting similar experiments for perceiving the upjier partial tones, the observer comes to discover them more and more easily, till he is finally able to dispense with any aids. But a certain amount of undisturbed concentration is always necessary for analysing musical tones by the ear alone, and hence the use of resonators is quite indispensable for an accurate comparison of different * Nmivemt SysUme de Musique thioriquc. Paris : 1726. Preface.

52 PROOF OF OHM'S LAW. part i.

qualities of tone, especially in respect to the weaker upper partials. At least, I must confess, that my own attempts to discover the upper partial tones in the human voice, and to determine their differences for different vowels, were most unsatisfactory until I applied the resonators.

We now proceed to prove that the human ear really does analyse musical tones according to the law of simple vibrations. Since it is not possible to insti- tute an exact comparison of the strength of our sensations for different simple tones, we must confine ourselves to proving that when an analysis of a composite tone into simple vibrations, effected by theoretic calculation or by sympathetic resonance, shows that certain upper partial tones are absent, the ear also does not perceive them.

The tones of strings are again best adapted for conducting this proof, because they admit of many alterations in their quality of tone, according to the manner H and the spot in which they are excited, and also because the theoretic or experi- mental analysis is most easily and completely performed for this case. Thomas Young* first showed that when a string is plucked or struck, or, as we may add, bowed at any point in its length which is the node of any of its so-called harmonics, those simple vibrational forms of the string which have a node in that point are not contained in the compound vibrational form. Hence, if we attack the string at its middle point, all the simple vibrations due to the evenly numbered partials, each of which has a note at that point, will be absent. This gives the sound of the string a peculiarly hollow or nasal twang. If Ave excite the string at 1 of its length, the vibrations corresponding to the third, sixth, and ninth partials will be absent ; if at }, then those corresponding to the fourth, eighth, and twelfth partials will fail ; and so on.f

This result of mathematical theory is confirmed, in the first place, by analys- ing the compound tone of the string by sympathetic resonance, either by the f resonators or by other strings. The experiments may be easily made on the pianoforte. Press down the digitals for the notes c and c, without allowing the hammer to strike, so as merely to free them from their dampers, and then pluck the string c with the nail till it sounds. On damping the c string the higher c will echo the sound, except in the particular case when the c string has been plucked exactly at its middle point, which is the point where it would have to be touched in order to give its first harmonic when struck by the hammer.

If yve touch the c string at i or | its length, and strike it with the luunmer, we obtain the harmonic g' ; and if the damper of the g is raised, this string echoes the sound. But if we pluck the c string with the nail, at either ^ or | its length, g' is not echoed, as it will be if the c string is plucked at any other spot.

In the same way observations with the resonators show that when the r string

is plucked at its middle the Octave c is missing, and when at i or | its length the

Twelfth g is absent. The analysis of the sound of a string by the sympathetic

H resonance of strings or resonators, consequently fully confirms Thomas Young's

law.

But for the vibration of strings we have a more direct means of analysis than that furnished by sympathetic resonance. If we, namely, touch a vibrating string gently for a moment with the finger or a camel's-hair pencil, we damp all those simple vibrations which have no node at the point touched. Those vibrations, however, which have a node there are not damped, and hence will continue to sound without the others. Consequently, if a string has been made to speak in any way whatever, and we wish to know whether there exists among its simple vibrations one corresponding to the Twelfth of the prime tone, we need only touch one of the nodes of this vibrational form at ^ or | the length of the string, in order to reduce to silence all simple tones which have no such node, and leave the Twelfth sounding, if it were there. If neither it, nor any of the sixth, ninth,

* London. Philosophical Transactions, 1800, vol. i. p. 137. t See Appendix III.

CHAP. IV. PROOF OF OHM'S LAAV. 53

twelfth, i\:c., of the partial tones were present, giving corresponding harmonics, the string will be reduced to absolute silence by this contact of the finger.

Press down one of the digitals of a piano, in order to free a string from its damper. Pluck the string at its middle point, and immediately touch it there. The string will be completely silenced, showing that plucking it in its middle excited none of the CA^enly numbered partials of its compound tone. Pluck it at }^ or 'f. its length, and immediately touch it in the same place ; the string will be silent, ])roving the absence of the third partial tone. Pluck the string anywhere else than in the points named, and the second partial will be heard when the middle is touched, the third when the string is touched at i or ^ of its length.

The agreement of this kind of proof with the results from sympathetic reso- nance, is well adapted for the experimental establishment of the proposition based in the last chapter solely upon the results of mathematical theory, namely, that sympathetic vibration occurs or not, according as the corresponding simple^ vibrations are or are not contained in the compound motion. In the last described method of analysing the tone of a string, we are quite independent of the theory of sympathetic vibration, and the simple vibrations of strings are exactl}" charac- terised and recognisable by their nodes. If the compound tones admitted of being analvsed hj sympathetic resonance according to any other vibrational forms except those of simple vibration, this agreement could not exist.

If, after having thus experimentally proved the correctness of Thomas Young's law, we try to analyse the tones of strings by the unassisted ear, we shall continue to find complete agreement.* If we pluck or strike a string in one of its nodes, all those upper partial tones of the compound tone of the string to which the node belongs, disappear for the ear also, bat they are heard if the string is plucked at any other place. Thus, if the string r be plucked at ^ its length, the partial tone f/' cannot be heard, but if the string be plucked at only a little distance from this point the partial tone (/' is distinctly audible. Hence the ear analyses the sovmd ^ of a string into precisely the same constituents as are found by sympathetic reso- nance, that is, into simple tones, according to Ohm's definition of this conception. These experiments are also well adapted to show that it is no mere play of imagina- tion when we hear upper partial tones, as some people believe on hearing them for the first time, for those tones are not heard when they do not exist.

The following modification of this process is also very well adapted to make the upper partial tones of strings audible. First, strike alternately in rhythmical sequence, the third and fourth partial tone of the string alone, by damping it in the corresponding nodes, and request the listener to observe the simple melody thus produced. Then strike the luidamped string alternately and in the same rhythmical sequence, in these nodes, and thus reproduce the same melody in the upper partials, which the listener will then easily recognise. Of course, in order to hear the third partial, we must strike the string in the node of the fourth, and conversely.

The compound tone of a plucked string is also a remarkably striking example H of the power of the ear to analyse into a long series of partial tones, a motion which the eye and the imagination are able to conceive in a much simpler manner. A string, which is pulled aside by a sharp point, or the finger nail, assumes the form, fig. 18, A (p. 54a), before it is released. It then passes through the series of forms, fig. 18, B, C, D, E, F, till it reaches G, which is the inversion of A, and then returns, through the same, to A again. Hence it alternates between the forms A and G. All these forms, as is clear, are composed of three straight lines, and on expressing the velocity of the individual points of the strings by vibrational curves, these would have the same form. Now the string scarcely imparts any })erceptible portion of its own motion directly to the air. Scarcely any audible tone results when both ends of a string are fastened to immovable supports, as metal bridges, which are again fastened to the walls of a room. The sound of

* See Brandt in PoggendorfE's Annalcn der Fhijsik, vol. cxii. p. 324, where this fact is proved.

,2_

V

<^i'

o4

PROOF OF OHM'S LAW.

tlie string reaches the air through that one of its extremities which rests upon a bridge standing on an elastic sounding board. Hence the sound of the .string essentially depends on the motion of this extremity, through the pressure which it exerts on the sounding board. The magni- tude of this pressure, as it alters periodically with the time, is shown in fig. 19, where the height of the line h h corresponds to the amount of pressure exerted on the bridge by that extremity of the string when the string is at rest. Along h h suppose lengths to be set off corresponding to con- secutive intervals of time, the vertical

^ heights of the broken line above or below h h represent the corresponding augmenta- tions or diminutions of pressure at those times. The pressure of the string on the sounding board consequently alternates, as the figure shows, between a higher and a lower value. For some time the greater pressure remains unaltered ; then the lower suddenly ensues, and likewise remains for a time unaltered. The letters a to g in fig. 19 correspond to the times at which the string assumes the forms A to G in fig. 18. It is this alteration between a greater and a smaller pressure which produces the sound in the air. We cannot but feel astonished that a motion produced by means so simple and so easy to comprehend,

^ should be analysed by the ear into such a complicated sum of simple tones. For the eye and the understanding the action of the string on the sounding board can be figured with extreme simplicity. What has the simple broken line of fig. 19 to do with wave-curves, which, in the course of one of their periods, show

h-|--l-

de fgfpde ha

1)

3, 4, 5, up to 16, and more, crests and troughs? This is one of the most striking examples of the different ways in which eye and ear comprehend a periodic motion.

There is no sonorous body whose motions imder varied conditions can be so ^completely calculated theoretically and contrasted with observation as a string. The following are examples in which theory can be compared with analysis by ear : —

I have discovered a means of exciting simple pendular vibrations in the air. A tuning-fork when struck gives no harmonic upper partial tones, or, at most, traces of them when it is brought into such excessively strong vibration that it no longer exactly follows the law of the pendulum.* On the other hand, tuning forks have some very high inharmonic secondary tones, which produce that peculiar shaq)

* [On all ordinary tuning-forks between a pitch numbers. But the prime can always be

and d" in pitch, I have been able to hear the second partial or Octave of the prime. In some low forks this Octave is so powerful that on pressing the liaudle of the fork against the table, the prime quite disappears and the Octave only is heard, and this has often proved a source of embarrassment in tuning the forks, or in counting beats to determine

heard when the fork is held to the ear or over a properly tuned resonance jar, as described in this paragraph. I tune such jars by pouring water in or out until the resonance is strongest, and then I register the height of the water and pitch of the fork for future use on a slip of paper gummed to the side of the jar. I have found that it is not at all necessary to

CHAP. IV. PROOF OF OHM'S LAW. 55

tinkling of the fork at the moment of being struck, and generally become rapidly inaudible. If the tuning fork is held in the fingers, it imparts very little of its tone to the air, and cannot be heard unless it is held close to the ear. Instead of holding it in the fingers, we may screw it into a thick board, on the inider side of which some pieces of india-rubber tubing have been fastened. When this is laid upon a table, the india-rubber tubes on which it is supported convey no sound to the table, and the tone of the tuning-fork is so weak that it may be considered in- audible. Now if the prongs of the fork be brought near a resonance chamber* of a bottle-form of such a size and shape that, when we blow over its mouth, the air it contains gives a tone of the same pitch as the fork's, the air within this chamber vibrates sympathetically, and the tone of the fork is thus conducted with great strength to the outer air. Now the higher secondary tones of such resonance chambers are also inharmonic to the prime tone, and in general the secondary tones of the chambers correspond neither with the harmonic nor the inharmonic H secondary tones of the forks ; this can be determined in each particular case by producing the secondary tones of the bottle by stronger blowing, and discovering those of the forks with the help of strings set into sympathetic vibration, as will be presently described. If, then, only one of the tones of the fork, namely, the prime tone, corresponds Avith one of the tones of the chamber, this alone will be reinforced by sympathetic vibration, and this alone will be communicated to the external air, and thus conducted to the observer's ear. The examination of the motion of the air by resonators shows that in this case, provided the tuning-fork be not set into too violent motion, no tone but the prime is present, and in such case the unassisted ear hears only a single simple tone, namely, the common prime of the tuning-fork and of the chamber, without any accompanying upper partial tones.

The tone of a tuning-fork can also be purified from secondary tones by placing its handle upon a string and moving it so near to the bridge that one of the proper tones of the section of string lying between the fork and the bridge is the same as ^ that of the tuning-fork. The string then begins to vibrate strongly, and conducts the tone of the tuning-fork with great power to the sounding board and surround- ing air, whereas the tone is scarcely, if at all, heard as long as the above-named section is not in unison with the tone of the fork. In this way it is easy to find the lengths of string which correspond to the prime and upper partial tones of the fork, and accurately determine the pitch of the latter. If this experiment is con- ducted with ordinary strings which are uniform throughout their length, we shield the ear from the inharmonic secondary tones of the fork, but not from the harmonic upper partials, which are sometimes faintly present when the fork is made to vibrate strongly. Hence to conduct this experiment in such a way as to create purely pendular vibrations of the air, it is best to weight one point of the string, if only so much as by letting a drop of melting sealing-wax fall upon it. This causes the upper proper tones of the string itself to be inharmonic to the prime tone, and hence there is a distinct interval between the points where the fork must be placed U to bring out the prime tone and its audible Octave, if it exists. '

In most other cases the mathematical analysis of the motions of soimd is not nearly far enough advanced to determine with certainty what upper partials will be present and what intensity they will possess. In circular plates and stretched membranes which are struck, it is theoretically possible to do so, but their inhar-

put the fork into excessively strong vibration of Chap. VII., and Prof. Preyer's in App. XX.

in order to make the Octave sensible. Thus, sect. L. art. 4, <% The conditions according

taking a fork of 232 and another of 468 vibra- to Koenig that tuning-forks should have no

tions, after striking them both, and letting the upper partials are given in App. XX. sect. L.

deeper fork spend most of its energy until I art. 2, «. — Translator.]

could not see the vibrations with the eye at all, * Either a bottle of a proper size, which the beats were heard distinctly, when I pressed can readily be more accurately tuned by pour- both on to a table, and continued to be heard ing oil or water into it, or a tube of pasteboard even after the forks themselves were separately quite closed at one end, and having a small inaudible. See also Prof. Helmholtz's experi- round opening at the other. See the proper ments on a fork of 64 vibrations at the close sizes of such resonance chambers in App. IV.

56

PROOF OF OHM'S LAW.

monic secondary tones are so numerous and so nearly of the same pitch that most observers would probably fail to separate them satisfactorily. On elastic rods, how- ever, the secondary tones are very distant from each other, and are inharmonic, so that they can be readily distinguished from each other by the ear. The following are the proper tones of a rod which is free at both ends ; the A'ibrational number of the prime tone taken to be c, is reckoned as 1 : —

Pitch Number

Cents *

Notation

Prime tone

Second proper tone .....

Third proper tone

Fourth proper tone

1-0000 2-7570 5-4041 13-3444

0

1200 + 556 2400 + 521 3600 + 886

/ +0-2 f" +0-1 V" -0-1

The notation is adapted to the equal temperament, and the appended fractious H are parts of the interval of a complete tone.

Where we are unable to execute the theoretical analysis of the motion, we can, at any rate, by means of resonators and other sympathetically vibrating bodies, analyse any individual musical tone that is produced, and then compare this analysis, which is determined by the laws of sympathetic vibration, with that effected by the unassisted ear. The latter is naturally much less sensitive than one armed with a resonator ; so that it is frequently impossible for the unarmed ear to recognise amongst a number of other stronger simple tones those which the resonator itself can only faintly indicate. On the other hand, so far as my ex- perience goes, there is complete agreement to this extent : the ear recognises with- out resonators the simple tones which the resonators greatly reinforce, and perceives no upper partial tone which the resonator does not indicate. To verify this con- clusion, I performed numerous experiments, both with the human voice and the harmonium, and they all confirmed it.+ ^ By the above experiments the proposition enunciated and defended by G. S. Ohm must be regarded as proved, viz. that the human ear perceives 2)endidar vibra- tions alone as simple tones, and resolves all other periodic motions of the air into a series of pendular viM'ations, hearing the series of simple tones which correspond, ivith these simple vibrations.

Calling, then, as already defined (in pp. 2.3, 24 and note), the sensation excited in the ear by any periodical motion of the air a musical tone, and the sensation excited by a simple pendular vibration a simp)le tone, the rule asserts that the serisation of a mtisical tone is compounded out of the sensations of several simple tones. In particular, we shall henceforth call the sound produced by a single sonorous body its (simple or compound) tone, and the sound produced by several musical instruments acting at the same time a composite tone, consisting generally of several (simple or compound) tones. If, then, a single note is sounded on a

reed tones, by the beats (Chap. VIII.) that their upper partials made with the primes of a set of Scheibler's tuning-forks. The correctness of the process was proved by the fact that the results obtained from different partials of the same reed tone, which were made to beat with different forks, gave the same pitch numbers for the primes, within one or two hundredths of a vibration in a second. I not only employed such low partials as 3, 4, 5 for one tone, and 4, 5, 6 for others, but I determine the pitch number 31-47, by partials 7, 8, 9, 10, 11, 12, 13, and the pitch number 15-94 by partials 25 and 27. The objective reality of these ex- tremely high upper partials, and their inde- pendence of resonators or resonance jars, was therefore conclusively shown. On the Har- monical the beats of the 16th partial of C 66, with c'", when slightly flattened bypressing the note lightly down, are very clear.- -Translator.]

H * [For cents see note p. i\d. As a Tone is 200 ct., 0-1 Tone = 20ct., these would give for the Author's notation /' + 40 ct., /" + 20 ct., «'" - 10 ct., whereas the column of cents shows that they are more accurately /' + 56 ct. , /" + 21 ct., a'" - 14 ct. For convenience, the cents for Octaves are separated, thus 1200 + 556 in place of 1756, but this separation is quite unnecessary. The cents again show the inter- vals of the inharmonic partial tones without any assumption as to the value of the prime. By a misprint in all the German editions, followed in the first English edition, the second proper tone was made /"' - 0-2 in place of /' + 0-2.— Translator.]

t [In my ' Notes of Observations on Musi- cal Beats,' Proceedings of the Royal Socirfi/, May, 1880, vol. xxx. p. 531, largely cited in App. XX. sect. B. No. 7, I showed that I was able to determine the pitch numbers of deep

CHAP. IV. DIFFICULTIES IX OBSERVING PARTIALS. 57

musical instruinent, as a violin, tninipot, organ, or by a singing voice, it must bo called in exact language a tone of the instrument in ([uestion. This is also the ordinary language, but it did not then imply that the tone migiit be mnipound. When the tone is, as usual, a compound tone, it will be distinguished by this term, or the abridgment, a compound ; while tone is a general term which includes both simple and compound tones.* The prime tone is generally louder than any of the upper partial tones, and hence it alone generally determines the ^>«'fcy^ of the com- pound. The tone produced by any sonorous body reduces to a migle simple tone in very few cases indeed, as the tone of tuning-forks imparted to the air by reso- nance chambers in the manner already described. The tones of wide-stopped organ pipes when gently blown are almost free from upper partials, and are accom- panied only by a rush of wind.

It is well known that this union of several simple tones into one compound tone, which is naturally effected in the tones produced by most musical instruments, ^ is artiricially imitated on the organ by peculiar mechanical contrivances. The tones of organ pipes are comparatively poor in upper partials. When it is desirable to use a stop of incisive penetrating quality of tone and great power, the wide pipes {principal re;/ister and weitgedacJd f) are not sufficient ; their tone is too soft, too defective in upper partials ; and the narrow-pipes {geigen-register and qidntaten %) are also unsuitable, because, although more incisive, their tone is weak. For such occasions, then, as in accompanying congregational singing, recourse is had to the compound stops.^ In these stops every key is connected with a larger or smaller series of pipes, which it opens simultaneously, and which give the prime tone and a certain number of the lower upper partials of the compound tone of the note in question. It is very usual to connect the upper Octave with the prime tone, and after that the Twelfth. The more complex compounds (cor/t^^i; i) give the first six partial tones, that is, in addition to the two Octaves of the prime tone and its Twelfth, the higher major Third, and the Octave of the Twelfth. This is as much H of the series of upper partials as belongs to the tones of a major chord. But to prevent these compound stops from being insupportably noisy, it is necessary to reinforce the deeper tones of each note by other rows of pipes, for in all natural tones which are suited for musical purposes the higher partials decrease in force as they rise in pitch. This has to be regarded in their imitation by compound stops. These compound stops were a monster in the path of the old musical theory, which was acquainted only with the prime tones of compounds ; but the practice of organ-builders and organists necessitated their retention, and when they are suitably arranged and properly applied, they form a very effective musical apparatus.

* [Here, again, as on pp. 23, 24, I have, in toned diapason, eight feet.' Hopkins, Organ,

the translation, been necessarily obliged to p. 445. ' A manual stop of eight feet, produ-

deviate slightly from the original. K/.aiuj, as cing a pungent tone very lilce that of the

here defined, embraces Ton as a particular Gamba, except that the pipes, being of larger

case. I use tunc for the general term, and scale, speak quicker and produce a fuller tone.

wmpoiuul tone and simple tone for the two Examples of the stop exist at Doncaster, If

particular cases. Thus, as ijresently mentioned the Temple Church, and in the Exchange

in the text, the tone produced by a tuning-fork Organ at Northampton.' Ibid. p. 138. For

held over a proper resonance chamber we know, quintaten, see supra, p. S3d, note. — Translator.^ on analysis, to be simple, but before analysis it § [As described in Hopkins, Organ, p. 142,

is to us only a (musical) tone like any other, these are the scsquialtera ' of five, four, three,

and hence in this case the Author's Klamj or two ranks of open metal pipes, tuned

becomes the Author's Ton. I believe that the in Thirds, Fifths, and Octaves to the Diapa-

language used in my translation is best adapted son'. The mixture, consisting of five to two

for the constant accurate distinction between ranks of open metal pipes smaller than the

compound and simple tones by English readers, last, is in England the second, in Germany the

as I leave nothing which runs counter to old first, compound stop (p. 143). The Furniture of

habits, and by the use of the words simple and five to two sets of small open pipes, is variable,

compound, constantly recall attention to this (1) The Cornet, vwnnted, has five ranks of very

newly discovered and extremely important rela- large and loudly voiced pipes, (2) the echo is

tion. — Transltdor.] similar, but light and delicate, and is enclosed

t [Pr//*rv>(/— double open diapason. Gross- in a box. In German organs the cornet is also

gedackt—dou.h\e stopped diapason. Hopkins, a pedal reed stop of four and two feet ((6/d). —

Organ, p. 444-5. — Translator.] Translator.]

t [' Oeigcn Principal — violin or crisp-

58 DIFFICULTIES IN OBSERVING PARTIALS. part i.

The nature of the case at the same time fully justifies their use. The musician is bound to regard the tones of all musical instruments as compomided in the same way as the compound stops of organs, and the important part this method of com- position plays in the construction of musical scales and chords will be made evident in subsequent chapters.

We have thus been led to an appreciation of upper partial tones, which diflers considerably from that previously entertained by musicians, and even physicists, and must therefore be prei)ared to meet the opposition which will be raised. The upper partial tones were indeed known, but almost only in such compound tones as those of strings, where there was a favourable opportunity for observing them ; but they appear in previous physical and musical works as an isolated accidental phenomenon of small intensity, a kind of curiosity, which was certainly occasion- ally adduced, in order to give some support to the opinion that nature had pre- H figured the construction of our major chord, but which on the whole remained almost entirely disregarded. In opposition to this we have to assert, and we shall prove the assertion in the next chapter, that upper partial tones are, with a few exceptions already named, a general constituent of all musical tones, and that a certain stock of upper partials is an essential condition for a good musical quality of tone. Finally, these upper partials have been erroneously considered as weak, because they are difficult to observe, while, in point of fact, for some of the best musical qualities of tone, the loudness of the first upper partials is not far inferior to that of the prime tone itself.

There is no difficulty in verifying this last fact by experiments on the tones of strings. Strike the string of a piano or monochord, and immediately touch one of its nodes for an instant with the finger ; the constituent partial tones having this node will remain with unaltered loudness, and the rest will disappear. We might also touch the node in the same way at the instant of striking, and thus obtain the f corresponding constituent partial tones from;, the first, in place of the complete compound tone of the note. In both ways we can readily convince ourselves that the first upper partials, as the Octave and Twelfth, are by no means weak and difficult to hear, but have a very appreciable strength. In some cases we are able to assign numerical values for the intensity of the upper partial tones, as will be shown in the next chapter. For tones not produced on strings this a posteriori proof is not so easy to conduct, because we are not able to make the upper partials speak separately. But even then by means of the resonator we can apjjreciate the in- tensity of these upper partials by producing the corresponding note on the same or some other instrument until its loudness, when heard through the resonator, agrees with that of the former.

The difficulty we experience in hearing upper partial tones is no reason for considering them to be weak ; for this difficulty does not depend on their intensity, but upon entirely diffi^rent circumstances, which could not be properly estimated 51 until the advances recently made in the physiology of the senses. On this diffi- culty of observing the upper partial tones have been founded the objections which A. Seebeck* has advanced against Ohm's law of the decomposition of a musical tone ; and perhaps many of my readers who are unacquainted with the physiology of the other senses, particularly with that of the eye, might be inclined to adopt Seebeck's opinions. I am therefore obliged to enter into some details concerning this difference of opinion, and the peculiarities of the perceptions of our senses, on which the solution of the difficulty depends.

Seebeck, although extremely accomplished in acoustical experiments and observations, was not always able to recognise upper partial tones, where Ohm's law required them to exist. But we are also bound to add that he did not apply the methods already indicated for directing the attention of his ear to the upper partials in question. In other cases when he did hear the theoretical upper * In PoggendorfE's Annaloi der Physik, vol. Ix. p. 449, vol. Ixiii. pp. 353 and 368.— Ohm, ibid. vol. lix. p. 513, and vol. Ixii. p. 1.

CHAP. IV. DIFFICULTIES IN OBSERVINO PARTIALS. 59

pavtials, they were weaker than the theory required. He conchided that the defi- nition of a simple tone as given by Ohm was too limited, and that not only pcn- dular vibrations, but other vibrational forms, provided they were not too widely separated from the pendular, were capable of exciting in the ear the sensation of a single simple tone, which, however, had a variable quality. He consequently asserted that when a musical tone was compounded of several simple tones, part of the intensity of the u])per constituent tones went to increase the intensity of the prime tone, with which it fused, and that at most a small remainder excited in the ear the sensation of an upper partial tone. He did not formulate any deter- minate law, assigning the vibrational forms which would give the impression of a simple and those which would give the impression of a compound tone. The experiments of Seebeck, on which he founded his assertions, need not be here described in detail. Their object was only to produce musical tones for which either the intensity of the simple vibrations corresponding to the upper partialsll could be theoretically calculated, or in which these upper partials could be rendered separately audible. For the latter purpose the siren was used. We have just described how the same object can be attained by means of strings. Seebeck shows in each case that the simple vibrations corresponding to the upi)er partials have considerable strength, but that the upper partials are either not heard at all, or heard with difficulty in the compound tone itself. This fact has been already mentioned in the present chapter. It may be perfectly true for an observer who has not applied the proper means for observing upper partials, while another, or even the first observer himself when properly assisted, can hear them perfectly well.* Now there are many circumstances which assist us first in separating the musical tones arising from different sources, and secondly, in keeping together the partial tones of each se[)arate source. Thus when one musical tone is heard for some time before being joined by the second, and then the second continues after the first has ceased, the separation in sound is facilitated by the succession of time. H We have already heard the first musical tone by itself, and hence know inune- diately what we have to deduct from the compound effect for the effect of this first tone. Even when several parts proceed in the same rhythm in polyphonic music, the mode in which the tones of different instruments and voices commence, the nature of their increase in force, the certainty with which they are held, and the manner in which they die off, are generally slightly different for each. Thus the tones of a pianoforte commence suddenly with a blow, and are consequently strongest at the first moment, and then rapidly decrease in power. The tones of brass instruments, on the other hand, commence sluggishly, and require a small but sensible time to develop their full strength. The tones of bowed instruments are distinguished by their extreme mobility, but when either the player o the instrument is not unusually perfect they are interrupted by little, vei-y short, pauses, producing in the ear the sensation of scraping, as will be described more in detail when we come to analyse the musical tone of a violin. When, then, such ^ instruments are sounded together there are generally points of time when one or the other is predominant, and it is consequently easily distinguished by the ear. Bat besides all this, in good part music, especial care is taken to facilitate the separation of the parts b}' the ear. In polyphonic music proper, where each part has its own distinct melody, a principal means of clearly separating the progres- sion of each part has always consisted in making them proceed in different rhythms and on different divisions of the bars ; or where this could not be done, or was at any rate only partly possible, as in four-part chorales, it is an old rule, contrived for this purpose, to let three parts, if possible, move by single degrees of the scale, and let the fourth leap over several. The small amount of alteration in the pitch makes it easier for the listener to keep the identity of the several voices distinctly in mind.

* [Here ivom ' Upper partial tones,' p. 94, to ' former analysis,' p. 100 of the 1st English edition are omitted, in accordance with the 4th German edition. — 'I'ranslafor.]

60

FUSION OF PARTIALS INTO A COMPOUND.

All these helps fail in the resolution of musical tones into their constituent partials. When a compound tone commences to sound, all its partial tones commence with the same comparative strength ; when it swells, all of them generally swell uniformly; when it ceases, all cease simultaneously. Hence no opportunity is generally given for hearing them separately and independently. In precisely the same manner as the naturally connected partial tones form a single source of sound, the partial tones in a compound stop on the organ fuse into one, as all are struck with the same digital, and all move in the same melodic progression as their prime tone.

Moreover, the tones of most instruments are usually accompanied by charac- teristic irregular noises, as the scratching and rubbing of the violin bow, the rush of wind in flutes and organ pipes, the grating of reeds, &c. These noises, with wdiich we are already familiar as characterising the instruments, materially ^ facilitate our power of distinguishing them in a composite mass of sounds. The partial tones in a compound have, of course, no such characteristic marks.

Hence we have no reason to be surprised that the resolution of a compound tone into its partials is not quite so easy for the ear to accomplish, as the resolu- tion of composite masses of the musical sounds of many instruments into their proximate constituents, and that even a trained musical ear requires the applica- tion of a considerable amount of attention when it undertakes the former problem.

It is easy to see that the auxiliary circumstances already named do not always sufhce for a correct separation of musical tones. In uniformly sustained musical tones, where one might be considered as an upper partial of another, onr judgment might readily make default. This is really the case. G. S. Ohm proposed a very instructive experiment to show this, using the tones of a violin. But it is more suitable for such an experiment to use simple tones, as those of a stopped organ pipe. The best instrument, however, is a glass bottle of the form H shown in fig. 20, which is easily procured and prepared for the experiment. A little rod c supports a guttapercha tube a in a proper position. The end of the tube, which is directed towards the bottle, is softened in warm water and pressed flat, forming a narrow chink, through which air can be made to rush over the mouth of the bottle. When the tube is fastened by an india-rubber pipe to the nozzle of a bellows, and wind is driven over the bottle, it produces a hollow obscure sound, like the vowel 00 in too, which is freer from upper partial tones than even the tone of a stopped pipe, and is only accompanied by a slight m noise of wind. I find that it is easier to keep the pitch unaltered in this instrument wliile the pressure of the wind is slightly changed, than in stopped pipes. We deepen the tone by fr partially shading the orifice of the bottle with t_^ a little wooden plate ; and we sharpen it by pouring in oil or melted wax. We are thus able tcj make any required little alterations in pitch. I tuned a large bottle to b'^ and a smaller one to b'b and united them with the same bellows, so that when used both began to speak at the same instant. When thus united they gave a musical tone of the pitch of the deeper />[?, but having the quality of tone of the vowel oa in toad, instead of oo in too. When, then, I compressed first one of the india-rubber tubes and then the other, so as to produce the tones alternately, separately, and in connection, I was at last able to hear them separately when sounded together, but I could not continue to hear them separately for long, for the upper tone gradually fused with

I

CHAP. IV. SEPARATION OF THE PARTIALS. lU

the lower. Tliis fusion takes place even when the upper tone is somewhat stronger than the lower. The alteration in the quality of tone which takes place during this fusion is characteristic. On producing the upper tone first and then letting the lower sound with it, I found that I at first continued to hear the upper tone with its full force, and the under tone sounding below it in its natural quality of oo in too. But by degrees, as my recollection of the sound of the isolated upper tone died away, it seemed to become more and more indistinct and weak, while the lower tone appeared to become stronger, and sounded like oa in toad. This weakening of the upper and strengthening of the lower tone was also observed by Ohm on the violin. As Seebeck remarks, it certainly does not always occur, and probably depends on the liveliness of our recollections of the tones as heard separately, and the greater or less uniformity in the simultaneous production of the tones. But where the experiment succeeds, it gives the best proof of the essential dependence of the result on varying activity of attention. With the tones H produced by bottles, in addition to the reinforcement of the lower tone, the altera- tion in its quality is very evident and is characteristic of the nature of the process. This alteration is less striking for the penetrating tones of the violin.*

This experiment has been appealed to both by Ohm and by Seebeck as ;i cDrroboratiou of their different opinions. When Ohm stated that it was an ' illusion of the ear ' to apprehend the upper partial tones wholly or partly as a reinforcement of the prime tone (or rather of the compound tone whose pitch is determined by that of its prime), he certainly vised a somewhat incorrect expression, although he meant what was correct, and Seebeck was justified in replying that the ear was the sole judge of auditory sensations, and that the mode in which it apprehended tones ought not to be called an ' illusion '. However, our experiments just described show that the judgment of the ear differs according to the liveliness of its recollection of the separate auditory impressions here fused into one whole, and according to the intensity of its attention. Hence we can certainly appeal from H the sensations of an ear directed without assistance to external objects, whose interests Seebeck represents, to the ear which is attentively observing itself and is suitably assisted in its observation. Such an ear really proceeds according to the law laid down by Ohm.

Another experiment should be adduced. Raise the dampers of a pianoforte so that all the strings can vibrate freely, then sing the vowel a in father, art, loudly to any note of the piano, directing the voice to the sounding-board ; the sym- pathetic resonance of the strings distinctly re-echoes the same a. On singing oe in toe, the same oe is re-echoed. On singing a in fare, this a is re-echoed. For ee in see the echo is not quite so good. The experiment does not succeed so well if the damper is removed only from the note on which the vowels are sung. The vowel character of the echo arises from the re-echoing of those upper partial tones which characterise the vowels. These, however, will echo better and more clearly when their corresponding higher strings are free and can vibrate sym-^ pathetically. In this case, then, in the last resort, the musical effect of the resonance is compounded of the tones of several strings, and several separate partial tones combine to produce a musical tone of a peculiar quality. In addition to the vowels of the human voice, the piano will also quite distinctly imitate the quality of tone ])roduced by a clarinet, when strongly blown on to the sounding- board.

Finally, we must remark, that although the pitch of a compound tone is, for

* [A very convenient form of this experi- The tone is also brighter and unaccompanied

nient, useful even for lecture purposes, is to by any windrush. By pressing the handle of

employ two tuning-forks, tuned as an Octave, the deeper fork on the table, we can excite its

say (■' and c", and held over separate resonance other upper partials, and thus produce a third

jars. By removing first one and then the other, quality of tone, which can be readily apprc-

or letting both sound together, the above effects elated; thus, simple c', simple c' -t- simple c",

can be made evident, and they even remain compound c'. — Translator.'] when the Octave is not tuned perfectly true.

62 ANALYSIS OF COMPOUND SENSATIONS. part i.

musical purposes, determined by that of its prime, the influence of the upper partial tones is by no means unfelt. They give the compound tone a brighter an d higher effect. Simple tones are dull. When they are compared with compound tones of the same pitch, we are inclined to estimate the compound as belonging to a higher Octave than the simple tones. The difference is of the same kind as that heard when first the vowel oo in too and then a in tar are simg to the same note. It is often extremely difficult to compare the pitches of compound tones of different qualities. It is very easy to make a mistake of an Octave. This has happened to the most celebrated musicians and acousticians. Thus it is well known that Tartini, who was celebrated as a violinist and theoretical musician, estimated all combinational tones (Chap. XI.) an Octave too high, and, on the other hand, Henrici * assigns a pitch too low by an Octave to the upper partial tones of tuning-forks.f

H The pi'oblem to be solved, then, in distinguishing the partials of a compound tone is that of analysing a given aggregate of sensations into elements which no longer admit of analysis. We are accustomed in a large number of cases whei*e sensations of different kinds or in different parts of the body, exist simultaneously, to recognise that they are distinct as soon as they are perceived, and to direct our attention at will to any one of them separately. Thus at any moment we can be separately conscious of what wc see, of what we hear, of what we feel, and dis- tinguish what we feel in a finger or in the great toe, whether pressure or a gentle touch, or warmth. So also in the field of vision. Indeed, as I shall endeavour to show in what follows, we readily distinguish our sensations from one another when we have a precise knowledge that they are composite, as, for example, when we have become certain, by frequently repeated and invariable experience, that our present sensation arises from the simultaneous action of many independent stimuli, each of which usually excites an equally well-known individual sensation. This induces

H us to think that nothing can be easier, when a number of different sensations are simultaneously excited, than to distinguish them individually from each other, and that this is an innate facility of our minds.

Thus we find, among others, that it is quite a matter of course to hear sepa- rately the different musical tones which come to our senses collectively, and expect that in every case when two of them occur together, we shall be able to do the like.

The matter is very different when we set to work at investigating the more un- usual cases of perception, and at more completely understanding the conditions under which the above-mentioned distinction can or cannot be made, as is the case in the physiology of the senses. We then become aware that two different kinds or grades must be distinguished in our becoming conscious of a sensation. The lower grade of this consciousness, is that where the influence of the sensation in question makes itself felt only in the conceptions we form of external things and processes, and assists

H in determining them. This can take place without our needing or indeed being able to ascertain to what particular part of our sensations we owe this or that relation of our perceptions. In this case we will say that the impression of the sensation in question is p^}xeived synthetically. The second and higher grade is when we immediately distinguish the sensation in question as an existing part of the sum of the sensations excited in us. We will say then that the sensation is perceived analytically. X The two cases must be carefully distinguished from each other.

* Poggd. Aim., vol. xcix. p. 506. The with ivahrgcnommen, and then restricting the

same difficulty is mentioned by Zamminer meaning of this very common German word.

{Die Mtisik und die niusikalischen I/istru- It appeared to me that it would be clearer to

m«nie, 1855, p. Ill) as well known to musicians. an English reader not to invent new words

+ [Here the passage from ' The problem or restrict the sense of old words, but to

to be solved,' p. 62h, to ' from its simple use perceived in both cases, and distinguish

tones,' p. 65b, is inserted in this edition from the them (for percipirt and appercipirt respectively)

4th German edition. — 'Translator.'] by the adjuncts syntlietically and analytically,

X [Prof. Helmholtz uses Leibnitz's terms the use of which is clear from the explanations

percipirt and appercipirt, alternating the latter given in the text.- — Traiislator.]

CHAP. IV. ANALYSIS OF COMPOUND SENSATIONS. Cy^^

Seebeck and Ohm are agreed that the upper partials of a musical tone are perceived synthetically. This is acknowledged by Seebeck when he admits that their action on the ear changes the force or quality of the sound examined. The dispute turns upon whether in all cases they can be perceived analytically in their individual existence ; that is, whether the ear when unaided by resonators or other physical auxiliaries, which themselves alter the mass of musical somid heard by the observer, can by mere direction and intensity of attention distinguish wlicther, and if so in what force, the Octave, the Twelfth, etc., of the prime exists in the given musical sound.

In the first place I will adduce a series of examples which show that the difficulty felt iu analysing musical tones exists also for other senses. Let us begin with the comparatively simple perceptions of the sense of taste. The ingredients of our dishes and the spices with which we flavour them, are not so complicated that they could not be readily learned by any one. And yet there are ^ very few people who have not themselves practically studied cookery, that are able readily and correctly to discover, by the taste alone, the ingredients of the dishes placed before them. How much practice, and perhaps also peculiar talent, belongs to wine tasting for the piu-pose of discovering adulterations is known in all wine- growing countries. Similarly for smell ; indeed the sensations of taste and smell may unite to form a single whole. Using our tongues constantly, we are scarcely aware that the peculiar character of many articles of food and drink, as vinegar or wine, depends also upon the sensation of smell, their vapours entering the back part of the nose through the gullet. It is not till we meet with persons in whom the sense of smell is deficient that we learn how essential a part it plays iu tasting. Such persons are constantly in fault when judging of food, as indeed any one can learn from his own experience, when he suffers from a heavy cold in the head without having a loaded tongue.

When our hand glides unawares along a cold and smooth piece of metal we^ are apt to imagine that we have wetted our hand. This shows that the sensation of wetness to the touch is compounded out of that of unresisting gliding and cold, which in one case results from the good heat-conducting properties of metal, and in the other from the cold of evaporation and the great specific heat of water. We can easily recognise both sensations in wetness, when we think over the matter, but it is the above-mentioned illusion which teaches us that the peculiar feeling of wetness is entirely resolvable into these two sensations.

The discovery of the stereoscope has taught us that the power of seeing the depths of a field of view, that is, the different distances at which objects and their parts lie from the eye of the spectator, essentially depends on the simul- taneous synthetical perceptions of two somewhat different perspective images of the same objects by the two eyes of the observer. If the difference of the tAvo images is sufficiently great it is not difficult to perceive them analytically as separate. For example, if we look intently at a distant object and hold one of ^ our fingers slightly in front of our nose we see two images of our finger against the background, one of which vanishes when we close the right eye, the other belonging to the left. But when the differences of distance are relatively small, and hence the differences of the two perspective images on the retina are so also, great practice and certainty in the observation of double images is necessary to keep them asiuider, yet the synthetical perception of their differences still exists, and makes itself felt in the apparent relief of the surface viewed. In this case also, as well as for upper partial tones, the ease and exactness of the analytical perception is far behind that of the synthetical perception.

In the conception which we form of the direction in which the objects viewed seem to lie, a considerable part must be played by those sensations, mainly muscular, which enable us to recognise the position of our body, of the head with regard to the body, and of the eye with regard to the head. If one of these is altered, for example, if the sensation of the proper position of the eye is changed by pressing

64 ANALYSIS OF COMPOUND SENSATIONS. part i.

a finger against the eyeball or by injury to one of the nuiscles of the eye, our per- ception of the position of visible objects is also changed. But it is only by such occasional illusions that we become aware of the fact that muscular sensations form part of the aggregate of sensations by which our conception of the position of a visible object is determined.

The phenomena of mixed colours preseiit considerable analogy to those of com- pound musical tones, only in the case of colour the number of sensations reduces to three, and the analysis of the composite sensations into their simple elements is still more difficult and imperfect than for musical tones. As early as 1686 R. Waller mentions in the Philosophical Transactimis the reduction of all colours to the mixture of three fundamental colours, as something already well known. This view could in earlier times only be founded on sensations and experiments arising from the mixture of pigments. In recent times we have discovered better methods,

H by mixing light of different colours, and hence have confirmed the correctness of that hypothesis by exact measurements, but at the same time we have learned that this confirmation only succeeds within a certain limit, conditioned by the fact that no kind of coloured light exists which can give us the sensation of a single one of the fundamental colours with exclusive purity. Even the most saturated and purest colours that the external world presents to us in the prismatic spectrum, may by the development of secondary images of the complementary colours in the eye be still freed as it were from a white veil, and hence cannot be considered as abso- lutel}^ pure. For this reason we are unable to show objectively the absolutely pure fundamental colours from a mixture of which all other colours without exception can be formed. We only know that among the colours of the spectrum scarlet-i^ed, yellow-green, and blue- violet approach to them nearer than any other oljjective colours.* Hence we are able to compound out of these three colours almost all the colours that usually occur in different natural bodies, but we cannot produce the

H yellow and blue of the spectrum in that complete degree of saturation which they reach when purest within the spectrum itself. Our mixtures are always a little whiter than the corresponding simple colours of the spectrum. Hence it follows that we never see the simple elements of our sensations of colour, or at least see them only for a very short time in particular experiments directed to this end, and consequently cannot have any such exact or certain image in our recollection, as would indisputably be necessary for accurately analysing every sensation of colour into its elementary sensations by inspection. Moreover we have relatively rare opportunities of observing the process of the composition of colours, and hence of recognising the constituents in the compound. It certainly appears to me very characteristic of this process, that for a century and a half, from Waller to Goethe, every one relied on the mixtures of pigments, and hence believed green to be a mixture of blue and yellow, whereas when sky-blue and sulphur-yellow beams of light, not pigments, are mixed together, the result is white. To this very cir-

Hcumstance is due the violent opposition of Goethe, who was only acquainted with the colours of pigments, to the assertion that white w^as a mixture of variously coloured beams of light. Hence we can have little doubt that the power of dis- tinguishing the different elementary constituents of the sensation is originally absent in the sense of sight, and that the little which exists in highly educated observers, has been attained by specially conducted experiments, through which of course, when wrongly planned, error may have ensued.

On the other hand every individual has an opportunity of experimenting on the

* [In Ms Physiological Optics, p. 227, E,' hence I translate span-griin by 'yellow- Prof. Helmholtz calls scarlet-red or vermilion green.' Maxwell's blue or third colour was the part of the spectrum before reaching between the lines F and G, but twice as far Fraunhofer's line C. He does not use span- from the latter as the former, This gives the g7'Un { = Griui-span or verdigris, literally colour which Prof. H. in his 0;y<a-s calls ' cya- ' Spanish-green ') in his Optics, but talks of nogen blue,' or Prussian blue. The violet green-yellow between the lines E and b, and proper does not begin till after the line G. It he says, on p. 844, that Maxwell took as one of is usual to speak of these throe colours, vaguely, the fundamental colours ' a green near the line as Red, Green, and Blue. — Translator.]

CHAPS. IV. V. ANALYSIS OF COMPOUND SENSATIONS. 65

composition of two or more musical sounds or noises on the most extended scale and the power of analysing even extremely involved compounds of musical tones, into the separate parts produced hy individual instruments, can readily be acquired by any one who directs his attention to the subject. But the ultimate simple elements of the sensation of tone, simple tones themselves, are rarely heard alone. Even those instruments by which they can be prodiiced, as tuning-forks before resonance chambers, when strongly excited, give rise to weak harmonic upper partials, jjartly within and partly without the ear, as Ave shall see in Chapters V. and VII. Hence in this case also, the opportunities are very scanty for impress- ing on our memory an exact and sure image of these simple elementary tones. But if the constituents to be added are only indefinitely and vaguely known, the analysis of the sum into those parts must be correspondingly^ uncertain. If we do not know with certainty how much of the musical tone under consideration is to be attributed to its prime, we cannot but be luicertain as to what belongs to the H partials. Consequently we must begin by making the individual elements which have to be distinguished, individually audible, so as to obtain an entirely fresh recollection of the corresponding sensation, and the whole business requires un- disturbed and concentrated attention. We are even without the ease that can be obtained by frequent repetitions of the experiment, such as we possess in the analysis of musical chords into their individual tones. In that case we hear the individual tones sufficiently often by tliemselves, whereas we rarely hear simple tones and may almost be said never to hear the building up of a compound from its simple tones.

The results of the preceding discussion may be summed up as follows : — ■

(1) The upper partial tones corresponding to the simple vibrations of a com- pound motion of the air, are perceived synthetically, even when they are not always perceived analytically.

(2) But they can be made objects of analytical perception withoxit any other U help than a proper direction of attention.

(3) Even in the case of their not being separately perceived, because they fuse into the whole mass of musical sound, their existence in our sensation is established by an alteration in the quality of tone, the impression of their higher pitch being characteristically marked by increased brightness and acuteness of quality.

In the next chapter we shall give details of the relations of the upper partials to the quality of compound tones.

CHAPTER V.

ON THE DIFFERENCES IN THE QUALITY OF MUSICAL TONES.

Towards the close of Chapter I (p. 21d), we found that differences in the quality of musical tones must depend on the form of the vibration of the air. The H reasons for this assertion were only negative. We have seen that force depended on amplitude, and pitch on rapidity of vibration : nothing else was left to distin- guish quality but vibrational form. We then proceeded to show that the existence and force of the upper partial tones which accompanied the prime depend also on the vibrational form, and hence we could not but conclude that musical tones of the same quality would always exhibit the same combination of partials, seeing that the peciiliar vibrational form which excites in the ear the sensation of a certain quality of tone, must always evoke the sensation of its corresponding upper partials. The question then arises, can, and if so, to what extent can the difterences of musical quality be reduced to the combination of difterent partial tones with dif- ferent intensities in different musical tones? At the conclusion of last chapter (p. QOd), we saw that even artificially combined simple tones were capable of fusing into a musical tone of a quality distinctly different from that of either of its con- stituents, and that consequently the existence of a new upper partial really altered

F

66 CONCEPTION OF MUSICAL QUALITY AND TONE. part i.

the quality of a tone. By this means we gained a clue to the hitherto enigmatical nature of quality of tone, and to the cause of its varieties.

There has been a general inclination to credit quality with all possible pecu- liarities of musical tones that were not evidently due to force and pitch. This was correct to the extent that quality of tone was merely a negative conception. But very slight consideration will suffice to show that many of these peculiarities of musical tones depend upon the way in which they begin and end. The methods of attacking and releasing tones are sometimes so characteristic that for the human voice they have been noted by a series of different letters. To these belong the ex- plosive consonants B, D, G, and P, T, K. The effects of these letters are produced by opening the closed, or closing the open passage through the mouth. For B and P the closure is made by the lips, for D and T by the tongue and upper teeth,* for G and K by the back of the tongue and soft palate. The series of the mediae

U B, D, G is distinguished from that of the tenues P, T, K, by the glottis being suffi- ciently narrowed, when the closure of the former is released, to produce voice, or at least the rustle of whisper, whereas for the latter or tenues the glottis is wide open,t and cannot sound. The mediae are therefore accompanied by voice, which is capable of counnencing at the beginning of a syllable an instant before the open- ing of the mouth, and of lasting at the end of a syllable a moment after the closure of the mouth, because some air can be still driven into the closed cavity of the mouth and the vibration of the vocal chords in the larynx can be still maintained. On account of the narrowing of the glottis the influx of air is more moderate, and the noise of the wind less sharp for the mediae than the tenues, which, being spoken with open glottis, allow of a great deal of wind being forced at once from the chest.;]: At the same time the resonance of the cavity of the mouth, which, as we shall more clearly understand further on, exercises a great influence on the vowels, varies its pitch, corresponding to the rapid alterations in the magnitude of its volume

H and orifice, and this brings about a corresponding rapid variation in the quality of the speech sound.

As with consonants, the difterences in the quality of tone of struck strings, also partly depends on the rapidity with which the tone dies away. When the strings have little mass (such as those of gut), and are fastened to a very mobile sounding board (as for a violin, guitar, or zither), or when the parts on which they rest or which they touch are but slightly elastic (as when the violin strings, for example, are pressed on the finger board by the soft point of the finger), their vibrations rapidly disappear after striking, and the tone is dry, short, and without ring, as in the pizzicato of a violin. But if the strings are of metal wire, and hence of greater weight and tension, and if they are attached to strong heavy bridges w^hich cannot be much shaken, they give out their vibrations slowly to the

* [This is true for German, and most Con- examples, it seemed better in the present case,

tinental languages, and for some dialectal where the author was speaking especially of

U English, especially in Cumberland, Westmore- the phenomena of speech to which he was

land, Yorkshire, Lancashire, the Peak of Derby- personally accustomed, to leave the text un-

shire, and Ireland, but even then only in con- altered and draw attention to English pecu-

nection with the trilled R. Throughout Eng- liarities in footnotes. — Translator.']

land generally, the tip of the tongue is quite \ [Observe again that this description of

free from the teeth, except for TH in thin and the rush of wind accompanying P, T, K,

then, and for T and D it only touches the hard although true for German habits of speech, is

palate, seldom advancing so far as the root of not true for the usual English habits, which

the gums. — Translator.'] require the windrush between the opening of

t [This again is true for German, but not the mouth and sounding of the vowel to be

for English, French, or Italian, and not even entirely suppressed. The English result is a

for the adjacent Slavonic languages. In these gliding vowel sound preceding the true vowel on

languages the glottis is quite closed for both commencing a syllable, and following the vowel

the mediae and the tenues in ordinary speech, on ending one. The difference between English

but the voice begins for the mediae before P and German Pis precisely the same (as I have

releasing the closure of the lips or tongue and verified by actual observation i a.^ ihat between

palate, and for the tenues at the moncnt of the simple Sanscrit tenuis P, and the postaspi-

releasc. Although in giving vowel sounds, &c., rated Sanscrit Ph, as now actually pronounced

I have generally contented myself with trans- by cultivated Bengalese. See raj Early English

lating the same into English symbols and Pronunciation, p. 1136, col. 1. — TroMslaior.']

CHAP. V. CONCEPTION OF MUSICAL QUALITY AND TONE. 67

air and the sounding board ; their vibrations continue longer, their tone is more durable and fuller, as in the pianoforte, but is comparatively less powerful and penetrating than that of gut strings, which give up their tone more readily when struck with the same force. Hence the pizzicato of bowed instruments when well executed is much more piercing than the tone of a pianoforte. Pianofortes with their strong and heavy supports for the strings have, consequently, for the same thickness of string, a less penetrating but a much more lasting tone than those instruments of which the suppoi'ts for the strings are lighter.

It is very characteristic of brass instruments, as trumpets and trombones, that their tones commence abruptly and sluggishly. The various tones in these instruments are produced by exciting different upper partials through different styles of blowing, which serve to throw the column of air into vibrating portions of different numbers and lengths similar to those on a string. It always requires a certain amount of effort to excite the new condition of vibration in place of the H old, but when once established it is maintained with less exertion. On the other hand, the transition from one tone to another is easy for wooden wind instruments, as the flute, oboe, and clarinet, where the length of the column of air is readily changed by application of the fingers to the side holes and keys, and where the style of blowing has not to be materially altered.

These examples will suffice to show how certain characteristic peculiarities in the tones of several instruments depend on the mode in which they begin and end. When we speak in what follows of musical quality of tone, we shall disregard these peculiarities of beginning and ending, and confine our attention to the peculiarities of the musical tone which continues uniformly.

But even when a musical tone continues with uniform or variable intensity, it is mixed up, in the general methods of excitement, with certain noises, which express greater or less irregularities in the motion of the air. In wind instruments where the tones are maintained by a stream of air, we generally hear more or less H whizzing and hissing of the air which breaks against the sharp edges of the mouthpiece. In strings, rods, or plates excited by a violin bow, we usually hear a good deal of noise from the rubbing. The hairs of the bow are naturally full of many minute irregularities, the resinous coating is not spread over it with absolute evenness, and there are also little inequalities in the motion of the arm which holds the bow and in the amount of pressure, all of which influence the motion of the string, and make the tone of a bad instrument or an unskilful performer rough, scraping, and variable. We shall not be able to explain the nature of the motions of the air and sensations of the ear which correspond to these noises till we have investigated the conception of heats. Those who listen to music make themselves deaf to these noises by purposely withdrawing attention from them, but a slight amount of attention generally makes them very evident for all tones pro- duced by blowing or rubbing. It is well known that most consonants in human speech are characterised by the maintenance of similar noises, as F, V ; S, Z ; TH H in thin and in then ; the Scotch and German guttural CH, and Dutch 0. For some the tone is made still more irregular by trilling parts of the mouth, as for R and L. In the case of R the stream of air is periodically entirely interrupted by trilling the uvula* or the tip of the tongue ; and we thus obtain an intermitting sound to which these interruptions give a ijeculiar jarring character. In the case of L the soft side edges of the tongue are moved by the stream of air, and, withoiit completely interrupting the tone, produce inequalities in its strength.

Even the vowels themselves are not free from such noises, although they are kept more in the background by the musical character of the tones of the voice. Donders first drew attention to these noises, which are partly identical with those which are produced when the corresponding vowels are indicated in low voiceless

* [In the northern parts of Germany and of There are also many other trills, into which, France, and in Northumberland, but not other- as into other phonetic details, it is not neces- wise in England, except as an organic defect. sary to enter. — Translator.']

F 2

68 CONCEPTION OF MUSICAL QUALITY AND TONE. part r.

speecli. They are strongest for ee in see, the French u in vu (which is nearly tlie same as the Norfolk and Devon oo in too), and for oo in too. For these vowels they can be made audible even when speaking aloud.*^ By simply increasing their force the vowel ee in see becomes the consonant y in yon, and the vowel oo in too the consonant w in wan.-\ For a in art, a in at, e in met, there, and o in more, the noises appear to me to be produced in the glottis alone when speaking gently, and to be absorbed into the voice when speaking aloud.:}: It is remarkable that in speaking, the vowels a in art, a in at, and e in met, there, are produced with less musical tone than in singing. It seems as if a feeling of greater compression in the larynx caused the tuneful tone of the voice to give way to one of a more jarring character which admits of more evident articulation. The greater intensity thus given to the noises, appeal's in this case to facilitate the characterisation of the peculiar vowel quality. In singing, on the contrary, we try to favour the musical

U part of its quality and hence often render the articulation somewhat obscui-e.§

Such accompanying noises and little inequalities in the motion of the air, furnish much that is characteristic in the tones of musical instruments, and in the vocal tones of speech which correspond to the different positions of the mouth ; but besides these there are numerous peculiarities of quality belonging to the musical tone proper, that is, to the perfectly regular portion of the motion of the air. The importance of these can be better appreciated by listening to musical instruments or human voices, from such a distance that the comparatively weaker noises are no longer audible. Notwithstanding the absence of these noises, it is generally possible to discriminate the different musical instruments, although it must be acknowledged that under such circumstances the tone of a French horn may be occasionally mistaken for that of the singing voice, or a violoncello may be confused with an harmonium. For the human voice, consonants first disappear at a distance, because they are characterised by noises, but M, N, and the vowels

f can be distinguished at a greater distance. The formation of M and N in so far resembles that of vowels, that no noise of wind is generated in any part of the cavity of the mouth, which is perfectly closed, and the sound of the voice escapes through the nose. The mouth merely forms a resonance chamber which alters the quality of tone. It is interesting in calm weather to listen to the voices of men who are descending from high hills to the plain. Words can no longer be recog- nised, or at most only such as are composed of M, N, and vowels, as Mamma, Ko, Noon. But the vowels contained in the spoken words are easily distinguished. Wanting the thread which connects them into words and sentences, they form a strange series of alternations of quality and singular inflections of tone.

In the present chapter we shall at first disregard all irregular portions of the motion of the air, and the mode in which sounds commence or terminate, directing our attention solely to the musical part of the tone, properly so called, which corresponds to a uniformly sustained and regularly periodic motion of the air,

^ and we shall endeavour to discover the relations between the quality of the sound

* [At the Comedie Fran(,'aise I have heard the important phonetic observations in the

M. Got pronounce the word oui and Mme. text. — Translator.]

Provost-Ponsin pronounce the last syllable of § [These observations must not be cou-

haehis entirely without voice tones, and yet sidered as exhausting the subject of the dif-

make them audible throughout the theatre. — ference between the singing and the speak-

Translator.] ing voice, which requires a peculiar study

t [That this is not the whole of the pheno- here merely indicated. See my Pronunciatimi

menon is shown by the words ye, v^oo. The fur Singers (Curwen) and S-jxcch in Soncj

whole subject is discussed at length in my (Novello). The difference between English and

Early English Prommciation, pp. 1092-1094, German habits of speaking and singing must

and 1149-1151 — Translator.] also be borne in mind, and allowed for by

% [By ' speaking gently ' [leise) seems to the reader. The English vowels given in the

be meant either speaking absolutely without text are not the perfect equivalents of Prof,

voice, that is with an open glottis, or in a Helmholtz's German sounds. The noises

whisper, with the glottis nearly closed. For which accompany the vowels are not nearly

voice the glottis is quite closed, and this is so marked in English as in German, but they

indicated by ' speaking aloud ' {hcim lantcn differ very much locally, even in England. —

Sprechen). It would lead too far to discuss Translator.]

CHAP. V. 1. TONES WITH XO UPPER PARTIALS. 69

and its composition out of individual simple tones. The peculiarities of ([uality of sound belonging to this division, we shall briefly call its musical qua/it i/.

The object of the present chapter is, therefore, to describe the difl;creut com- position of musical tones as produced by diflferent instruments, for the purpose of showing how different modes of combining the upper partial tones correspond to characteristic varieties of musical quality. Certain general rules will result for the arrangement of the upper partials which answer to such species of musical quality as are called, soft, jnercinff, fmii/mg, hoUoic or poor, full or rich, dull, hright, cfisp, jmngent, and so on. Independently of our immediate object (the determination of the physiological action of the ear in the discrimination of musical quality, which is reserved for the following chapter), the results of this investigation are important for the resolution of purely musical (piestions in later chapters, because they show us how rich in upper partials, good musical qualities of tone are found to be, and also point out the peculiarities of musical quality^ favoured on those musical instruments, for which the quality of tone has been to some extent abandoned to the caprice of the maker.

Since physicists have worked comparatively little at this subject 1 shall be forced to enter somewhat more minutely into the mechanism by which the tones of several instruments are produced, than will be, perhaps, agreeable to many of my readers. For such the principal results collected at the end of this chapter will suffice. On the other hand, I must ask indulgence for leaving many large gaps in this almost -unexplored region, and for confining myself principally to instru- ments sufficiently well known for us to obtain a tolerably satisfactory view of the source of their tones. In this inquiry lie rich materials - for interesting acovistical work. But I have felt bound to confine myself to what was necessary for the continuation of the present investigation.

1. Musical Tones withoxit Upper Partials. ^

We begin with such musical tones as are not decomposable, but consist of a single simple tone. These are most readily and purely produced by holding a struck tuning-fork over the mouth of a resonance tube, as has been described in the last chapter (p. 54^/).* These tones are uncommonly soft and free from all shrillness and roughness. As already remarked, they appear to lie comparatively deep, so that such as correspond to the deep tones of a bass voice produce the impression of a most remarkable and unusual depth. The musical quality of such deep simple tones is also rather dull. The simple tones of the soprano pitch sound bright, but even those corresponding to the highest tones of a soprano voice are very soft, without a trace of that cutting, rasping shrillness which is displayed by most instruments at such pitches, with the exception, perhaps, of the flute, for which the tones are very nearly simple, being accompanied with very few and faint upper partials. Among vowels, the oo in too comes nearest to a simple tone, but even this vowel is not entirely free from upper partials. On comparing the H musical quality of a simple tone thus produced with that of a compound tone in which the first harmonic upper partial tones are developed, the latter will be found to be more tuneful, metallic, and brilliant. Even the vowel oo in too, although the dullest and least tuneful of all vowels, is sensibly more brilliant and less dull than a simple tone of the same pitch. The series of the first six partials of a compound tone may be regarded musically as a major chord with a very predominant fundamental tone, and in fact the musical quality of a compound tone possessing these partials, as, for example, a fine singing voice, when heard beside a simple tone, very distinctly produces the agreeable ettect of a consonant chord.

Since the form of simple waves of known periodic time is completely given when their amplitude is given, simple tones of the same pitch can only difter in force and not in musical quality. In fact, the difference of quality remains * On possible sources of disturbance, see Appendix IV.

70 TONES WITH INHARMONIC UPPER PARTIALS. part i.

perfectly indistinguishable, whether the simple tone is conducted to the external air in the preceding methods by a tuning-fork and a resonance tube of any given material, glass, metal, or pasteboard, or by a string, provided only that we guard against any chattering in the apparatus.

Simple tones accompanied only by the noise of rushing wind can also be pro- duced, as already mentioned, by blowing over the mouth of bottles with necks (p. 60c). If we disregard the friction of the air, the proper musical cpiality of such tones is really the same as that produced by tuning-forks.

2. Musical I'ones with Inharmonic Up'per Partlah.

Nearest to musical tones without any upper partials are those with secondary tones which are inharmonic to the prime, and such tones, therefore, in strictness, ^should not be reckoned as musical tones at all. They are exceptionally used in artistic music, but only when it is contrived that the prime tone should be so much more powerful than the secondary tones, that the existence of the latter may be ignored. Hence they are placed here next to the simple tones, because musically they are available only for the more or less good simple tones which they represent. The first of these are tuning-forks themselves, when they are struck and applied to a sounding board, or brought very near the ear. The [inharmonic] upper partials of tuning-forks lie very high. In those which I have examined, the first made from 5-8 to 6-6 as many vibrations in the same time as the prime tone, and hence lay between its third diminished Fifth and major Sixth. The pitch numbers of these high upper partial tones were to one another as the squares of the odd numbers. In the time that the first upper partial would execute 3x3 = 9 vibra- tions, the next would execute 5x5 = 25, and the next 7 x 7 = 49, and so on. Their pitch, therefore, increases with extraoi'dinary rapidity, and they are usually all

^inharmonic with the prime, though some of them may exceptionally become harmonic. If we call the prime tone of the fork c, the next succeeding tones are nearly a"\), cV, c'it.* These high secondary tones produce a bright inharmonic clink, which is easily heard at a considerable distance when the fork is first struck, whereas when it is brought close to the ear, the prime tone alone is heard. The ear readily separates the prime from the upper tones and has no inclination to fuse them. The high simple tones usually die off rapidly, while the prime tone remains audible for a long time. It should be remarked, however, that the nuitual relations of the proper tones of tuning-forks differ somewhat according to the form of the fork, and hence the above indications must be looked upon as merely approximate. In theoretical determinations of the upper partial tones, each prong of the fork may be regarded as a rod fixed at one end.

The same relations hold for straight elastic rods, which, as already mentioned, when struck, give rather high inharmonic upper partial tones. When such a rod

f is firmly supported at the two nodal lines of its prime tone, the continuance of that tone is favo\ired in preference to the other higher tones, and hence the latter disturb the effect very slightly, more especially as they rapidly die away after the rod has been struck. Such rods, however, are not suitable for real artistic music,

* [On calculating the number of cents (as hence it is called d'^' in the text. The interval

in App. XX. sect. C), we find tliat the first to the next tone is 25 : 49 or 1165 cents.

tone mentioned, which vibrates from 5-8 to Adding this to the former numbers the uiterval

6-6 as fast as tlie prime, makes an interval with the prime must be between 5977 and

with it of from .3043 to 3267 ct., so that if C201 cents, or between ?/^' + 77 and (f ' - 3, for

the prime is called c, the note lies between which in the text c"Jf is selected. The inde-

g"\y + 43, and a" - 33, where <i"r, and a" are terminacy arises from the difftculty of finding

the third diminished Fifth and major Sixth of the pitch of the first inharmonic upper partial,

the prime c mentioned in the text. This Prof. The intervals between that and the next upper

Helmholtz calls a"% or 3200 cents. Then the partials are 9 : 25 or 1769 ct., 9 : 49 or 2934

interval between this partial and the next is ct., 9 : 81 or 3699 ct., and so on. The word

9 : 25 or 1769 ct. , and hence the interval ' inharmonic ' has been inserted m the text,

with the prime is between 4812 and 5036 as tuning-forks have also generally harmonic

cents, or lies between c"' + 12 and d"' + 36, and upper partials. See p. 54(7, noie.^Trunslatoi:']

CHAP. V. 2. TONES WITH INHARMONIC UPPER PARTIALS. 71

although tlicy have hitely been introduced for military and dance music on accoimt of their penetrating qualities of tone. Glass rods or plates, and wooden rods, were foi'inei'ly used in this way for the <7/a.s.s harnionicon and the xti'dir-ficldle or icocxl- iMvmonicon. Tlie rods were inserted between two pairs of intertwisted strings^ which grasped them at their two nodal lines. The wooden rods in the German strmr-jiddle were simply laid on straw cylinders. 'I'hey were struck with hanmiers of wood or cork.

The only eflect of the material of the rods on the quality of tone in these cases, consists in the greater or less length of time that it allows the proper tones, at difterent pitches to continue. These secondary tones, including the higher ones, usually continue to sound longest in elastic metal of fine uniform consistency, because its greater mass gives it a greater tendency to continue in any state of motion which it has once assumed, and among metals the most perfect elasticity is found in steel, and the better alloys of copper and zinc, or copper and tin. InH slightly alloyed precious metals, their greater specific gravity lengthens the dura- tion of the tone, notwithstanding their inferior elasticity. Superior elasticity a])pears to favoiu- the continuance of the higher proper tones, because imperfect elasticity and friction generally seems to damp rapid more (piickly than slow vibra- tions. Hence I think that I may describe the general characteristic of what is. usually called a nietallic quality of tone, as the comparatively continuous and uniform maintenance of higher upper partial tones. The quality of tone for glass is similar : but as it breaks when violently agitated, the tone is always weak and

soft, and it is also comparatively high, and dies rapidly away, on account of the smaller mass of the vibrating body. In wood the mass is small, the internal structure comparatively rough, being full of countless interstices, and the elasticity also comparatively imperfect, so that the proper tones, especially the higher ones, rapidly die away. And for this reason the straw-fiddle or wood harmonicon is per- haps more satisfactory to a musical ear, than harmonicons formed of steel or glass rods or plates, with their piercing inharmonic upper partial tones,— at least so far as simple tones are suitable for music at all, of which I shall have to speak later on. "■=

For all of these instruments which have to be struck, the hammers are made of wood or cork, and covered with leather. This renders the highest \ipper partials much weaker than if only hard metal hammers were employed. Greater H hardness of the striking mass produces greater discontinuities in the original motion of the plate. The influence exerted by the manner of striking will be considered more in detail, in reference to strings, where it is also of much impor- taiice.

Acconling to Ghladni's discoveries, elasitic plates, cut in circular, oval, scpiare, o])long, triangular, or hexagonal forms, will sound in a great numV)er of diff"erent vibrational forms, usually producing simple tones which are mutually inharmonic. Fig. 21 gives the iriore simple vibrational forms of a circular plate. Much more complicated forms occur when several circles or additional diameters appear as nodal lines, or where both circles and diameters occur. Supposing the vibrational form A to give the tone c, the others give the following proper tones : —

* [In Java the principal music is produced the rods are laid on the edges of boat-shaped

by harmonicons of metal or wooden rods and vessels, like old fashion cheese-trays, and kept

kettle-shaped gongs. The wooden harmonicons in position by nails passing loosely through

are frequent also in Asia and Africa. In Java holes. See App. XX. sect. K.— Translator.]

72

TONES WITH INHARMONIC UPPER PARTIALS.

Number of Nodal Circles

Number of Diameters

0

1

2 3 j 4

5

0

1 2

f'h

^'b

C 1 d'

y" 1

[

c" j

1

'f-o"^

This shoAvs that many proper tones of nearly the same pitch are produced by a plate of this kind. When a plate is struck, those proper tones which have no node at the point struck, will all sound together. To obtain a particular deter- minate tone it is of advantage to support the plate in points which lie in the nodal lines of that tone ; because those proper tones which have no node in those points will then die off more rapidly. For example, if a circular plate is supported at

H 3 points in the nodal circle of fig. 21, C (p. 71c), and is struck exactly in its middle, the simple tone called (A in the table, which belongs to that form, will be heard, and all those other proper tones which have diameters as some of their nodal lines* will be very weak, for example, c, d', c", g", b'\) in the table. In the same way the tone g"^ with two nodal circles, dies off immediately, because the points of support fall on one of its ventral segments, and the first proper tone which can sound loudly at the same time is that corresponding to three nodal circles, one of its nodal lines being near to that of No. 2. But this is 3 Octaves and more than a whole Tone higher than the proper tone of No. 2, and on account of this great interval does not disturb the latter. Hence a disc thus struck gives a tolerably good musical tone, whereas plates in general j)roduce sounds composed of many in- harmonic proper tones of nearly the same pitch, giving an empty tin-kettle sort of quality, which cannot be used in music. But even when the disc is ])roperly sup- ported the tone dies away rapidly, at least in the case of glass plates, because

^ contact at many points, even when nodal, sensibly impedes the freedom of vibra- tion.

The sound of hells is also accompanied by inharmonic secondary tones, which, however, do not lie so close to one another as those of flat plates. The vibrations which usually arise have 4, 6, 8, 10, ifec, nodal lines extending from the vertex of the bell to its margin, at equal intervals from each other. The corresponding proper tones for glass bells which have approximativcly the same thickness throughout, are nearly as the squares of the numbers 2, 3, 4, 5, so that if we call the lowest tone <\ wo have for the

Number of nodal Hues .

. 1 4

1

G

8

10

12

Tones

Cenfe

• 1 ^' 0

d' + 1404

c" ' 2400

3173

d"' + 3804

The tones, however, vary with the greater or less thickness of the wail of the Hbell towards the margin, and it appears to be an essential point in the ait of casting bells, to make the deeper proper tones mutually harmonic by giving the bell a certain empirical form. According to the observations of the organist Gleitz.t the bell cast for the cathedral at Erfurt in 1477 has the following proper tones : E, e, g^, h, f-', r/'jj, //, c"|. The [former] bell of St. Paul's, London, gave a and c'lf. Hemony of Ziitphen, a master in the seventeenth century, required a good bell to have three Octaves, two Fifths, one major and one minor Third. The deepest tone is not the strongest. The body of the bell when struck gives a deeper tone than the 'sound bow,' but the latter gives the loudest tone. Probably other vibrational forms of bells are also possible in which nodal circles are formed

* Provided that the supported points do not happen to belong to a system of diameters making equal angles with each other.

t ' Historical Notes on the Great Bell and the other Bells in Erfurt Cathedral'

{Geschichtliche-s iibcr die grosse Glockc und die ilbrigen Glocken des Domes zu Erfurt). Erfurt, 1867.— See also Schafh;iutl in the Kunst und Gewerbcblatt fur das Konigreich Bay cm, 1868, liv. 325 to 350 ; 385 to 427.

CHAP. V. Z.

tonp:s with inharmonic upper partials.

73

parallel to the margin. But these seem to be produced with ditliculty autl have not yet been examined.

If a bell is not perfectly symmetrical in respect to its axis, if, for example, the wall is a little thicker at one point of its circumference than at another, it will give, on being struck, two different tones of very nearly the same pitch, which will ''beat' together. Four points on the margin will be found, separated from each other by quarter-circles, in which only one of these tones can be heard without accompanying beats, and four others, half-way between the pairs of the others, where the second tone only sounds. If the l)ell is struck elsewhere both tones are heard, producing beats, and such beats may be perceived in most bells as their tone dies gradually away.

Stretclied membranes have also inharmonic proper tones of nearly the same pitch. For a circular membrane, of which the deepest tone is c, these are, in a vacuiim and arranged in order of pitch, as follows : —

Number of Nodal Lines

Tone

Diameters

Circles

0

0

c

1

. 0

«ba

2

0

/# + 0-l*

0

1

d'^ + 0-2

1

1

<j' -0-2

"

2

b'\f +0-1

These tones rapidly die out. If the membranes sound in air,t or are associated with an air chamber, as in the kettledrum, the relation of the proper tones may be altered. No detailed investigations have yet been made on the secondary tones of the kettledrum. The kettledrum is used in artistic music, but only to mark H certain accents. It is tmied, indeed, but only to prevent injury to the harmony, not for the purpose of filling up chords.

The common character of