Why are the harmonics of a piano tone not multiples of the base frequency?

I was trying to figure out which piano keys were being played in an audio recording using spectral analysis, and I noticed that the harmonics are not integer multiple of the base note. What is the reason for this? Take a look at the spectrogram of a clean sample from a single piano key. I am using Piano.ff.A4 from here. The following is the same as above, with a superimposed reference grid of $ 440 ~\mathrm{Hz}$. As you can see, the harmonics have increasingly higher frequencies than integer multiples of $440 ~\mathrm{Hz}$. At this point you might think that the actual base frequency is just slightly higher than $440 ~\mathrm{Hz}$. So let us make a different reference grid, which lines up with the harmonic at ~$5060 ~\mathrm{Hz}$. You can now clearly see that they aren't actually integer multiples of a base frequency. Question: What is the explanation for this? I am looking both for simple high-level explanations of what is happening, and more in-depth, instrument specific ones, which could maybe allow me to attempt to calculate the harmonics. My first reaction was that this must be some non-linear effect. But you can see that the harmonics do not change frequency at all as time passes and the sound gets quieter. I would expect a non-linear effect to be pronounced only in the loudest part of the sample. Update – I measured the frequencies using peak detection on the Fourier transform from 0.3 to 0.4 seconds in the sample. This table compares the measured values with integer multiples of 440: meas. int. mult. 440. 440. 880. 880. 1330. 1320. 1780. 1760. 2230. 2200. 2680. 2640. 3140. 3080. 3610. 3520. 4090. 3960. 4570. 4400. 5060. 4840. 5570. 5280.