Looking at this “grand piano” synth sample, I see that toward the beginning (19 cycles just after the attack has settled down a bit):
there’s a fair bit of white (or rather pink) noise between the harmonics;
its fundamental is at 443 Hz at -13dB,
its second overtone at 883 Hz is 6.8dB lower,
its third overtone at 1326 Hz is the same (well, only 1.1dB lower),
its fourth overtone at 1775 Hz is a bit lower at 16.1dB below the fundamental,
its fifth overtone at 2223 Hz is substantially depressed at 23.1 dB below the fundamental (almost as low as the second subharmonic),
its sixth overtone at 2678 Hz is about 13 cents sharp (twice the inharmonicity of anything so far) and is 19.3dB below the fundamental;
its supposedly undesirable seventh overtone at 3129 Hz is almost 16 cents sharp and is 23.4dB below the fundamental;
its eighth overtone at 3597 Hz is 25.9dB below the fundamental, and inharmonicity continues to increase sharply to 26 cents;
its ninth overtone at 4072 Hz is 31dB below the fundamental;
higher overtones (which probably don't matter) continue to diminish in amplitude exponentially with their frequency.
Overall this spectrum looks a lot like an impulse train has had its fifth harmonic attenuated by about 6dB, has been low-pass filtered at about 9 dB per octave with a cutoff below the note frequency, and has had its overtones stretched out a bit, by about 9 cents per octave.
(It seems strange that amplitude would diminish exponentially with frequency, though. 9 dB per octave would give you amplitude diminishing cubically with frequency. Might be hard to tell the difference with this much noise, though.)
This suggests that my attempt to synthesize a piano with triangle waves didn’t sound very piano-like because it was too low-pass filtered, not the opposite, which I guess I should have figured out just by looking at the waveform.
Looking at it about 500ms later, the peaks are much more separated (much less pink noise in between), everything is quite attenuated (the fundamental, which has moved down to the more correct 441Hz, is down to -30.9 dB, a decay of 27.9dB, indicating a half-life of about 55ms or about 24 cycles) and the rolloff is sharper: the second overtone is now 13.8dB below the fundamental, and the third 25dB below the fundamental. This is consonant with my understanding of the low-pass-filtering nature of the KS model, but it is being generated from a very realistic synthesized sample.
Looking at a somewhat noisy and MPEG-4-artifacted sample from an upright piano shortly after the attack, I see peaks at
119 Hz -46.7dB
247 Hz -19.2dB
370 Hz -28.4dB
491 Hz -34.0dB
614 Hz -40.5dB
739 Hz -32.1dB
866 Hz -34.2dB
986 Hz -41.6dB
1110 Hz -47.2dB
1236 Hz -44.5dB
It also has a notably high peak at 1867Hz at -45.7dB.
(Those are from a 2048-sample window. Later I got better numbers with a 4096-sample window but didn't update it, but in particular the first peak really peaks closer to 124Hz.)
These are roughly harmonics of about 123.47 Hz, which would be B2 in A440 pitch (although as I recall, the guy was tuning the piano a little flat) and which is nearly missing from the sound, which is probably why Débora says her upright piano sounds shitty compared to a grand.
If we figure that the two octaves from the second overtone 247Hz at -19dB up to 986Hz at -42dB represent the normal falloff, that’s about 11.5dB per octave. The fifth overtone at 614Hz is attenuated some 8dB below this line, just like in the synthesized sample, and the seventh overtone at 866Hz is not particularly attenuated at all.
Looking at it after a second or two of decay, there’s less noise between the overtone peaks, everything is of course much quieter (the first overtone is down to -61.8dB and the second down to -38.3), and the third and seventh harmonics have gotten much stronger relatively — stronger, in fact, than anything else! And there’s a second subharmonic peak at 55Hz at a barely-detectable -82dB, presumably due to nonlinearities in the instrument. This is after about 850ms, or about 105 cycles. If we figure the overall attenuation is about 12dB, that’s a half-life of about 26 cycles, which is quite similar to the half-life of the synthesized piano signal.
The initial attack is about 35ms long, and seems to be pretty similar across all the frequencies; it doesn’t show the phenomenon I saw in the synthesized sample where the high frequencies start later.
So, overall, our piano recipe is:
There’s also some stuff about sympathetic strings and energy transfer among different modes of vibration, but that stuff doesn’t really show up in my analyses. You can definitely hear it at times in the highest strings on the actual physical piano.