Del Vento ust curious on what this is. Can you please define "overlapping"? And assuming what temperament?
So here's how I understand it. For each note, the strings vibrate at the primary frequency (eg A4 = 440 Hz), and then also vibrate at the higher harmonics. These higher harmonics all follow the same interval pattern relative to the induced note. So A4 on a piano is actually many frequencies, 440hz, 880hz, 1318 hz etc…. (aka the fundamental frequencies of A4, A5, E6, etc.; AKA midi note numbers 69, 81, 88 )
This interval pattern of harmonics comes down to these known intervals (translated into midi note number intervals): 0, 12, 19, 24, 28, 31, 34, 36, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52. When given 2 midi note numbers, you can just make arrays of the harmonics for each note and compare the "overlapping" ones - the harmonics that are in common between the two notes. Anything overlapping is a sympathetic resonance.
So, for C4, and C5, which are midi notes 60 and 72 (see picture), the common overtones are midi notes 72, 84, 91, 96, 100, 103, 106. These are the frequencies to simulate sympathetic resonance using sine waves.
Now, I haven't fully considered the temperament as I'm mainly working in the center of the piano. I know (think) that the harmonics are pure physics and math (equal temperament), but pianos I work with are stretch tuned. So that means that if I induce a harmonic of a low string, which is stretched flat, the sine waves I'm working with will not be exactly the right frequencies. I haven't figured this out yet - but maybe like my half pedaling scripts - better to have something that is 90% there than nothing given my coding skills and limitations of Kontakt.
Its good to get my thoughts on this down somewhere for reference and as a reality check that others can read/comment. I did study vibrations in college, and went on to intern at a defense company studying how to dampen vibrations in missile airframes. But that was a long time ago, I've forgotten many of the details, and those vibrations were highly non-linear. I think I realize why no one has directed me on the relative power between the harmonics now after writing this - its because
- each piano has different damping characteristics
- because the distance between the strings matters in how much power is transferred from the struck string to the open string.
- The mass of the string
- Stretch tuning vs equal temperament mucks up the power induced since the frequencies don't overlap perfectly
And maybe some other contributing factors. I just need to do as was recommended. Record some and figure out a close-enough correlation.