...exactly!!
The here well known Steven M. Martin movie An Electronic Odyssey 1991 is a very exciting film with and about Lev Termen, Clara Rockmore and Robert Moog themselves. Clara Rockmore is shown in a scene using her own RCA Theremin before a concert. While testing it she plays a certain note over three octaves. So I have used this record to separating the notes and prepairing them in the manner described above for clarinet and duck sound. Here are the results.
What you can see is the harmonics remain their amplitude ratios over the first 9 harmonics, the patterns of higher octaves only are influenced by the low pass filtering. And the time signal shapes differ strong due to phase shifts of harmonics. As we have already explained.
The Rockmore theremin has not the behavior of an classical instrument because of absence of fix resonances and formants e.g. due to a hollow body. Because Clara Rockmore was a brilliant viola artist she makes - with vibrato and laudness effects - that it sounds like a wunderful cello or violin.
Assumed, I will be able to reconstruct the RCA sound with the proper harmonic ratios, without Claras experience it would be sound only like a - ...No, that will be a furthermore basic experiment.