zlacker

>>u1hcw9+(OP)
With real instruments, you must also manage inharmonicity.

A real piano string, for instance, is made of metal and resists bending slightly unlike an idealised string. This affects higher harmonics more than lower ones (think of all the bends in the string on the 7th harmonic, for example). This increases the harmonic frequencies slightly above exact integer multiples of the fundamental.

As a result, pianos require "stretched tuning" so the harmonics better match the higher notes. It's always a bit of a compromise. The higher harmonics will be more "off" than the lower ones.

So even if you were to tune the fundamental frequency of all the keys on a piano "perfectly" in a given key (so-called Just Intonation), the harmonics would not perfectly match up.

>>HPsqua+a4d
There is actually an interesting mathematical method for piano tuning that takes harmonics into account [1]. The core idea is to minimize the integral over the logarithm of the sum of all spectra. This basically favors spectra that are smeared out less. Instead of the logarithm, one could also use another sublinear function, such as the square root, but I guess it just makes for a better story to call it „piano entropy tuner“. The paper also shows a nice plot about the „stretched tuning“ that you mentioned.

[1] https://arxiv.org/abs/1203.5101