Would an instrument that tuned itself harmonically on the fly sound in tune?
Pianos can't be tuned perfectly using harmonic tuning, because they have a string for every note. See this MinutePhysics video for a good explanation of why.
But with current technology, we could theoretically create an instrument that tuned every note harmonically according to the previous note. For example, if one played A440 and then D5, D5 would sound at 440 * 4/3 = 586.333... Hz. But playing A440, then E5, and then D5 would cause the D to sound at 440 * 3/2 * 8/9 = 586.666... Hz.
How would this instrument sound, and would it sound in tune playing an equal-tempered piece?
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There is a fundamental problem with this idea - it only works if the "magic tuning system" understands what key the music is in. If not, the overall pitch will drift up or down over time.
It's easy to see why that happens if you do the math and chain some intervals together. Let's start with A = 440.
Go down a 4th to E: E = 440 x (3/4) = 330.
Up a 5th to B: B = 330 x (3/2) = 495.
Down a 4th to F sharp: F# = 495 x (3/4) = 371.25
Down a major 3rd to D: D = 371.25 x (4/5) = 297.
Up a 5th and back to A: A = 297 x 3/2 = 445.5.
Oops! A used to be 440, but now it's drifted higher by a factor of 81/80, which is much too big to ignore.
If we "know" that all these notes are in A major, we could fudge the numbers so we do get back to A = 440. But most interesting music doesn't stay in one key for ever.
This is one of the reasons why unaccompanied choirs (which usually try to sing chords with "pure intonation") tend to drift up or down in pitch over time.
Any good wind or string ensemble or choir tunes itself harmonically on the fly. For example in a major chord, whoever is playing the third will (probably subconsciously) play a bit flatter to be in tune according to just intonation.
Of interest may be "adaptive tuning" by William A. Sethares
sethares.engr.wisc.edu/mp3s/three_ears.html
which aims at achieving maximum consonance via microtonal adjustments based on the spectrum. Whether the resulting music "sounds in tune" is subjective:
``The chords sounded smooth and nondissonant but strange and somewhat
eerie. The effect was so different from the tempered scale that there
was no tendency to judge in-tuneness or out-of-tuneness. It seemed
like a peek into a new and unfamiliar musical world, in which none of
the old rules applied, and the new ones, if any, were undiscovered."
- F. H. Slaymaker
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