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I don't believe that's it. The problem of tuning higher notes to the stretched harmonics of lower notes would still be present even in a piano tuned using just

I don't believe that's it. The problem of tuning higher notes to the stretched harmonics of lower notes would still be present even in a piano tuned using just intonation. Suppose for instance the 2nd harmonic of C2 is 2 cents sharp of where it "ought" to be (which is twice the frequency of C2 itself). Then if you tune C3 (the C in the next octave) to be similarly 2 cents sharp of where it ought to be, it will not clash against the 2nd harmonic in C2 -- but the result is that the distance from C2 to C3 is a little more than an octave. (I don't know whether 2 cents is anywhere ballpark -- I've studied the theory but haven't performed measurements.) Incidentally, if you want to play microtonal music on an equally tuned piano, this is an argument for selecting a sharp system, such as 27-edo or 58-edo -- that is, a system whose approximations to the harmonics you're interested in are sharps. (12-edo gives an approximation 2 cents flat of the 3rd harmonic and 14 cents sharp of the 5th harmonic. It ignores the other harmonics. 27-edo gives a sharp approximation to harmonics 3, 5 and 7. 58-edo gives a sharp approximation to 3, 5, 7, 11, and 13 -- nirvana, as far as I'm concerned.)

pclmulqdq 3y ago
While it is true that the harmonics will be off for any tuning system, octave stretching is relatively new and arises only on equal tempered instruments because it needs to be applied equally to every note, so it sounds terrible if you apply it to unequal tunings where (for example) the distance between C2 and G3 can be 20 cents different than the distance between C#2 and G#3. In equal temperament, the distance is ~2 cents between the 2nd harmonic of C2 and G3 (and every other pair a 12th apart), so stretching an octave by ~1 cent spreads that distance out while being pretty much imperceptible. In a decent quarter-comma tuning system like Werckmeister (a very common Baroque tuning), four of the 5ths are off by about 5 cents, and the others are pure. As a result, the 2nd harmonic differs from the note a 12th above by a variable amount. Stretching every octave by 2.5 cents to balance this becomes very much audible across a keyboard, and sounds very odd on the harmonic series' that are nearly perfect. Baroque tunings also pay a lot of attention to the major 3rd (the 4th harmonic), which is a notable weakness of equal temperament - it is far too narrow. Regardless, octave stretching doesn't have to do with piano dynamics, it has to do with tuning systems.
- mrob 3y ago
  As I understand it, stretched tuning in a piano is specifically intended to compensate for inharmonicity, not to compensate for the weaknesses of equal temperament. You can apply stretch to unevenly tempered pianos too, and it should also make those sound better. A formal description of stretched piano tuning is relatively recent (1938, with Railsback's publication of his eponymous curve), but some kind of stretched tuning likely goes back to the first piano, as it happens automatically if you tune the piano against itself by ear.
- Jeff_Brown 3y ago
  12-edo's approximation to the 5th harmonic (a.k.a. 4th overtone) is 14 cents sharp, not flat -- it's at 400 cents plus two octaves, whereas the 5th harmonic is at 386 cents plus two octaves. I admit to being unfamiliar with unequal temperaments. As a jazz player they seems like a complete bummer to me, as I love to transpose on a whim and don't want the intervallic structure to depend on what key I'm in.
  - pclmulqdq 3y ago
    Yeah, I had a sign error in my head there (unequal major thirds are the relatively flat ones). I know that there is some unequal tempered jazz out there from very early recordings, and I assume they thought the different character of the keys added extra character and possibly some extra "grunge," but it would be interesting to compare the improvisational style to equal tempered jazz.
    - Jeff_Brown 3y ago
      I know of at least one person who's very fond of a 12-tone JI scale she came up with, which she happily plays in every key. She tattooed it onto her arm. I've played in 58-edo, which to me is basically indistinguishable from JI. It's just a little too big to use comfortably on the Lumatone. I can use it on the monome comfortably, but the monome doesn't have velocity sensitivity, which for me is a deal killer. On the Lumatone the biggest I feel comfortable playing is 46-edo. Fortunately, 46-edo sounds incredible. It doesn't have the best approximation to 5/4 -- it's 5 cents sharp -- but that error goes in the same direction as the error in 12-edo that we're used to, so (because we're all used to 12-edo) it sounds world better than, say, the 5/4 that 41-edo gives you, which is 6 cents flat.

pclmulqdq 3y ago

While it is true that the harmonics will be off for any tuning system, octave stretching is relatively new and arises only on equal tempered instruments because it needs to be applied equally to every note, so it sounds terrible if you apply it to unequal tunings where (for example) the distance between C2 and G3 can be 20 cents different than the distance between C#2 and G#3. In equal temperament, the distance is ~2 cents between the 2nd harmonic of C2 and G3 (and every other pair a 12th apart), so stretching an octave by ~1 cent spreads that distance out while being pretty much imperceptible. In a decent quarter-comma tuning system like Werckmeister (a very common Baroque tuning), four of the 5ths are off by about 5 cents, and the others are pure. As a result, the 2nd harmonic differs from the note a 12th above by a variable amount. Stretching every octave by 2.5 cents to balance this becomes very much audible across a keyboard, and sounds very odd on the harmonic series' that are nearly perfect. Baroque tunings also pay a lot of attention to the major 3rd (the 4th harmonic), which is a notable weakness of equal temperament - it is far too narrow. Regardless, octave stretching doesn't have to do with piano dynamics, it has to do with tuning systems.

mrob 3y ago
As I understand it, stretched tuning in a piano is specifically intended to compensate for inharmonicity, not to compensate for the weaknesses of equal temperament. You can apply stretch to unevenly tempered pianos too, and it should also make those sound better. A formal description of stretched piano tuning is relatively recent (1938, with Railsback's publication of his eponymous curve), but some kind of stretched tuning likely goes back to the first piano, as it happens automatically if you tune the piano against itself by ear.
Jeff_Brown 3y ago
12-edo's approximation to the 5th harmonic (a.k.a. 4th overtone) is 14 cents sharp, not flat -- it's at 400 cents plus two octaves, whereas the 5th harmonic is at 386 cents plus two octaves. I admit to being unfamiliar with unequal temperaments. As a jazz player they seems like a complete bummer to me, as I love to transpose on a whim and don't want the intervallic structure to depend on what key I'm in.
- pclmulqdq 3y ago
  Yeah, I had a sign error in my head there (unequal major thirds are the relatively flat ones). I know that there is some unequal tempered jazz out there from very early recordings, and I assume they thought the different character of the keys added extra character and possibly some extra "grunge," but it would be interesting to compare the improvisational style to equal tempered jazz.
  - Jeff_Brown 3y ago
    I know of at least one person who's very fond of a 12-tone JI scale she came up with, which she happily plays in every key. She tattooed it onto her arm. I've played in 58-edo, which to me is basically indistinguishable from JI. It's just a little too big to use comfortably on the Lumatone. I can use it on the monome comfortably, but the monome doesn't have velocity sensitivity, which for me is a deal killer. On the Lumatone the biggest I feel comfortable playing is 46-edo. Fortunately, 46-edo sounds incredible. It doesn't have the best approximation to 5/4 -- it's 5 cents sharp -- but that error goes in the same direction as the error in 12-edo that we're used to, so (because we're all used to 12-edo) it sounds world better than, say, the 5/4 that 41-edo gives you, which is 6 cents flat.