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I learnt this week that if we divide an octave into 53 notes (53-TET), then all basic intervals approximations will be improved. I am trying to understand how one would play a normal song (for 12-TET) given an n-TET instrument.

Suppose I came across a hypothetical n-TET keyboard, which has keys labelled C♯, C♯♯ etc. If I were to play any song written in 12-TET, would I just press the corresponding keys on the keyboard? For example, if there is a triad C-E-G, then press the keys labelled as C-E-G on the n-TET keyboard.

I suppose this cannot be right, since if none of the new notes in the keyboard are used, then we are basically using a 12 tone keyboard, tuned using 53-TET. And yet, any intervals that we play in consecutive notes, chords, etc. will be closer to their just intervals. What am I missing? For context, I'm learning about why just intonation is impractical and how equal temperament is used to solve the issue.

Aaron
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aiwl
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    I'd ask for better labels on your n-TET keyboard, especially since you imply that the labels highly influence the correct answer when you say, "For example, if there is a triad C-E-G, then press the keys labelled as C-E-G on the n-TET keyboard." For example, a 19TET keyboard can be labelled by assigning each zero-to-one-accidental note name its own note, and a 24TET keyboard should be labelled with quarter tone names in between all the regular 12TET note names. – Dekkadeci Apr 24 '21 at 15:51
  • Also, I'm having a stronger hunch that playing pieces on n-TET keyboards quickly becomes a matter of opinion. People are going to quite possibly arbitrarily flip between how often they play "B#" vs. "B" on a 13TET keyboard, for example, and with a, say, 100TET keyboard, you can easily switch notes to flip between sounding more like 12TET equal temperament or closer-to-just-intonation barbershop quartets. – Dekkadeci Apr 24 '21 at 15:54
  • @Dekkadeci I'm not sure how enharmonics will translate; I guess that is what I'm missing. On Wikipedia it says that "Western music maps unambiguously onto [19 TET] (unless it presupposes 12-EDO enharmonic equivalences)", but I'm not sure how that would theoretically work since I'm used to treating C♯ and D♭ as the same when playing the piano. – aiwl Apr 24 '21 at 16:39
  • @Dekkadeci how do you handle double sharps and flats in 19-tone equal temperament? – phoog Apr 25 '21 at 16:10
  • @phoog - The same way as in 24TET - I'd probably have to translate to 12TET note names, but others might vary. – Dekkadeci Apr 26 '21 at 11:10

3 Answers3

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There's no single, universal answer to this.

Most Western music is based on a combination of diatonic melody (which is arguably best rendered in Pythagorean tuning, i.e. 9:8 whole-tone steps), and 5-limit JI harmony. It immediately follows that there's a conflict between the ditone 81:64 (≈1.266) and the just major third 5:4 (=1.25). So you either need to make a deliberate distinction between these, i.e. have two notes that are only a syntonic comma apart, or else approximate both major-third–candidates by the same ratio. The latter is the idea behind meantone temperaments, which includes several edo-tunings, most noteworthy 12-edo, 19-edo and 31-edo. So in these tunings, it is in fact always quite clear how to translate existing music. For example, you can indeed tune the white keys of a piano to a subset of 31-edo, and then any piece in C-major will sound pretty much just fine. (Of course, modulations are another story.)

53-edo is not a meantone tuning. You can still tune a keyboard to the subset that approximates the Ptolemaic scale. In that scale, the main triads C, F and G sound great, but there are a couple of things that will sound strange. The fifth D-A is a wolf fifth, and the intervals D-E and G-A will be 10:9 steps – still whole tones, but notably narrower then the 9:8 major tones C-D, F-G and A-B.

leftaroundabout
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    As a supplement to this excellent explanation, n-tet tunings are generally developed to be used in their own right and not as refinements of, or supplements to, 12-tet. In that regard, one wouldn't play music written for 12-tet in those other "tets", except perhaps as a novelty. If one is going to play in (n<>12)-tet, then the music should take advantage of the unique attributes of that temperament. – Aaron Apr 24 '21 at 19:17
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    @Aaron well, but what is “music written for 12-tet”? Most Western music isn't really written for 12-edo, but rather for _some tuning that supports major and minor chords and diatonic melody_. Which basically fits _any meantone tuning_, of which 12-edo happens to be one. And most performances aren't 100% in 12-edo anyway, because of expressive freedom, the fact that JI thirds do often sound better than 12-edo ones, as well as unintentional intonation inaccuracy. – leftaroundabout Apr 24 '21 at 19:35
  • leftaroundabout: a lot of 12-tone music works in non-meantone tunings, too. @Aaron since 53-tone equal temperament is favored because of its resemblance to 5-limit just tuning, it's entirely reasonable to use it for all sorts of music of all periods from the renaissance to the present. Problems do of course arise, with comma pumps and the like, but they can be resolved. – phoog Apr 25 '21 at 16:07
  • @phoog I would say most 12-tone music works just about the same no matter the tuning system, whether meantone or not... (that is, it works as bad in 12-edo as in any other tuning). But the central paradigm of 12-tone of having symmetry between all the possible notes is certainly specific to 12-edo. – leftaroundabout Apr 25 '21 at 17:25
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    @leftaroundabout by 12-tone I was responding to "music written for 12-edo," so I didn't mean the compositional technique associated with Schönberg but, more broadly, music written for the standard European notational system, even where it uses only a subset of the 12 tones. Now I realize that the standard European notational system doesn't actually require a 12-tone system, but for the vast majority of its existence it has been mapped to 12-tone keyboards. – phoog Apr 26 '21 at 02:10
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By offering better approximations to just intervals, 53-tone equal temperament gives you the opportunity to choose different pitches not only for accidentals but also for so-called "white" notes. For example, the A that is a major third above F, when F is a perfect fifth below C, is not the same A that is a perfect fifth above D when C, G, and D are all a perfect fifth apart.

Therefore, choosing any one-to-one correspondence between pitches specified in the twelve-tone system and those of 53-tone equal temperament is just as impractical as tuning a keyboard in 5-limit just intonation: it's not possible to have even a single diatonic scale with perfect fourths and fifths where all the major thirds are pure.

The premise of the question is therefore questionable: you are unlikely to "come across a hypothetical n-TET keyboard, which has keys labelled C♯, C♯♯ etc." If you did, the utility of those labels would be limited.

phoog
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  • Well, for a keyboard in 19-edo or 31-edo, keys labelled C♯, C would be perfectly sensible. But not for 22-edo, 34-edo or 53-edo, that's true. – leftaroundabout Apr 25 '21 at 19:34
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It seems like the answer would be to simply determine the desired JI interval (there may be multiple candidate intervals), then find the 53-EDO note that best approximates that note.

But that is a very simplistic answer that ignores the vast differences between 12-EDO and other equal temperaments. An entirely new tuning system could (should) be treated as an entirely new type of music, and converting 12-TET music to the 53-TET system seems like an extreme limitation of potential. It should come as no surprise that 53-TET doesn't do 12-TET's music justice; the music in question was designed under the 12-TET system, and not optimized for other tuning systems.

With some rare exceptions, composers do not write their music solely seeking a good approximation of just intonation; they write their music so that it sounds good in their native tuning system. If composers intend for justly-intonated sound, they would probably not use 12-EDO in the first place. The "out-of-tuneness" of 12-TET is integral to how the music itself is composed and perceived. Thus, this conversion out of the native temperament only loses compositional intent in a vain quest for mathematically aesthetic harmonies.

user45266
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  • I disagree, it is by no means rare for composers to write mostly for _a meantone temperament_, rather than specifically for 12-edo. (Excepting piano, it won't end up being played in 12-edo anyway.) It's mainly in enharmonic-crossing modulations that going to a different tuning can pose problems, but these doesn't happen all that often. Mere chromatic passages meanwhile tend to work _ok_ to other meantone tunings, and certainly everything diatonic. [{See also}](https://music.stackexchange.com/questions/113964/how-would-you-play-a-normal-song-on-an-n-tet-instrument/113966#comment198441_113966) – leftaroundabout Apr 26 '21 at 00:17