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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.

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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, 2021 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, 2021 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, 2021 at 16:39
  • @Dekkadeci how do you handle double sharps and flats in 19-tone equal temperament?
    – phoog
    Apr 25, 2021 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, 2021 at 11:10

6 Answers 6

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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.

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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, 2021 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. Apr 24, 2021 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, 2021 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. Apr 25, 2021 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, 2021 at 2:10
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By offering better approximations to (5-limit, 7-limit, or even higher) 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.

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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. Apr 25, 2021 at 19:34
  • I don't understand. (3/2)^4/4=81/64. Where is the discrepancy?
    – Emil
    Apr 14, 2023 at 6:36
  • @Emil I don't understand what you don't understand. Perhaps this will help: by approximating 5-limit just intonation, 53-tone ET provides approximations to both the Pythagorean major third of 81:64 and the 5-limit major third of 5:4. If you're playing a piece written in the normal 12-tone system, you might sometimes want one and sometimes the other. But a static set of labels would require you to decide beforehand whether E is 17/53 of an octave above C or 18/53. In other words, the labels effectively pick a subset of the 53 tones that is an unequal temperament of fewer tones.
    – phoog
    Apr 17, 2023 at 12:46
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    @Emil the classic example is the fifth between the major second and major sixth, which in 5-limit JI is 40:27 (= 80:54 instead of 3:2 = 81:54) or 30/53 of an octave instead of 31/53. To avoid this interval in C major you may need your A to be sometimes the 30th and sometimes the 31st step above C, or your D can be sometimes the 8th and sometimes the 9th step above C. (In JI, A above F is 5:4*4:3 = 5:3, while A above D is 3:2*9:8 = 27:16; the discrepancy of course is 81:80.)
    – phoog
    Apr 17, 2023 at 12:55
  • @phoog I did not know you did not mean the pythagorean major third. I also do not know what unequal temperament is.
    – Emil
    Apr 18, 2023 at 4:45
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You cannot. The reason is that 12-TET contains an entire highly developed theory that even at best cannot exceed itself in the sense that 12-TET is an approximation of something that can be clarified with higher-TET. All issues that are coming from having "errors" within 12-TET regarding pure ratios that would be required are annulled by the musical composition and by people simply adjusting to it. We are listening for almost 5 centuries 12-TET, at least the western hemisphere, and there is so much of musical material, that you cannot just sort of transpose any of it to higher-TET as much as you cannot switch to just intonation anything that is written for 12-TET. It would start sounding out of tune, either because you would try to clarify something somewhere just to ruin something else somewhere else in the composition or orchestration, or because people are used to all that you could correct the way it is. Locally, it is not unusual that a player would switch to just intonation with, for example, strong vibrato so A# and Bb are not quite the same with that particular instrument, yet this is very often spelled out or assumed or inferred. Mozart Piano Concerto No. 21 cannot be moved to anything out of what conductor may occasionally require for dynamic or other purposes. You cannot rewrite it into any other TET because you would write something else, variation on Piano Concerto No. 21 in 53-TET at best, but definitely not a correction of 12-TET version.

You can take acrylic colors and make a copy of Mona Lisa, but that is not Mona Lisa even if you make a perfect copy the way Leonardo meant it to be. It is a copy of Mona Lisa with acrylic colors. Are there errors in Mona Lisa painting? That does not have any sense even to ask because Mona Lisa is exactly how Leonardo wanted it to be, there was no a better version of it he was striving for but could not achieve it due to technical limitations.

Orchestration is always fully aware of the limitations of 12-TET, and it is one of the composer's tasks to avoid potential clashes for example in overtones of various instruments. Even if you compose everything in 72-TET you have the same set of problems, because no instrument is producing anything like pure overtones, all that we are actually doing by marking every tone within a specified frequency and duration is a blunt approximation. This is why the execution of each instrument is crucial. Most of the passages in any composition are trivial and you would believe that you could take a trumpet and produce those two tones that appear somewhere, right? Well, right there is the problem of how we think about music. It is a player of a specific instrument that hears all limitations of 12-TET right there at the rehearsals and bit by bit adjust all that so it would be perfect. Otherwise you would hear no difference between a school band and a professional full orchestra. It is all about ear.

In that sense, from musical perspective 12-TET is a tool, a well developed tool. If you want to use 72-TET, you can definitely use the same principles as in 12-TET, alike modulation and obtain a system that follows comparable mathematical modeling, but you cannot get Am7 and say that it can be made more precise in 72-TET, because there, with 72-TET, each of the replacements or choices is a chord on its own, that first of all you have to be able to hear and understand. If you cannot hear 72-TET, any replacement from what you got used to in 12-TET for Am7 would sound out of tune because you would not understand what you are hearing.

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    "We are listening for almost 5 centuries 12-TET": no we're not, at least not exclusively; equal temperament was not widely adopted until maybe 200 years ago, and close-to-equal temperaments around 300 years ago. Plenty of modern-day performers use unequal temperaments to perform music of those periods and earlier (as well as newer music, of course). Just as it's possible to take any piece of music written in the standard European system and map it to JI, it's possible to map it to 53-TET as an approximation of JI; you're just likely to find yourself using more than 12 of the 53 divisions.
    – phoog
    Apr 27, 2023 at 7:37
  • @phoog: Not sure what you have read but I said the same, that some people are using something else nowadays and that it is even acceptable, but our ears during 5 centuries of slow introduction has accustomed to 12-TET. You can use something else here and there, but you cannot rewrite something that is written FOR 12-TET completely into anything else. I am not sure what problem you had that you had to write an elaborate comment about nothing that is not said in the article. 53-TET and 12-TET cannot be mapped out of the box, when the composer had 12-TET in mind or just intonation, and that is it
    – Alex Peter
    May 8, 2023 at 6:02
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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.

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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} Apr 26, 2021 at 0:17
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I made a little cheat sheet of which intervals are closest to the 53-TET intervals, maybe you could see something in it interval names close to 53-tet

I think chords use just intonation ("classical" in table) but I do not have enough accuracy and precision to experiment which ones sound best together (yet) so I can't confirm.

I tried a bit to create chords with a pure sine generator and the classic intervals wobbled very little, while the 53tet approximation was a little bit wobbly, and the 53tet approx of pythagorean was very wobbly. If I could choose I would probably go with the exact ratios instead since they were most pleasant but I think 53tet can be nice as a mental model at least.

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    Sawtooth waves are a better approximation of musical tones than sine waves. They also make it easier to hear when two pitches are out of tune.
    – phoog
    Apr 13, 2023 at 12:38
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Actually I've seen keyboards tuned for (n>>12)TeTs. Even heard one in the Berlin museum for music instruments. There are multiple keyboards rolled in one, and as far as I could follow the interpreter played on the higher keyboards if the tonality was high into shaprs and conversely down with flats. I shudder to think what would have happened switching from F sharp to G flat though. As some previous answers mentioned, having a larger choice of notes enables to make distinctions between (say) F sharp and G flat. And which is the highest would depend on the temperament you choose to enjoy (yes, in meantone F sharp can be lower). To put it bluntly, there is no recipe. One has to decide what one is doing and use the multiple variants of pitches with some rationale. You may be interested in Jon Wild's paper on Vicentino's tuning (in JMT).

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  • I don't understand why this approach would seem right. All of the keyboards are equal tempered, so there is no clear reason to suspect that any one would sound better in certain keys (e.g. sharp keys) than others.
    – Edward
    Apr 13, 2023 at 17:41

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