So I always wanted to compose a song in 31-TET inspired by a song I heard on musescore some months ago, the thing is that I didnt know how to notate in 31-TET, so obviously i want to learn.


Another way of looking at it is the distinction between diatonic and chromatic semitones. Obviously 12-EDO* doesn't distinguish between them, but 31-EDO (and 19-EDO) does.

A diatonic semitone ("between tones") is between two note names in the standard major scale, represented by B-C and E-F, and also by implication between F♯-G, A-B♭, etc.

A chromatic semitone ("colour", i.e. black/white on a keyboard), is what single accidentals do on the same note letter. C-C♯, E♭-E, etc.

The fundamental rule is: diatonic semitone + chromatic semitone = whole tone - simply think about C-C♯-D (or C-D♭-D as well).

Along with the fact that 31-EDO has a whole tone = 5/31 steps, diatonic semitone = 3/31 steps (5x5+2x3=31), this is enough to derive all notation.

A single accidental mark is therefore 5 - 3 = 2 steps, so the 5 steps between C and D can be written as C-D𝄫-C♯-D♭-C𝄪-D. Between a diatonic semitone, we have B-C♭-B♯-C.

With the accidental being exactly 2 steps, the concept of a half-sharp or flat makes perfect sense, so it may also be written B-Bhalf sharp-Chalf flat-C-Chalf sharp-C♯-D♭-Dhalf flat-D.

Normal major/minor scales work the same way as in any meantone system, as long as you keep the fully spelled out accidentals when double or triple sharps/flats exist. If you play around, you can find the enharmonic equivalences of 31-EDO, for example B♯ = Chalf flat = D𝄫♭ (compare B♯ = C = D𝄫).

*: "equal divisions of the octave" instead of just "equal temperament". See Bohlen-Pierce scale for why.

PS (comparison):

  • 31-EDO has diatonic:chromatic = 3:2
  • 19-EDO has 2:1 (B-B♯/C♭-C-C♯-D♭-D; 5x3+2x2=19)
  • 12-EDO has 1:1 (B/C♭-B♯/C-C♯/D♭-D; 5x2+2x1=12)

There are two alternative notations for 31-TET. One of them uses double sharps and double flats:

Note name A B♭♭ A♯ B♭ A♯♯ B C♭ B♯ C D♭♭ C♯ D♭ C♯♯ D E♭♭ D♯ E♭ D♯♯ E F♭ E♯ F G♭♭ F♯ G♭ F♯♯ G A♭♭ G♯ A♭ G♯♯ A
Note (cents) 0 39 77 116 155 194 232 271 310 348 387 426 465 503 542 581 619 658 697 735 774 813 852 890 929 968 1006 1045 1084 1123 1161 1200

The other one uses half sharps and half flats. There's more detail at the Wikipedia page for 31-TET

  • This is really useful, thank you, but i still wonder how enharmonics and key signatures work in 31-TET
    – arcioko
    Jun 5 at 21:47
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    @arcioko The key signatures are all explained in the Wikipedia article.
    – PiedPiper
    Jun 5 at 21:49

Generally, you don't notate for 31-edo. Recall that we don't really notate for 12-edo either, otherwise there would be no such thing as enharmonics.

Rather, as long as you intend to use it for tonal 5-limit music, you should mostly notate for meantone tuning, which can apply to both 12-edo and 31-edo. I.e., you notate in diatonic tonality, just as you're used to.

That shifts the question to: how do these notes actually map to 31-edo? But because it's meantone, that's actually straightforward: you can construct everything from the knowledge that a perfect fifth has 18 steps and a major third 10. For example, to figure out how many steps there are between C and D♯, you note that you can stack a perfect fifth and two major thirds to get from one to the other, i.e. 18+10+10 = 38 steps, which mod 31 is 7 steps.

Of course, you do not get the same enharmonic equivalences as you would in 12-edo. (Note that the minor third has 8 steps in 31-edo, unlike our augmented second C-D♯!)

Another matter is when you want to make use of the 7-limit or odd-11-limit capabilities that 31-edo offers. This could be expressed with quarter-flats and -sharps, or you can again base it on just intonation, using a notation like Ben Johnston's.

  • +1 in general. Also, specifically in 31-EDO the harmonic 7th (which is extremely close) is 5 whole tones, so that particular equivalence to 10 fifths can be used to reach it, the same way 4 fifths reaches the major third. I wouldn't really consider 9.4 cents in 38.7 error to allow use of the 11th harmonic; after all the 12-EDO harmonic 7th is is 31/100 cents bad.
    – obscurans
    Jun 6 at 11:28
  • Fair point about 31-EDO, which means a division of the octave into 31 equal intervals but does not imply any harmonic meaning of any of them. But the OP said 31-TET, not 31-EDO, and so is presumably considering a particular interpretation of 31-EDO, no? Seeing as that interpretation which has the best fifths and thirds is the meantone one, I agree with your 2nd paragraph.
    – Rosie F
    Jun 7 at 17:17
  • @RosieF right. Actually the first half of my answer is rather about TET than EDO... though I tend to just always say EDO, regardless of how it's used. Jun 7 at 19:26

You just notate with the usual manner of notation. The whole point of TET-31 is that your normal scale is represented by 5 whole steps of 5/31th of an octave and 2 half steps of 3/31th of an octave. An accidental causes a change of 2/31th of an octave, a double accidental twice that (so an accidental is "less" than a "natural" half step, and a natural whole step is larger than the difference by a double accidental, but smaller than the difference by two natural half steps).

The main difference is that there are no enharmonics: accidentals don't go as far as the natural distance.

Notation/transposition work in a rather mathematically pleasing coherent and regular manner: any scale interval of a particular kind (major/minor xth) is identical in width to any other of the same kind. Standard notation is fully adequate for the representation, and yet everything is built from 31 rather than 12 notes in an octave, all of them being accessible to notation.

  • But suppose you want to approximate just intonation with a major third of 9/31 of an octave? How do you notate that?
    – phoog
    Jun 5 at 23:44
  • 1
    @phoog that's not a major third, it's a neutral third. How you'd notate that depends on what “meaning” this neutral third has – if it's the difference between two stacked major thirds and a diminished octave, that would correspond to a triple sharp sign. If you mean an undecimal neutral third, you'd probably want to write it with a half-flat sign. Jun 5 at 23:51
  • @leftaroundabout I think I was confusing 31-tone ET and 53-tone ET.
    – phoog
    Jun 6 at 1:50
  • 1
    "there are no enharmonics" is always wrong in an equally divided scale. You get different enharmonics. At the very least, C-31sharps = C regardless of what a sharp represents in 31-EDO.
    – obscurans
    Jun 6 at 11:34
  • 3
    @phoog remember also that 53-EDO is no longer meantone, and therefore 4 fifths is not the major third. If you treat note names as Pythagorean, C-E-G is not a major triad. Notation is a lot more annoying with the syntonic comma floating around.
    – obscurans
    Jun 6 at 11:40

There are various different conventions. One of the more common ones is to incorporate symbols for

  • quarter sharp: quarter sharp
  • quarter flat: quarter flat
  • three-quarter sharp: three-quarter sharp
  • three-quarter flat: three-quarter flat

(Images are screenshots from MuseScore)

Of course, to compose in any "style" (using the term loosely here), it's highly instructive to study the scores of existing pieces. That's particularly true in learning how to notate 31-tone music.

The Huygens-Fokker Foundation: Centre for Microtonal Music has a list of 31-tone compositions. The scores are likely all under copyright (short of a composer deciding to make one public domain), but some might be available at a university library. Also, articles about those composers or their compositions may include excerpts that would provide notation examples. For instance, the article "Six American Composers on Nonstandard Tunings" by Douglas Keislar (Perspectives of New Music Vol. 29, No. 1 [Winter, 1991], pp. 176-211) includes an excerpt from Joel Mandelbaum's Four Miniatures for Archiphone, which demonstrates with 31-tone scale notation.

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