# The definition of scales in 53edo

The common division of the octave, 12edo, doesn't have a problem to define sharps and flats:

[C] [C♯/D♭] [D] [D♯/E♭] [E] [F] [F♯/G♭] [G] [G♯/A♭] [A] [A♯/B♭] [B]

Bold texts indicate black keys.

19edo doesn't, either:

[C] [C♯] [D♭] [D] [D♯] [E♭] [E] [E♯/F♭] [F] [F♯] [G♭] [G] [G♯] [A♭] [A] [A♯] [B♭] [B] [B♯/C♭]

It looks like white keys are to approximate dedicated just ratios:

    C = 1/1
D = 9/8
E = 5/4
F = 4/3
G = 3/2
A = 5/3
B = 15/8


And it looks like sharps and flats are to approximate the chromatic semitone, just ratio 25/24.

This logic applies to 12edo and 19edo, but let's apply it to 53edo. 53edo would have the following names for keys:

[C] [C+] [C++] [C♯] [C♯+] [D♭-] [D♭] [D--] [D-]
[D] [D+] [D++] [D♯] [D♯+/E♭-] [E♭] [E--] [E-]
[E] [E+] [E++/F♭] [E♯/F--] [F-]
[F] [F+] [F++] [F♯] [F♯+] [G♭-] [G♭] [G--] [G-]
[G] [G+] [G++] [G♯] [G♯+/A♭-] [A♭] [A--] [A-]
[A] [A+] [A++] [A♯] [A♯+] [B♭-] [B♭] [B--] [B-]
[B] [B+] [B++/C♭] [B♯/C--] [C-]

The keys represented with italic texts would have a hypothetical 3rd color (say, green).

Though these names fit C Major scale and C natural minor scale, they won't to scales based on other keys. Say, G Major scale. The usual definition of G Major scale is:

    G A B C D E F♯


But shifting the C Major scale to G actually gives:

    G A+ B C D E F♯+


Furthermore, since this definition of names of keys gives accidentals smaller than ♯ and ♭, the usual definition of scales cannot be applied to scales based on "green" keys.

So my questions in summary are:

1. How did sheet musics for instruments in 53edo overcome this problem?

2. If I choose the usual definition of scales, would that mean I have to choose the tonic according to musical key characteristics by Schubart?

3. If I shift the C Major/minor scale to another note, would it be too hard to play on a hypothetical 53edo keyboard due to the "green" keys?

4. Would it be inappropriate for a scale to have a "green" key as the tonic?

• In an effort to help people interested in alternative tunings beyond 12 EDO understand where all the notes lie, I've created scale diagrams for the four most important non-12EDO historical tunings (1/3, 1/4, and 1/6-comma meantone, plus Pythagorean tuning) using their respective EDOs (19, 31, 55, and 53) for standardisation. This should work for the vast majority of music written in 12 EDO. Hope this helps.
– user59346
May 27, 2023 at 5:38

Basically the problem is that the standard notation system always implicitly assumes a meantone temperament, i.e. that two whole steps sum to a major third or equivalently, the whole scale can be constructed by going around the circle of fifths. That is true in 12-edo, 17-edo, 19-edo and 31-edo, but not in most other tunings.

In a non-meantone tuning like 53, or more practically 22-edo, what you find is that the major third splits into two unequal divisions: the major tone approximating 9:8, and minor tone 10:9. The difference between them is called syntonic comma, and it's what is meant by the + symbol in your 53-edo listing (as well as in just-intonation scores by e.g. Ben Johnston).

How did sheet musics for instruments in 53edo overcome this problem?

Well, one option are precisely those + and − symbols. Another option is to stubbornly follow Pythagorean terminology, but that would look very awkward for 5-limit music.

The syntonic comma could also just be left for the performer to figure out. That's basically what always happens when free-intonation instruments play 5-limit music.

If I choose the usual definition of scales, would that mean I have to choose the tonic according to musical key characteristics by Schubart?

No. Those key characteristics are linked to completely non-EDO tunings. You could certainly take a WerckmeisterⅡ-ish subset of 53-edo or so, but then you might as well just use Werckmeister Ⅱ.

The point of equal temperaments is that you can modulate around without changing the characteristic.

If I shift the C Major/minor scale to another note, would it be too hard to play on a hypothetical 53edo keyboard due to the "green" keys?

Well, a 53-edo keyboard is pretty much too hard to play anyway, regardless of key. Practically, what you should probably do is use a smaller keyboard that only can access the key-relevant notes at a given time, and then add a modulation feature in some other way.

Would it be inappropriate for a scale to have a "green" key as the tonic?

Well, it would certainly mean that most musicians wouldn't be able to play your piece. But that's true for any more involved microtonal stuff. Whether it's appropriate you need to decide yourself. You should certainly not do it without a clear musical reason why you want to.