How to notate scores when the keys don't admit a heptatonic scale?

Question

I was playing with microtonality. Specifically, equal temperaments other than 12edo. Since the point of equal temperaments is versatility, I wanted them to admit a notion of key signatures similar to those of the standard 12edo. My requirements were:

• The white keys shall represent the default scale.

• The perfect fifth (C-to-G) shall approximate Pythagorean, and the white keys shall be along the generated circle of fifths.

• No monotonic scales.

• Sharps (resp. flats) shall be positive (resp. negative).

• One unit of sharp (that is, for example, half-sharp if appropriate) on an appropriate white key shall transpose the scale (whatever the default scale was).

After some investigation, it turns out many of EDOs don't admit a heptatonic scale. For 22edo as an example, it doesn't admit a heptatonic scale, but a pentatonic scale. Picking the pentatonic major mode as an example, some examples of key signatures are (+/- for one step, #/b for two steps):

(White key position in steps: C=0, D=4, E=8, G=13, A=17)
C pentatonic major: C, D, E, G, A
C# pentatonic major: C#, D#, E#, G#, A#
D pentatonic major: D, E, G-, A, C-
E pentatonic major: E, G-, A-, C-, D-
E+ pentatonic major: E+, G, A, C, D
G pentatonic major: G, A, C-, D, E
A pentatonic major: A, C-, D-, E, G-

For 41edo as another example, it doesn't admit a heptatonic scale either, but a dodecatonic scale. That is, we need to take extra characters for some white keys. Let me notate them by Cyrillic (+/- for one step):

(White key position in steps: C=0, Д=3, D=7, Э=10, E=14, F=17, Ф=21, G=24, Я=27, A=31, Б=34, B=38)
C chromatic: C, Д, D, Э, E, F, Ф, G, Я, A, Б, B
Ф chromatic: Ф, G, Я+, A, Б+, B, C+, Д+, D, Э+, E, F+
Ф+ chromatic: Ф+, G+, Я++, A+, Б++, B+, C++, Д++, D+, Э++, E+, F++
F chromatic: F, Ф-, G, Я, A, Б, B, C, Д, D, Э, E
G chromatic: G, Я, A, Б, B, C, Д+, D, Э, E, F, Ф

So far, what I've defined are the white key names, the scale they represent, and how key signatures work for them. This arises some questions. The standard five-lines score is designed for the major scale, which is heptatonic. Would it be appropriate to modify the number of lines, so pentatonic/dodecatonic/whatever scale could be notated over? If so, how would clefs work?

Answer 1

Your question is based on a partial misconception. The five-line staff is not based on the major scale (it predates major/minor tonality) and has more to do with readability: fewer lines admit too few notes; more lines are visually confusing.

The staff lines and spaces represent white keys. As such, they can retain that meaning, and the five-line staff, within any system you develop. The clefs just assign a particular line or space to a particular pitch. So you could create clefs that correspond to your "new" white keys.

This staff, new clefs, plus appropriately defined accidentals, should allow for your scales to be fully notated in a way that looks like the standard heptatonic scales: that is, one line or space per note-name.

Answer 2

It took me a bit to figure out what you mean by "admit a heptatonic scale". The sharp sign results in going up 7 5ths and down 4 octaves. Your last requirement ("One unit of sharp...") requires the sharp to equal one edostep, correct? Assuming heptatonicism, it only does so for the sharp-1 edos, which are edos 5, 12, 19, 26, 33, 40 and 47. Whereas 22edo is a sharp-3 edo, because from C to C# is 3 edosteps. And 41edo is a sharp-4 edo.

The Stern-Brocot tree (with improper fractions such as 9/15 included) in the region around 4/7 and 3/5 makes it easy to find an edo's sharpness. The sharp-1 edos define a sharp-1 line. The sharp-2 line is roughly parallel to the sharp-1 line.

The S-B tree also tells you what "framework" allows your last requirement to be met. 22edo's 5th is 13 edosteps, and one of 13/22's S-B ancestors is 3/5, which implies a pentatonic framework. Likewise one of 24/41's ancestors is 7/12, implying a dodecatonic framework.

You can indeed notate 22edo pentatonically and 41edo dodecatonically. You can change the number of lines in the staff from 5 to 4 or 7 or whatever. You can use the standard clefs, which are actually just giant letters. The treble clef is a G, and the low G note is circled. The bass clef is an F, and the high F lies between the two dots.

But there are many edos that cannot be notated by only changing the framework. Consider edos 15, 21, 24 and 34. These are improper fractions on the Stern-Brocot tree. The circle of 5ths does not include every note, and microtonal accidentals are absolutely unavoidable.

In addition there are edos for which the appropriate framework will be too large to be practical. For example, 49edo requires a 22-note framework. 59 requires 27 notes, but there are only 26 letters!

So alternate frameworks are not a complete answer. Might as well stick with the heptatonic framework and standard staffs and clefs. Then the years that musicians have spent learning to effortlessly sight-read will not be for naught!

So assuming the heptatonic framework, which microtonal accidentals to use? This paper explains how to notate edos 5-72 using only two new accidentals: https://tallkite.com/misc_files/notation%20guide%20for%20edos%205-72.pdf. 22-edo would be notated as

C, ^C/Db, vC#/^Db, C#/vD, D, ^D/Eb, vD#/^Eb, D#/vE, E, F, ^F/Gb, vF#/^Gb, F#/vG, G, ^G/Ab, vG#/^Ab, G#/vA, A, ^A/Bb, vA#/^Bb, A#/vB, B, C