Extended meantone notation

Standard meantone notation uses 7 base note letters, plus sharps and flats.

... Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# F## C## G## D## A## ...

However, when transferred onto a 31edo scale, it looks like this:

C Dbb C# Db C## D Ebb D# Eb D## E Fb E# F Gbb F# Gb F## G Abb G# Ab G## A Bbb A# Bb A## B Cb B# C

Note that the base note letters alternate.

The 31edo sharp can be split in half, so in 31edo this is solved by semisharps and semiflats, sometimes notated with ups and downs.

The meantone circle of fifths however, has no single semisharp/semiflat. In extended meantone notation, a sharp is split into 2 different parts that can be added to produce a sharp.

# — sharpen by meantone chromatic semitone, 7 fifths up
b — flatten by meantone chromatic semitone, 7 fifths down
Y — sharpen by meantone diesis, 12 fifths down
Z — flatten by meantone diesis, 12 fifths up
y — sharpen by meantone kleisma, 19 fifths up
z — flatten by meantone kleisma, 19 fifths down

Unlike a single semisharp/semiflat, this can be generalized to other meantone tunings:

• 7edo (chromatic semitone is tempered out, diesis is positive and kleisma is negative)
• 12edo (chromatic semitone is tempered equal to kleisma, diesis is tempered out)
• 19edo (chromatic semitone is tempered equal to diesis, kleisma is tempered out)
• 26edo (diesis is larger than chromatic semitone, kleisma is negative)
• 31edo (kleisma is tempered equal to diesis)
• 43edo (diesis is smaller than kleisma)
• 50edo (diesis is larger than kleisma)

There are of course notational equivalences.

• B#Y and B##z are equal to C
• CyY is equal to C# (because the two semisharps add up)
• DbbZ and Dbbby are equal to C

The meantone diesis can be considered to be 36/35, 50/49 64/63, or 128/125, while the meantone kleisma is 49/48 or 15625/15552 assuming septimal meantone.

9–odd–limit intervals and their notation relative to C:

 1/1 — C

 3/2 — G
 4/3 — F

 5/4 — E
 5/3 — A
 8/5 — Ab
 6/5 — Eb

 7/4 — A#, or BbZ
 7/6 — D#, or EbZ
 7/5 — F#, or GbZ
 8/7 — Ebb, or DY
12/7 — Bbb, or AY
10/7 — Gb, or F#Y

 9/8 — D
 9/5 — Bb
 9/7 — Fb, or EY
16/9 — Bb
10/9 — D
14/9 — G#, or AbZ

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit and 13–limit notation can vary.

Y, Z, y and z is placeholder notation and should be replaced with better notation. It should be in ASCII, so that it can be easily written on a keyboard, like # and b are.