Extended meantone notation

Extending the chain of fifths

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
^ — sharpen by meantone diesis, 12 fifths down
v — flatten by meantone diesis, 12 fifths up
+ — sharpen by meantone kleisma, 19 fifths up
- — flatten by meantone kleisma, 19 fifths down```

A diesis plus a kleisma, added together, equals a meantone chromatic semitone. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone.

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#^ and B##- are equal to C
• C+^ is equal to C# (because the two semisharps add up)
• Dbbv and Dbbb- 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, 245/243, 3125/3072 or 15625/15552 assuming septimal meantone. An octave is made of 19 dieses and 12 kleisma.

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 Bbv
7/6 — D#, or Ebv
7/5 — F#, or Gbv
8/7 — Ebb, or D^
12/7 — Bbb, or A^
10/7 — Gb, or F#^

9/8 — D
9/5 — Bb
9/7 — Fb, or E^
16/9 — Bb
10/9 — D
14/9 — G#, or Abv```

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone. 11–limit and 13–limit notation can vary (see meantone vs meanpop).

Extended meantone notation was created as a way to notate 43edo with only a base letter with one symbol.

True half-sharps and half-flats

If true half-sharps and half-flats are desired, which exactly bisect the chromatic semitone, this mathematically implies that the meantone fifth is split in half. This creates a 2D new tuning system which is generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.

While Middle Eastern maqam music is far too complex in real life to be represented by either of these temperaments (although one can certainly try, see Maqamat in maqamic temperament), it is commonly notated using half sharps and half flats. If we take these to be exactly equal to 1/2 of a chromatic semitone, then mathematically, this notation system results in a 2D lattice that is generated by a neutral third and an octave. If we furthermore decide that C# and Db are enharmonically equal, this 2D lattice collapses further to the 1D lattice of 24edo, which has sometimes been suggested as a simplified framework for maqam music. But the usual written notation typically lets you notate them as two distinct entities if you want, so if we instead decide to leave them unequal, we get the 2D lattice above.

The chain-of-neutral thirds tuning system is not a true "temperament," because it is contorted: the neutral third does not have any JI interval mapping to it in the 7-limit. But, if we go to the 11-limit, and add 121/120 to the kernel, we obtain mohajira, an exceptionally good 11-limit temperament. The neutral third becomes equal to 11/9, and two of them make 3/2. Furthermore, if you take a minor third and flatten it by a half-flat, you obtain a good representation of 7/6. Conversely if you take a major third and sharpen it by a half-sharp, you obtain a good representation for 9/7. 31edo is a very good tuning for mohajira.

Although mohajira may not be a great tuning to reflect the way maqam music is played in practice, which often uses multiple unequal neutral thirds and exhibits significant regional variations, it is a highly interesting regular temperament of its own merit, and deserves further study.