Meantone notation

Meantone notation is a standard notation based on fifths which also has an extended version.

Definition for 3-limit

The meantone notation can be defined in terms of one reference note and 3-limit intervals.

Any one note in the meantone notation can be assigned a reference frequency; the most common is to assign A4 to 440Hz, but other reference notes such as C4 being 264Hz are possible.

The ratios between base notes are defined as the following (think of prime factors 2 and 3 as being abstract factors rather than exact numbers):

C=F*3/2/2, G=C*3/2, D=G*3/2/2, A=D*3/2, E=A*3/2/2, B=E*3/2

The sharp modifier (♯) is a factor of 3*3*3*3*3*3*3/2/2/2/2/2/2/2/2/2/2/2 (3^7 * 2^-11), and flat modifier (♭) is the inverse factor.

The base note (any of CDEFGAB) followed by any non-negative integer amount of sharp or flat modifiers, is then followed by integer octave discriminator. Incrementing the octave discriminator is a factor of 2, and decrementing the octave discriminator is dividing by 2.

Replacing prime factors of 2 and 3 with any two respective constants defines a meantone system in terms of the two approximations of primes being generators, and all ratios between frequencies are defined. Furthermore, by defining one note as the reference frequency, all other frequencies can be derived by applying the appropriate frequency ratio, defined in the respective amounts of factors of approximations of primes 2 and 3.

Definition for 5-limit

This is the same as the 3-limit definition, but also defines an approximation of the prime 5 as being equal to 3*3*3*3/2/2/2/2 (with 2 and 3 themselves being approximate).

Circle of fifths

... 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## ...

Therefore going over right, it's +1 fifth, left is -1 fifth. Corresponds to the interval of 3/2 or 3/4, etc. . Octave equivalence assumed, as it is assumed that for absolute pitch the Meantone notation will be followed by the octave discriminator. Sharp is +7 fifths, flat is -7 fifths, they may stack (the extended version delays the stacking by using more symbols).

The 3-odd-limit in meantone notation:

4/3 — -1 fifths
3/2 — +1 fifths

Extending to 5-odd-limit requires tempering out 81/80, but when that is done, it proceeds naturally:

6/5 — -3 fifths
5/4 — +4 fifths
4/3 — -1 fifths
3/2 — +1 fifths
8/5 — -4 fifths
5/3 — +3 fifths

Intervals with nine demonstrate the tempered out effect:

10/9 — +2 fifths
 9/8 — +2 fifths
16/9 — -2 fifths
 9/5 — -2 fifths

Meantone systems generally blend the two whole tones, resulting in a mean tone.

Extending to full 7-odd-limit and 9-odd-limit requires tempering out 225/224 and 126/125, but when that is done, it proceeds naturally:

10/9 — + 2 fifths
 9/8 — + 2 fifths
 8/7 — -10 fifths
 7/6 — + 9 fifths
 6/5 — - 3 fifths
 5/4 — + 4 fifths
 9/7 — - 8 fifths
 4/3 — - 1 fifths
 7/5 — + 6 fifths
10/7 — - 6 fifths
 3/2 — + 1 fifths
14/9 — + 8 fifths
 8/5 — - 4 fifths
 5/3 — + 3 fifths
12/7 — - 9 fifths
 7/4 — +10 fifths
16/9 — - 2 fifths
 9/5 — - 2 fifths

To extend to 11-limit is that there are two conflicting systems. One called undecimal meantone where 99/98 must be tempered, and the other called undecimal meanpop where 385/384 must be tempered. See Meantone vs meanpop. They have the 11 differ by 31 fifths, so 31edo with a fifth of 18/31 octave or 696.77419354839 cents follows both in the patent val. In general sharper fifth meantones like in 12edo, 43edo, 74edo follow undecimal meantone, while the flatter fifth meantones like in 19edo, 50edo, 81edo follow undecimal meanpop. Some meantones like 55edo and 69edo don't follow septimal meantone in their patent val in the first place.

Meantone: 12/11 — -17 fifths 11/10 — +14 fifths 10/9 — + 2 fifths 9/8 — + 2 fifths 8/7 — -10 fifths 7/6 — + 9 fifths 6/5 — - 3 fifths 11/9 — +16 fifths 5/4 — + 4 fifths 14/11 — - 8 fifths 9/7 — - 8 fifths 4/3 — - 1 fifths 11/8 — +18 fifths 7/5 — + 6 fifths 10/7 — - 6 fifths 16/11 — -18 fifths 3/2 — + 1 fifths 14/9 — + 8 fifths 11/7 — + 8 fifths 8/5 — - 4 fifths 18/11 — -16 fifths 5/3 — + 3 fifths 12/7 — - 9 fifths 7/4 — +10 fifths 16/9 — - 2 fifths 9/5 — - 2 fifths 20/11 — -14 fifths 11/6 — +17 fifths

Meanpop: 12/11 — +14 fifths 11/10 — -17 fifths 10/9 — + 2 fifths 9/8 — + 2 fifths 8/7 — -10 fifths 7/6 — + 9 fifths 6/5 — - 3 fifths 11/9 — -15 fifths 5/4 — + 4 fifths 14/11 — +23 fifths 9/7 — - 8 fifths 4/3 — - 1 fifths 11/8 — -13 fifths 7/5 — + 6 fifths 10/7 — - 6 fifths 16/11 — +13 fifths 3/2 — + 1 fifths 14/9 — + 8 fifths 11/7 — -23 fifths 8/5 — - 4 fifths 18/11 — +15 fifths 5/3 — + 3 fifths 12/7 — - 9 fifths 7/4 — +10 fifths 16/9 — - 2 fifths 9/5 — - 2 fifths 20/11 — +17 fifths 11/6 — -14 fifths

When extending to 13-odd-limit and 15-odd-limit, 13-limit splits it into at least three conflicting systems. Again meantone with the 13 being +15 fifths (99/98, 105/104), grosstone with the 13 being -16 fifths (144/143, 99/98), and meanpop with the 13 being +15 fifths but different 11 than meantone (105/104, 144/143). Most whopping is 13/11 with three different mappings between the systems, yet all unified when 31edo is used. The reason for conflicting systems is that intervals more far away in the circle of fifths are more sensitive to variations in fifth size.

Meantone: 16/15 — - 5 fifths 15/14 — - 5 fifths 14/13 — - 5 fifths 13/12 — +14 fifths 12/11 — -17 fifths 11/10 — +14 fifths 10/9 — + 2 fifths 9/8 — + 2 fifths 8/7 — -10 fifths 15/13 — -10 fifths 7/6 — + 9 fifths 13/11 — - 3 fifths 6/5 — - 3 fifths 11/9 — +16 fifths 16/13 — -15 fifths 5/4 — + 4 fifths 14/11 — - 8 fifths 9/7 — - 8 fifths 13/10 — +11 fifths 4/3 — - 1 fifths 15/11 — -13 fifths 11/8 — +18 fifths 18/13 — -13 fifths 7/5 — + 6 fifths 10/7 — - 6 fifths 13/9 — +13 fifths 16/11 — -18 fifths 22/15 — +13 fifths 3/2 — + 1 fifths 20/13 — -11 fifths 14/9 — + 8 fifths 11/7 — + 8 fifths 8/5 — - 4 fifths 13/8 — +15 fifths 18/11 — -16 fifths 5/3 — + 3 fifths 22/13 — + 3 fifths 12/7 — - 9 fifths 26/15 — +10 fifths 7/4 — +10 fifths 16/9 — - 2 fifths 9/5 — - 2 fifths 20/11 — -14 fifths 11/6 — +17 fifths 24/13 — -14 fifths 13/7 — + 5 fifths 28/15 — + 5 fifths 15/8 — + 5 fifths

Grosstone: 16/15 — - 5 fifths 15/14 — - 5 fifths 14/13 — +26 fifths 13/12 — -17 fifths 12/11 — -17 fifths 11/10 — +14 fifths 10/9 — + 2 fifths 9/8 — + 2 fifths 8/7 — -10 fifths 15/13 — +21 fifths 7/6 — + 9 fifths 13/11 — -34 fifths 6/5 — - 3 fifths 11/9 — +16 fifths 16/13 — +16 fifths 5/4 — + 4 fifths 14/11 — - 8 fifths 9/7 — - 8 fifths 13/10 — -20 fifths 4/3 — - 1 fifths 15/11 — -13 fifths 11/8 — +18 fifths 18/13 — +18 fifths 7/5 — + 6 fifths 10/7 — - 6 fifths 13/9 — -18 fifths 16/11 — -18 fifths 22/15 — +13 fifths 3/2 — + 1 fifths 20/13 — +20 fifths 14/9 — + 8 fifths 11/7 — + 8 fifths 8/5 — - 4 fifths 13/8 — -16 fifths 18/11 — -16 fifths 5/3 — + 3 fifths 22/13 — +34 fifths 12/7 — - 9 fifths 26/15 — -21 fifths 7/4 — +10 fifths 16/9 — - 2 fifths 9/5 — - 2 fifths 20/11 — -14 fifths 11/6 — +17 fifths 24/13 — +17 fifths 13/7 — -26 fifths 28/15 — + 5 fifths 15/8 — + 5 fifths

Meanpop: 16/15 — - 5 fifths 15/14 — - 5 fifths 14/13 — - 5 fifths 13/12 — +14 fifths 12/11 — +14 fifths 11/10 — -17 fifths 10/9 — + 2 fifths 9/8 — + 2 fifths 8/7 — -10 fifths 15/13 — -10 fifths 7/6 — + 9 fifths 13/11 — +28 fifths 6/5 — - 3 fifths 11/9 — -15 fifths 16/13 — -15 fifths 5/4 — + 4 fifths 14/11 — +23 fifths 9/7 — - 8 fifths 13/10 — +11 fifths 4/3 — - 1 fifths 15/11 — +18 fifths 11/8 — -13 fifths 18/13 — -13 fifths 7/5 — + 6 fifths 10/7 — - 6 fifths 13/9 — +13 fifths 16/11 — +13 fifths 22/15 — -18 fifths 3/2 — + 1 fifths 20/13 — -11 fifths 14/9 — + 8 fifths 11/7 — -23 fifths 8/5 — - 4 fifths 13/8 — +15 fifths 18/11 — +15 fifths 5/3 — + 3 fifths 22/13 — -28 fifths 12/7 — - 9 fifths 26/15 — +10 fifths 7/4 — +10 fifths 16/9 — - 2 fifths 9/5 — - 2 fifths 20/11 — +17 fifths 11/6 — -14 fifths 24/13 — -14 fifths 13/7 — + 5 fifths 28/15 — + 5 fifths 15/8 — + 5 fifths