Ups and Downs Notation for Rank-3 JI

Ups and downs can be used to notate certain edos and certain rank-2 temperaments. It can also be used in place of color notation for certain rank-3 just intonation subgroups such as 2.3.5 or 2.3.7 or 2.3.11. It can also notate certain rank-3 temperaments such as 2.3.5.11 with 81/80 tempered out, which looks identical to 2.3.11 JI.

For 2.3.5, ^1 = 81/80. For 2.3.7, ^1 = 64/63. But for 2.3.11, ^1 could be either 33/32 or 729/704. Here the former is used. This makes ^F flatter than vF#, and makes 243/242 a vvA1. The latter would make this important comma be a ^^d1, more confusing. Likewise for 2.3.13, ^1 should be 1053/1024 not 27/26, to make 512/507 be a vvA1 not a ^^d1.

The 2.3.11 subgroup

This formula converts a 2.3.11 monzo to an upped or downed pythagorean interval:

   (a, b, c) = c * (-5, 1, 1) + (a + 5c, b - c) = c ups (or -c downs) + pythagorean interval

The pythagorean intervals are named conventionally as M2, m3, etc. So 11/9 = (0, -2, 1) = ^m3. Knowing that ^1 is slightly less than half a sharp, the size of any interval can easily be estimated. These formulas convert an upped or downed pythagorean interval to a monzo:

   x ups + (a, b) = (a - 5x, b + x, x)
   x downs + (a, b) = (a + 5x, b - x, -x)

To add together two upped/downed intervals, just add up the pythagorean intervals as usual, then add in the ups and downs. Adding an interval to a note works the same way, as does finding the interval between two notes.

• ^m3 + M2 = ^4 (11/9 x 9/8 = 11/8)
• D + ^4 = ^G
• from D to vF# = vM3 = 27/22

Here's the 2.3.11 lattice, with a vertical step of ^1 or 33/32. Each row is a chain of 5ths. Each row is a different height. The top row is the dup row, next is the up row, next plain, next down, next dud.

^^F  ^^C  ^^G  ^^D  ^^A  ^^E  ^^B

 ^F   ^C   ^G   ^D   ^A   ^E   ^B

  F    C    G    D    A    E    B

 vF   vC   vG   vD   vA   vE   vB

vvF  vvC  vvG  vvD  vvA  vvE  vvB

Another version of the lattice, with vertical steps of ^4 or 11/8:

^^Eb ^^Bb ^^F  ^^C  ^^G  ^^D  ^^A

 ^Bb  ^F   ^C   ^G   ^D   ^A   ^E

  F    C    G    D    A    E    B

 vC   vG   vD   vA   vE   vB   ^F#

vvG  vvD  vvA  vvE  vvB  vvF# vvC#

Another version, with the 1/1 - 11/9 - 3/2 and 1/1 - 27/22 - 3/2 triads forming triangles. The vertical step from ^F down to vF# is 243/242 = vvA1.

^^Fb ^^Cb ^^Gb ^^Db ^^Ab ^^Eb ^^Bb

    ^Ab  ^Eb  ^Bb  ^F   ^C   ^G   ^D

  F    C    G    D    A    E    B

    vA   vE   vB   vF#  vC#  vG#  vD#

vvF# vvC# vvG# vvD# vvA# vvE# vvB#