# 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.
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^^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#

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