(Greek: "off-cut", plural: apotomai)
The pythagorean chromatic semitone, 2187/2048 [ratio], 114 cents, the difference between 9/8 [ratio, the whole-tone] and the leimma or diatonic semitone, 256/243 [ratio], 90 cents.
In prime factor notation the apotome is written 2-1137; thus, it has the 2,3-monzo [-11 7>.
The apotome can be calculated thus by regular fractional math:
9 256 9 243 2187 - ÷ --- = - * --- = ---- 8 243 8 256 2048
or by vector addition:
2,3-monzo [ -3 2] 9/8 - [ 8 -5] ÷ 256/243 ---------- = ---------- [-11 7] 2187/2048
Below is a diagram illustrating this description, on an approximate logarithmic scale:
ratio monzo
2 3
A 1/1 -+- [ 0 0]
/ | \
/ | \
/ | 32/27 [ 5 -3] = trihemitone
81/64 [-6 4] = ditone G 9/8 -+- [-3 2] /
\ | /
\ F# 32/27 + [ 5 -3] \
\ | \
/ F 81/64 -+- [-6 4] 9/8 [-3 2] = tone
256/243 [ 8 -5] = limma | /
\ E 4/3 -+- [ 2 -1] /
. . . . . . . . . . . . . . .
F# 32/27 + [ 5 -3] \
| 2187/2048 [-11 7] = apotome
F 81/64 -+- [-6 4] /
|
E 4/3 -+- [ 2 -1]
A more accurate logarithmic value for the apotome is ~113.6850061 cents.
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