# apotome

(Greek: "off-cut", plural: apotomai)

[John Chalmers, Divsions of the Tetrachord]

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.

. . . . . . . . .
[Joe Monzo]

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
```

``` 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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