## Definitions of tuning terms © 1998 by Joseph L. Monzo All definitions by Joe Monzo unless otherwise cited

apotome

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

The Pythagorean chromatic semitone, 2187/2048, 114 cents, the difference between 9/8 and the Leimma or diatonic semitone, 256 /243, 90 cents.

. . . . . . . . . . . . . . . . . . . .

In prime factor notation this interval is written 2-1137.

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

[ -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      vector
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 it is ~113.6850061 cents.

[from Joe Monzo, JustMusic: A New Harmony]

see also

limma,
anomaly,
diesis,
comma,
kleisma,
skhisma,
5-limit intervals, 100 cents and under
Tutorial on ancient Greek tetrachord-theory

updated:

2002.09.12 - added diagrams and math illustrations
1999.12.19

 (to download a zip file of the entire Dictionary, click here)

 For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here

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