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.





[from John Chalmers, Divisions of the Tetrachord]

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


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

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updated:

2002.09.12 - added diagrams and math illustrations
1999.12.19

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