trihemitone

(Greek: "3 half-tones")

The Pythagorean minor 3rd, composed of 3 semitones, with ratio 32/27, ~294.1349974 cents.

The semitones are of two different sizes, because the trihemitone is composed of a tone and a limma; the tone in turn is composed of a limma and an apotome; thus the trihemitone equals 2 limmata and 1 apotome.

The trihemitone may also be found as a perfect 4th minus a tone.


In prime factor notation this interval is written 2⁵3^-3.

The trihemitone can be calculated thus by regular fractional math:

4   9       4   8       32
- ÷ -   =   - * -   =   --
3   8       3   9       27

or by vector addition:

    2   3

  [ 2 -1]          4/3
- [-3  2]       ÷  9/8
---------   =   -------
  [ 5 -3]         32/27

Below is a diagram illustrating these descriptions, on an approximate logarithmic scale:

                             ratio      vector
                                         2  3

                        /  A  1/1  -+- [ 0  0]   
                       /            |          \
  9/8   [-3  2] = tone              |           \
                       \            |            32/27  [ 5 -3] = trihemitone
                        /  G  9/8  -+- [-3  2]  /
256/243 [ 8 -5] = limma             |          /
                        \ F# 32/27  +  [ 5 -3] \
                                    |           \
                           F 81/64 -+- [-6  4]   9/8  [-3  2] = tone
                                    |           /
                           E  4/3  -+- [ 2 -1] /


[from Joe Monzo, JustMusic: A New Harmony]


see also

limma,
minor 3rd,
major 3rd,
Tutorial on ancient Greek tetrachord-theory

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updated:

2003.06.09 -- fixed error: link to "perfect 4th" was incorrectly written.
2002.10.05 -- fixed error: ratio had been given as 2187/2048 instead of 32/27.
2002.09.12 -- created

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