(Greek: "3 half-tones")
The pythagorean minor 3rd musical interval, 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; thus, its 2,3-monzo is [5 -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 monzo (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]
The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.