# trihemitone

(Greek: "3 half-tones")

[Joe Monzo]

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
```

```    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]
```
. . . . . . . . .

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