# limma / leimma

(Greek: "remnant", plural: leimmata, limmata)

###### 1. the ancient pythagorean limma
[John Chalmers, Divisions of the Tetrachord]

The Pythagorean diatonic semitone, the interval left after subtracting two Whole Tones from the Perfect Fourth. It has a ratio of 256/243 and 90 cents.

. . . . . . . . .
[Joe Monzo]

In prime-factor notation this interval is written 283-5 -- thus, the 2,3-monzo is [8 -5, >. Two whole-tones form the pythagorean interval of the ditone, so Chalmers's description of the limma can be calculated thus by regular fractional math:

```
4   81       4   64       256
- ÷ --   =   - * --   =   ---
3   64       3   81       243
```

or by vector addition:

```    2  3

[ 2 -1]           4/3
- [-6  4]       ÷  81/64
---------   =   ---------
[ 8 -5]         256/243
```

Below is a schematic diagram illustrating this description, on an approximate logarithmic scale:

```ratio        monzo
2  3

A  1/1  -+- [ 0  0, >
|              \
|               \
|                \
G  9/8  -+- [-3  2, >      81/64   [-6  4]  =  ditone
|                /
|               /
|              /
F 81/64 -+- [-6  4, >    \
|                256/243  [ 8 -5]  =  limma
E  4/3  -+- [ 2 -1, >    /
```

A more accurate logarithmic value for the limma is ~90.22499567 cents.

. . . . . . . . .
###### 2. the 5-limit JI limma of Woolhouse
[Joe Monzo]

The term used by W. S. B. Woolhouse (in his Essay on Musical Intervals, Harmonics, and Temperament) to refer 16/15 [= ~ 111.731 cents], the ratio of the 5-limit diatonic semitone.

As the size of this 16/15 interval resembles another Pythagorean semitone -- the apotome -- much more closely, Woolhouse perhaps should have used that name instead. Apparently he based his terminology on the function of this semitone, for the apotome is the Pythagorean chromatic semitone while the limma is the Pythagorean diatonic semitone.

. . . . . . . . .

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