# 10th-octave temperaments

10edo is notable for having close approximations of 15/14 to one step, and 13/8 to 7 steps. 10th-octave temperaments naturally occur between any equal divisions of the octave whose greatest common divisor is 10.

Temperaments discussed elsewhere include: decoid, deca, decal, decimetra, decistearn, and decavish.

## Linus (rank-3)

Tempering out the linus comma, 578509309952 / 576650390625 = [11 -10 -10 10 leads a number of regular temperaments, some of which are listed above. Linus rank three temperament can be described as the 130 & 190 & 270 temperament, which tempers out 9801/9800 and 391314/390625 in the 11-limit; 1001/1000, 4225/4224, and 4459/4455 in the 13-limit. The 1/10-octave period interval represents 15/14, three of which represents 16/13, and five of which represents both 99/70 and 140/99.

Subgroup: 2.3.5.7

Comma list: 578509309952/576650390625

Mapping: [10 0 0 -11], 0 1 0 1], 0 0 1 1]]

mapping generators: ~15/14 = 1\10, ~3/2 = 702.095, ~5/4 = 386.574

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 391314/390625

Mapping: [10 0 0 -11 4], 0 1 0 1 -1], 0 0 1 1 2]]

mapping generators: ~15/14 = 1\10, ~3/2 = 702.034, ~5/4 = 386.648

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 4225/4224, 4459/4455

Mapping: [10 0 0 -11 4 37], 0 1 0 1 -1 0], 0 0 1 1 2 0]]

mapping generators: ~15/14 = 1\10, ~3/2 = 701.931, ~5/4 = 386.473

## Neon

Neon tempers out the [21 60 -50 comma in the 5-limit. Neon extensions discussed elsewhere include deca, calcium, and zinc.

Subgroup: 2.3.5

Comma list: [21 60 -50

Mapping[10 4 9], 0 5 6]]

mapping generators: ~3125/2916 = 1\10, [10 29 -24 = 284.389

Optimal tuning (CTE): ~[10 29 -24 = 284.389