15/14 equal-step tuning

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15/14 equal temperament (AS15/14) is an equal multiplication of 15/14-wide semitone. The step size is about 119.4428 cents, corresponding to 10.0466 EDO.

Using a JI 15/14-wide semitone as the basis of an equal temperament tuning results in an interesting non-octave tuning. As every interval is a multiple of 15/14, the resultant tuning would be a subset of 7-limit just intonation. This can be also viewed as generating a subset of marvel, decoid, or septidiasemi temperament.

Lookalikes: 16ED3, 26ED6, 10EDO

Intervals as 7-limit ratios

Ratio Cents
(15/14)0 1/1 0.0000
(15/14)1 15/14 119.4428
(15/14)2 225/196 238.8856
(15/14)3 3375/2744 358.3284
(15/14)4 50625/38416 477.7712
(15/14)5 759375/537824 597.2140
(15/14)6 11390625/7529536 716.6568
(15/14)7 170859375/105413504 836.0997
(15/14)8 2562890625/1475789056 955.5425
(15/14)9 38443359375/20661046784 1074.9853
(15/14)10 576650390625/289254654976 1194.4281
(15/14)11 8649755859375/4049565169664 1313.8709

Related temperament

15/14 equal temperament is related to various regular temperaments including marvel, septidiasemi, subsedia, and linus temperaments.

Marvel (22&31&41)

Marvel temperament equates 16/15, 77/72, and 15/14 as a diatonic semitone, tempering out 225/224 and 385/384 in the 11-limit. Rank-two temperaments which temper out 225/224 and 385/384 include miracle, orwell, magic, meanpop, wizard, slender, garibaldi, catakleismic, septimin, and pajarous.

Septidiasemi (10&171)

15/14 equal temperament is closely related to the septidiasemi temperament, which tempers out 2401/2400 and 2152828125/2147483648 in the 7-limit. This temperament is supported by 10, 161, 171, 181, 332, 352, and 503 EDOs.

Subsedia (10&111)

The subsedia temperament is a temperament for the 17-limit, which tempers out 256/255, 352/351, 442/441, 540/539, and 715/714. The generator is ~15/14 = 118.97¢, 16 of them equals ~3/1, and 45 of them equals ~22/1.

Linus temperaments

Tempering out the linus comma, 578509309952 / 576650390625 = [11 -10 -10 10 leads a number of regular temperaments including decoid, deca, decal, decimetra, decistearn, and decavish. These have a period of 1/10 octave. 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.


7-limit linus (130&190&270)
Comma: 578509309952/576650390625
Mapping: [10 0 0 -11], 0 1 0 1], 0 0 1 1]]
POTE generators: ~15/14 = 120.000, ~3/2 = 702.095, ~5/4 = 386.574
Vals: 10, 40, 50, 80, 130, 140, 190, 270, 1270, 1400, 1540, 1670, 1810, 1940

11-limit linus (130&190&270)
Comma list: 9801/9800, 391314/390625
Mapping: [10 0 0 -11 4], 0 1 0 1 -1], 0 0 1 1 2]]
POTE generators: ~15/14 = 120.000, ~3/2 = 702.034, ~5/4 = 386.648
Vals: 10e, 50, 80, 130, 190, 270, 860, 940, 1130, 1400, 1670

13-limit linus (130&190&270)
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]]
POTE generators: ~15/14 = 120.000, ~3/2 = 701.931, ~5/4 = 386.473
Vals: 10e, 50, 80, 130, 190, 270, 590, 730, 860, 1130