15/14ths equal temperament

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15/14 equal temperament

Using a Just 15/14 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.

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, and linus temperaments (decoid, deca, decal).

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 10edo, 161edo, 171edo, 181edo, 332edo, 352edo, and 503edo.

Linus temperaments

Tempering out the linus comma, 578509309952 / 576650390625 = |11 -10 -10 10> leads a number of regular temperaments including decoid, deca, and decal. 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.

7-limit linus (130&190&270)
Comma: 578509309952/576650390625

POTE generators: ~15/14 = 120.000, ~3/2 = 702.095, ~5/4 = 386.574

Map: [<10 0 0 -11|, <0 1 0 1|, <0 0 1 1|]

EDOs: 10, 50, 60, 80, 130, 140, 190, 270, 400, 410

11-limit linus (130&190&270)
Commas: 9801/9800, 391314/390625

POTE generators: ~15/14 = 120.000, ~3/2 = 702.034, ~5/4 = 386.648

Map: [<10 0 0 -11 4|, <0 1 0 1 -1|, <0 0 1 1 2|]

EDOs: 80, 130, 140, 190, 270, 320, 400, 460

13-limit linus (130&190&270)
Commas: 1001/1000, 4225/4224, 4459/4455

POTE generators: ~15/14 = 120.000, ~3/2 = 701.931, ~5/4 = 386.473

Map: [<10 0 0 -11 4 37|, <0 1 0 1 -1 0|, <0 0 1 1 2 0|]

EDOs: 80, 130, 140, 190, 270, 320, 400, 460