1ed15/14
1 equal division of 15/14 (1ed15/14), also known as ambitonal sequence of 15/14 (AS15/14) or 15/14 equal-step tuning, is an equal multiplication of 15/14-wide semitone. The step size is about 119.4428 cents, corresponding to 10.0466edo.
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: 16edt, 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.
Harmonics
1ed15/14 offers a good approximation of the no-5s 17-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.6 | +9.1 | -39.1 | -24.4 | +29.2 | -21.1 | -7.8 | +38.5 | -53.3 | +23.1 | +27.1 |
| Relative (%) | -4.7 | +7.6 | -32.8 | -20.5 | +24.4 | -17.7 | -6.5 | +32.3 | -44.7 | +19.4 | +22.7 | |
| Step | 10 | 16 | 23 | 28 | 35 | 37 | 41 | 43 | 45 | 49 | 50 | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -40.3 | +20.8 | +57.8 | +23.3 | +54.2 | -12.0 | +49.7 | +6.7 | +25.8 | -22.3 | -39.6 |
| Relative (%) | -33.8 | +17.5 | +48.4 | +19.5 | +45.4 | -10.1 | +41.6 | +5.6 | +21.6 | -18.7 | -33.2 | |
| Step | 52 | 54 | 55 | 56 | 58 | 59 | 60 | 61 | 62 | 62 | 63 | |