1ed16/15
1 equal division of 16/15 (1ed16/15), also known as ambitonal sequence of 16/15 (AS16/15) or 16/15 equal-step tuning, is an equal-step tuning with a step size of a 16/15-wide semitone. The step size is about 111.7313 cents, corresponding to 10.7401edo.
Using a JI 16/15-wide semitone as the basis of an equal temperament tuning results in an interesting non-octave tuning. As every interval is a multiple of 16/15, the resultant tuning would be a subset of 5-limit just intonation. This can be also viewed as generating a subset of vavoom, stockhausenic, quintosec, or tertiosec temperaments.
Lookalikes: 11edo, 25ed5, 17edt
Intervals as 5-limit ratios
| Ratio | Cents | |
|---|---|---|
| (16/15)0 | 1/1 | 0.0000 |
| (16/15)1 | 16/15 | 111.7313 |
| (16/15)2 | 256/225 | 223.4626 |
| (16/15)3 | 4096/3375 | 335.1939 |
| (16/15)4 | 65536/50625 | 446.9251 |
| (16/15)5 | 1048576/759375 | 558.6564 |
| (16/15)6 | 16777216/11390625 | 670.3877 |
| (16/15)7 | 268435456/170859375 | 782.1190 |
| (16/15)8 | 4294967296/2562890625 | 893.8503 |
| (16/15)9 | 68719476736/38443359375 | 1005.5816 |
| (16/15)10 | 1099511627776/576650390625 | 1117.3129 |
| (16/15)11 | 17592186044416/8649755859375 | 1229.0441 |
Related temperament
16/15 equal temperament is related to various regular temperaments including vavoom, stockhausenic, quintosec, tertiosec, hendecatonic, or meridic.
Vavoom (118&783)
16/15 equal temperament is closely related to the vavoom temperament, which tempers out [-68 18 17⟩. The generator is ~16/15 = 111.9¢ (0.15 cents sharp 16/15-wide semitone), 17 of them equals ~3/1, and 18 of them equals ~16/5.
Stockhausenic (140&183)
The stockhausenic is a temperament for the 17-limit, which tempers out 442/441, 561/560, 676/675, 715/714, and 4096/4095. The generator is ~16/15 = 111.46¢, 25 of them equals ~5/1, and 44 of them equals ~17/1.
Quintosec (10&75)
Adding five equal divisions of the octave as a period, 16/15 equal temperament leads to the quintosec temperament, tempering out the quintosec comma, 140737488355328 / 140126044921875 = [47 -15 -10⟩.
Tertiosec (75&171)
Adding three equal divisions of the octave as a period, 16/15 equal temperament leads the tertiosec temperament, tempering out the tertiosec comma, [-89 21 24⟩. This temperament is supported by 21EDO, 75EDO, 96EDO, and 171EDO.
Hendecatonic (22&77)
Tempering out the hendecatonic comma, 8796093022208 / 8649755859375 = [43 -11 -11⟩ leads the hendecatonic temperament, which tempers out 6144/6125 and 10976/10935 in the 7-limit; 121/120, 176/175, and 24057/24010 in the 11-limit. This temperament has a period of 1/11 octave, which represents 16/15, four of which represents 9/7, and five of which represents 11/8.
Meridic (258&301)
Tempering out a 43-15-comma, [168 -43 -43⟩ leads the meridic temperament, which tempers out 5250987/5242880 and 2202927104/2197265625 in the 7-limit; 41503/41472, 172032/171875, and 1240029/1239040 in the 11-limit; 2200/2197, 4096/4095, 21168/21125, and 39366/39325 in the 13-limit. This temperaments has a period of 1/43 octave, which represents 16/15 (0.103 cents flat).
Harmonics
1ed16/15 offers a good approximation of the full 17-limit, and of the no-19s, no-23s 37-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +29.0 | -2.5 | +7.0 | -16.9 | -17.3 | +28.7 | +11.2 | +42.1 | +46.6 | -19.6 | -23.3 |
| Relative (%) | +26.0 | -2.3 | +6.2 | -15.1 | -15.4 | +25.7 | +10.0 | +37.7 | +41.7 | -17.5 | -20.8 | |
| Step | 11 | 17 | 25 | 30 | 37 | 40 | 44 | 46 | 49 | 52 | 53 | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.6 | +51.4 | -31.1 | +38.4 | +53.8 | -20.1 | +33.9 | -16.8 | -5.4 | -53.5 | +33.2 |
| Relative (%) | +5.0 | +46.0 | -27.8 | +34.3 | +48.2 | -18.0 | +30.4 | -15.0 | -4.9 | -47.9 | +29.7 | |
| Step | 56 | 58 | 58 | 60 | 62 | 63 | 64 | 65 | 66 | 66 | 68 | |