16/15 equal-step tuning

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16/15 equal temperament (AS16/15) is an equal multiplication of 16/15-wide semitone. The step size is about 111.7313 cents, corresponding to 10.7401 EDO.

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, qintosec, or tertiosec temperament.

Lookalikes: 11EDO, 25ED5, 17ED3

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, qintosec, 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.

Qintosec (10&75)

Adding five equal divisions of the octave as a period, 16/15 equal temperament leads the qintosec temperament, tempering out the qintosec 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).