16/15ths equal temperament

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16/15 equal temperament

Using a Just 16/15 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 hendecatonic, qintosec, or tertiosec temperament.

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

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 is supported by 22edo, 55edo, 77edo, and 99edo among others.

Qintosec (10&75)

Adding five equal divisions of the octave as a generator, 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 generator, 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.