# 16/15 equal-step tuning

**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).