# 18/17 equal-step tuning

**18/17s equal temperament** (AS18/17) is an equal multiplication of 18/17-wide semitone. The step size is about 98.9546 cents, corresponding to 12.1268 EDO.

Using a JI 18/17-wide semitone as the basis of an equal temperament tuning results in an interesting non-octave tuning. As every interval is a multiple of 18/17, the resultant tuning would be a subset of 17-limit just intonation. This can be also viewed as generating a subset of Galilei tuning or quintaleap temperament.

Lookalikes: 42ED11, 34ED7, 16ED5/2, 28ED5, 5ED4/3 (quintilipyth scale), 40ED10, 12EDO

## Intervals as 17-limit ratios

Ratio | Cents | |
---|---|---|

(18/17)0 | 1 / 1 | 0.0000 |

(18/17)1 | 18 / 17 | 98.9546 |

(18/17)2 | 324 / 289 | 197.9092 |

(18/17)3 | 5832 / 4913 | 296.8638 |

(18/17)4 | 104976 / 83521 | 395.8184 |

(18/17)5 | 1889568 / 1419857 | 494.7730 |

(18/17)6 | 34012224 / 24137569 | 593.7276 |

(18/17)7 | 612220032 / 410338673 | 692.6821 |

(18/17)8 | 11019960576 / 6975757441 | 791.6367 |

(18/17)9 | 198359290368 / 118587876497 | 890.5913 |

(18/17)10 | 3570467226624 / 2015993900449 | 989.5459 |

(18/17)11 | 64268410079232 / 34271896307633 | 1088.5005 |

(18/17)12 | 1156831381426176 / 582622237229761 | 1187.4551 |

(18/17)13 | 20822964865671168 / 9904578032905937 | 1286.4097 |

## Related temperament

Using a 18/17-wide semitone as a generator leads a number of regular temperaments including *quintaleap*, *quindromeda*, and *quintilischis*. These are cluster temperaments with 12 clusters of notes in an octave.

### Galilei tuning

**Galilei tuning** is a system of musical tuning in which the frequency ratios of all intervals are based on the ratios 2/1 (octave) and 18/17. This tuning system is named after an Italian music theorist Vincenzo Galilei. In this tuning, MOS of 12, 13, 25, 37, 49, 61, 73, 85, or 97 notes are available.

### Quintaleap (12&121)

*See also: Quintaleap family*

The *quintaleap* temperament is strongly related to Galilei tuning and tempers out 256/255, 361/360, and 4624/4617 in the 2.3.5.17.19 subgroup. It equates three 18/17s with 19/16, five 18/17s with 4/3, and sixteen 18/17s with 5/2. There are some extensions for full 19-limit including *quintupole* (12f&121) and *quinticosiennic* (12f&145).

### Quindromeda (12&205)

*See also: Quindromeda family*

The *quindromeda* temperament is a temperament for 2.3.5.17.19 subgroup, which tempers out 1216/1215, 1445/1444, and 6144/6137. It equates three 18/17s with 19/16, five 18/17s with 4/3, and twenty-eight 18/17s with the fifth harmonic. There are many extensions for full 19-limit including *quintakwai* (12&193), *quinkwai* (12f&181), *quintakwoid* (12f&193), *quindro* (205&217), *quintoneum* (12f&217), *quintoneoid* (12&217), *quinsandra* (217&229), *quinsandric* (12f&229), and *quinsandro* (12&229).

### Quintilischis (12&289)

*See also: 40ed10 #Regular temperaments*

The *quintilischis* temperament is a temperament for 2.3.5.17.19 subgroup, which tempers out 4624/4617, 6144/6137, and 6885/6859. It equates three 18/17s with 19/16, five 18/17s with 4/3, and forty 18/17s with the tenth harmonic. There are some extensions for full 19-limit including *quintilipyth* (12f&253), *quintaschis* (12f&289), *quintahelenic* (12f&301), and *quintahelenoid* (12&301).