# 18/17 equal-step tuning

18/17 equal-step tuning (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 (Marpurg-G 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)

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).