18/17s equal temperament

From Xenharmonic Wiki
Jump to navigation Jump to search

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.

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: 16ed5/2 #16ed5/2 as a generator

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: 28ed5 #28ed5 as a generator

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 some extensions for full 19-limit including quintakwai (12&193), quinkwai (12f&181), quintakwoid (12f&193), quintoneum (12f&217), quintoneoid (12&217), quintasandra (217&229), quintasandroid (12f&229), and quintasand (12&229).

Quintilischis (12&289)

See also: 40ed10 #40ed10 as a generator

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).