1ed18/17
1 equal division of 18/17 (1ed18/17), also known as ambitonal sequence of 18/17 (AS18/17) or 18/17 equal-step tuning, 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).
Quindromeda (12 & 205)
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)
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).
Gregorian leap day (97 & 400)
Gregorian leap day temperament is described in the full 19-limit and it equates three 18/17s with 19/16, eleven 18/17s with 15/8, eighteen 18/17s with 7/5 and fourty of them with 16/13.
Harmonics
1ed18/17 offers a good no-3s approximation of the 13-limit.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -12.5 | -21.8 | -15.6 | -4.4 | +4.8 | +12.4 | +42.8 | +48.1 | +14.2 | +8.7 | -7.8 |
| Relative (%) | -12.7 | -22.0 | -15.7 | -4.4 | +4.8 | +12.6 | +43.2 | +48.6 | +14.4 | +8.8 | -7.8 | |
| Step | 12 | 19 | 28 | 34 | 42 | 45 | 50 | 52 | 55 | 59 | 60 | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -17.2 | +3.0 | +19.5 | -35.5 | -45.6 | -33.4 | +7.8 | +43.3 | +41.9 | -6.2 | -44.0 |
| Relative (%) | -17.4 | +3.0 | +19.7 | -35.9 | -46.1 | -33.7 | +7.9 | +43.8 | +42.3 | -6.3 | -44.5 | |
| Step | 63 | 65 | 66 | 67 | 69 | 71 | 72 | 74 | 75 | 75 | 76 | |