159edo
159edo is the 159 equal division of the octave into equal parts of 7.547 cents each.
Theory
A salient fact about 159edo is that 159 = 3*53, so that it shares the same 5-limit thirds and fifths with 53edo. However, compared to 53edo, the patent vals differ on the mapping for 7. In the 7-limit it tempers out 1029/1024 and 10976/10935 in addition to the 5-limit commas 32805/32768 and 15625/15552. This makes it among other things an excellent tuning for guiron and tritikleismic temperaments. It has a very accurate 11, and in the 11-limit tempers out not only 385/384, 441/440, and 4000/3993, but - in a first for EDOs that are multiples of 53 - 117440512/117406179 as well. In the 13-limit it tempers out 325/324, 364/363, and 10985/10976. It also has an accurate 17, and in the 17-limit tempers out 273/272 and 375/374. In the 19-limit it tempers out 343/342 and 361/360. It also provides the optimal patent val for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament portending.
Another and notable temperament supported by 159 is yarman temperament, with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however.
Yarman temperament has MOS of 79 and 80 notes to the octave, and the 79-note MOS has been proposed by Ozan Yarman as a tuning standard for arabic/turkish/persian music.
Articles
- 79-Tone Tuning & Theory for Turkish Maqam Music - Ozan Yarman's dissertation
- Search For A Theoretical Model Conforming To Turkish Maqam Music Practice: A Selection Of Fixed-Pitch Settings From 34-tone Equal Temperament To The 79-tone Tuning - also by Ozan Yarman, gives a summary.
- Letter to Ozan Yarman by Margo Schulter (permalink)
Just approximation
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 17 | prime 19 | prime 23 | prime 29 | prime 31 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | 0.00 | -0.07 | -1.41 | -2.79 | -0.37 | -2.79 | +0.70 | -3.17 | -1.86 | -3.16 | +2.13 |
| relative (%) | 0.0 | -0.9 | -18.7 | -36.9 | -5.0 | -37.0 | +9.3 | -42.0 | -24.6 | -41.9 | +28.3 | |