159edo
159edo is the 159 equal division of the octave into equal parts of 7.547 cents each.
Theory
As the step size of 159edo is simultaneously above the average peak JND of human pitch perception and small enough to be well within the margin of error between Just 5-limit intervals and their 12edo counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having so many steps as to have individual steps blend completely into one another. Furthermore, the septimal kleisma, 225/224, maps to a single step in 159edo- a third the size of the tempered version of 81/80- which allows not only for the septimal kleisma to be easily accounted for in notation systems, but also for easy distinctions between certain fairly important intervals such as 25/16 and 14/9 that are tempered out in other EDOs. As a bonus, 159edo is consistent up to the 17 odd-limit- though it proves to be inconsistent in the 19-limit.
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, 10976/10935, 117649/116640 and 235298/234375 in addition to the 5-limit commas 32805/32768, 15625/15552, 1600000/1594323 and 2109375/2097152. 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, 1029/1024, 3025/3024 4000/3993, 4375/4356, 6250/6237 and 10976/10935, 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, 625/624, 1001/1000, 10985/10976 and 13720/13689. It also has an accurate 17, and in the 17-limit tempers out 273/272, 375/374, 595/594, 715/714, 936/935, 8624/8619 and 15379/15300. In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas. 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.
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 | |
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)