159edo: Difference between revisions
Added a few details about certain advantages and disadvantages compared to 94edo. |
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== Theory == | == Theory == | ||
Compared to [[94edo]], 159edo offers both advantages and disadvantages. On one hand is the disadvantage of 159edo being [[consistent]] only up to the 17 odd-limit- with it proving to be inconsistent in the 19-limit. On the other hand, the septimal kleisma, [[225/224]], maps to a single step in 159edo- a third the size of the tempered version of [[81/80]]- which not only allows for the septimal kleisma to be easily accounted for in notation systems, but also for | Compared to [[94edo]], 159edo offers both advantages and disadvantages. On one hand is the disadvantage of 159edo being [[consistent]] only up to the 17 odd-limit- with it proving to be inconsistent in the 19-limit. On the other hand, the septimal kleisma, [[225/224]], maps to a single step in 159edo- a third the size of the tempered version of [[81/80]]- which not only allows 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 otherwise tempered out in 94edo. | ||
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 [[Gamelismic_clan #Guiron|guiron]] and [[Gamelismic_clan #Tritikleismic|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 [[Gamelismic_family #Portending|portending]]. | 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 [[Gamelismic_clan #Guiron|guiron]] and [[Gamelismic_clan #Tritikleismic|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 [[Gamelismic_family #Portending|portending]]. | ||
Revision as of 17:35, 7 September 2020
159edo is the 159 equal division of the octave into equal parts of 7.547 cents each.
Theory
Compared to 94edo, 159edo offers both advantages and disadvantages. On one hand is the disadvantage of 159edo being consistent only up to the 17 odd-limit- with it proving to be inconsistent in the 19-limit. On the other hand, the septimal kleisma, 225/224, maps to a single step in 159edo- a third the size of the tempered version of 81/80- which not only allows 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 otherwise tempered out in 94edo.
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 | |