159edo: Difference between revisions

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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]], [[amity comma|1600000/1594323]] and 2109375/2097152. This makes it among other things an excellent tuning for [[Schismatic_family#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, 1029/1024, 3025/3024 4000/3993, 4375/4356, 6250/6237 and 10976/10935, but- in a first for EDOs that are multiples of 53- [[Quartisma|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 [[Gamelismic_family #Portending|portending]].

Another ~~and~~ notable temperament supported by 159 is [[Turkish_maqam_music_temperaments|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|arabic/turkish/persian]] music.

Another notable temperament supported by 159 is [[Turkish_maqam_music_temperaments|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|arabic/turkish/persian]] music.

: ''See also: [[Table of 159edo Intervals]]''

@@ Line 7: / Line 7: @@
 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]], [[amity comma|1600000/1594323]] and 2109375/2097152. This makes it among other things an excellent tuning for [[Schismatic_family#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, 1029/1024, 3025/3024 4000/3993, 4375/4356, 6250/6237 and 10976/10935, but- in a first for EDOs that are multiples of 53- [[Quartisma|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 [[Gamelismic_family #Portending|portending]].
-Another and notable temperament supported by 159 is [[Turkish_maqam_music_temperaments|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|arabic/turkish/persian]] music.
+Another notable temperament supported by 159 is [[Turkish_maqam_music_temperaments|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|arabic/turkish/persian]] music.
 : ''See also: [[Table of 159edo Intervals]]''