159edo: Difference between revisions

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'''159edo''' is the ''159 equal division of the octave'' into equal parts of 7.547 [[cent]]s each. A salient fact about it 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]].

'''159edo''' is the '''159 equal division of the octave''' into equal parts of 7.547 [[cent|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 [[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]].

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.

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-'''159edo''' is the ''159 equal division of the octave'' into equal parts of 7.547 [[cent]]s each. A salient fact about it 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]].
+'''159edo''' is the '''159 equal division of the octave''' into equal parts of 7.547 [[cent|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 [[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]].
 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.

159edo: Difference between revisions

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