159edo: Difference between revisions

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== Theory ==

As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 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.

As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 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. 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]], [[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]].

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, 235298/234375 and [[Meter_comma|703125/702464]] 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, as well as a possible tuning for metric temperament. This EDO 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 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.

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 == Theory ==
-As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 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.
+As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 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.  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]], [[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]].
+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, 235298/234375 and [[Meter_comma|703125/702464]] 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, as well as a possible tuning for metric temperament.  This EDO 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 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.