159edo: Difference between revisions

Line 3:

== 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.

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 step size of 159edo itself allows for a decent balance between allowing the possibility of seamless modulation to keys that are not on the same circle 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 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]].

@@ Line 3: / Line 3: @@
 == 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.
+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 step size of 159edo itself allows for a  decent balance between allowing the possibility of seamless modulation to keys that are not on the same circle 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 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]].