159edo: Difference between revisions

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== Theory ==

Compared to [[94edo]], 159edo offers both potential advantages and potential disadvantages. On one hand is the potential 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- due to being~~ 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- 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 ~~otherwise~~ tempered out in ~~94edo~~.

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.

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 - [[Quartismic temperaments|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]].

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 == Theory ==
-Compared to [[94edo]], 159edo offers both potential advantages and potential disadvantages.  On one hand is the potential 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- due to being 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- 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 otherwise tempered out in 94edo.
+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.
 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 - [[Quartismic temperaments|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]].