113edo: Difference between revisions

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== Theory ==

113edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

113edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency (and in the [[15-odd-limit]], only [[15/11]] and its complement are inconsistent). As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.

113edo is also notable as a ~~no-fives~~ [[31-limit]] system, ~~where~~ it is consistent in the [[29-odd-limit]] ~~(additionally~~, ~~5 itself, but not the patent 15 or 25, can also be added~~ to ~~this diamond with no inconsistent intervals), and nearly~~ the [[33-odd-limit]], as only [[31/21]] ~~and its complement are inconsistent, being~~ 50.8% off - and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].

113edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]], which nearly extends to the [[33-odd-limit]], with only two inconsistent interval pairs that both involve 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.

=== Prime harmonics ===

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| [[Gaster temperament|Gaster]]

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

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 == Theory ==
-edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.
+edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency (and in the [[15-odd-limit]], only [[15/11]] and its complement are inconsistent). As an equal temperament, it [[tempering out|tempers out]] the [[amity comma]] and the [[ampersand comma]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and 1071875/1062882 in the [[7-limit]]; [[243/242]], [[385/384]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[325/324]], [[364/363]], [[729/728]], and 1625/1617 in the [[13-limit]]. It notably [[support]]s the 5-limit [[amity]] temperament, 7-limit [[amicable]] temperament, 7- and 11-limit [[miracle]] temperament, and 13-limit [[manna]] temperament.
-edo is also notable as a no-fives [[31-limit]] system, where it is consistent in the [[29-odd-limit]] (additionally, 5 itself, but not the patent 15 or 25, can also be added to this diamond with no inconsistent intervals), and nearly the [[33-odd-limit]], as only [[31/21]] and its complement are inconsistent, being 50.8% off - and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]].
+edo is also notable as a [[31-limit]] system, especially if error on prime 5 is tolerable. In fact, it is consistent in the no-15 no-25 [[29-odd-limit]], which nearly extends to the [[33-odd-limit]], with only two inconsistent interval pairs that both involve 31, being [[31/21]] (50.8% off) and [[31/20]] (55.4% off) and their complements – and serves as a nearly optimal tuning for [[slendric]], in particular a 2.3.7.13.17(.19.23).29 extension of slendric harmonies known as [[euslendric]]. Notably as a slendric system, it is the largest EDO that maps [[64/49]] and [[21/16]] to the same interval consistently.
 === Prime harmonics ===
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 | [[Gaster temperament|Gaster]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct