113edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
113edo is [[consistency|distinctly consistent]] in the [[13-odd-limit]] with a flat tendency. As an equal temperament, it [[tempers out]] the [[amity comma]] and the [[ampersand]] 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 [[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 === | === Prime harmonics === | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3 | | 2.3 | ||
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| 0.643 | | 0.643 | ||
| 6.05 | | 6.05 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
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| 27/25 | | 27/25 | ||
| [[Quartemka]] | | [[Quartemka]] | ||
|- | |||
| 1 | |||
| 20\113 | |||
| 212.39 | |||
| 26/23 | |||
| [[Shoal]] | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 90: | Line 114: | ||
| 233.63 | | 233.63 | ||
| 8/7 | | 8/7 | ||
| [[Slendric]] | | [[Slendric]] / [[euslendric]] | ||
|- | |- | ||
| 1 | | 1 | ||
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| 1 | | 1 | ||
| 34\113 | | 34\113 | ||
| | | 361.06 | ||
| 16/13 | | 16/13 | ||
| [[Phicordial]] | | [[Phicordial]] | ||
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| 55/39 | | 55/39 | ||
| [[Gaster temperament|Gaster]] | | [[Gaster temperament|Gaster]] | ||
|} | |||
<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 | |||