961edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | |||
== Theory == | |||
===Odd harmonics=== | 961edo has a reasonable 7-limit interpretation. The equal temperament [[Tempering out|tempers out]] the [[schisma]] in the 5-limit; [[4375/4374]], [[65625/65536]], and [[14348907/14336000]] in the 7-limit, supporting [[pontiac]], the {{nowrap|395 & 566}} temperament. | ||
In the 11-limit, the 961e [[val]] {{val| 961 1523 2231 '''2698''' '''3324''' }} scores the best, which tempers out 102487/102400 and 234375/234256. It prompts us to consider the 961de val {{val| 961 1523 2231 '''2697''' '''3324''' }}, which tempers out [[3025/3024]] and 184877/184320. The [[patent val]] {{val| 961 1523 2231 '''2698''' '''3325''' }} tempers out [[4000/3993]] and 46656/46585. | |||
It works much better as a 2.3.5.7.13.17 [[subgroup temperament]], in which case it tempers out [[10985/10976]], [[1275/1274]], [[2025/2023]] and [[4914/4913]]. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|961}} | {{Harmonics in equal|961}} | ||
===Subsets and supersets=== | |||
961 factors into | === Subsets and supersets === | ||
==Regular temperament properties== | Since 961 factors into {{factorization|961}}, 961edo has [[31edo]] as its subset edo. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.3 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-1523 961}} | ! rowspan="2" | [[Comma list]] | ||
|{{ | ! rowspan="2" | [[Mapping]] | ||
| 0.0587 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo|-1523 961}} | |||
| {{mapping| 961 1523 }} | |||
| +0.0587 | |||
| 0.0587 | | 0.0587 | ||
| 4.70 | | 4.70 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|32805/32768, {{monzo|-22 -137 103}} | | 32805/32768, {{monzo| -22 -137 103 }} | ||
|{{ | | {{mapping| 961 1523 2231 }} | ||
| 0.1060 | | +0.1060 | ||
| 0.0823 | | 0.0823 | ||
| 6.59 | | 6.59 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|4375/4374, 32805/32768, | | 4375/4374, 32805/32768, {{monzo| 15 9 14 -22 }} | ||
|{{ | | {{mapping| 961 1523 2231 2698 }} | ||
| 0.0648 | | +0.0648 | ||
| 0.1008 | | 0.1008 | ||
| 8.01 | | 8.01 | ||
|} | |} | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|399\961 | | 399\961 | ||
|498.231 | | 498.231 | ||
|4/3 | | 4/3 | ||
| | | [[Pontiac]] | ||
|} | |} | ||
==Scales== | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | |||
* [[Haumea5]] | * [[Haumea5]] | ||
* [[Haumea9]] | * [[Haumea9]] | ||
* [[Haumea14]] | * [[Haumea14]] | ||
* [[Haumea19]] | * [[Haumea19]] | ||