986edo: Difference between revisions
create page, also octave-reduced links per discussion in #wiki, - 17 not 17 as an octave-reduced harmonic is more helpful to the reader. |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
986edo is a good 2.3.7.11 subgroup tuning, but it is in[[consistent]] to the 5-odd-limit and larger due to a high error on the [[5/4|5th harmonic]]. 986edo has an excellent [[11/8|11th harmonic]], being the denominator of a [[convergent]] to log<sub>2</sub>11, after [[949edo|949]] and before [[1935edo|1935]]. In the 2.3.7.11 subgroup, 986edo can be used with optional additions of either [[17/16|17]], [[23/16|23]], [[29/16|29]], or [[31/16|31]]. | 986edo is a good 2.3.7.11 subgroup tuning, but it is in[[consistent]] to the 5-odd-limit and larger due to a high error on the [[5/4|5th harmonic]]. 986edo has an excellent [[11/8|11th harmonic]], being the denominator of a [[convergent]] to log<sub>2</sub>11, after [[949edo|949]] and before [[1935edo|1935]]. In the 2.3.7.11 subgroup, 986edo can be used with optional additions of either [[17/16|17]], [[23/16|23]], [[29/16|29]], or [[31/16|31]]. | ||
In the 2.3.7 subgroup, 986edo tempers out the [[garischisma]], and is a strong tuning for 2.3.7.11-subgroup [[gary]]. It also tempers out | In the 2.3.7 subgroup, 986edo tempers out the [[garischisma]], and is a strong tuning for 2.3.7.11-subgroup [[gary]]. It also tempers out, 131072/130977, 3195731/3188646, 33554432/33480783, 67110351/67108864, and {{monzo|5 4 0 28 -26}} in the 2.3.7.11 subgroup. | ||
=== Prime harmonics === | |||
{{harmonics in equal|986}} | {{harmonics in equal|986}} | ||
=== Subsets and supersets === | |||
Since 986 factors as {{Factorization|986}}, 986edo has subset edos {{EDOs|1, 2, 17, 29, 34, 58, 493}}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
=== 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 | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Gary]] | | [[Gary]] | ||
|} | |} | ||
<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 | |||