412edo: 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 == | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
412 factors into | 412 factors into {{factorisation|412}}, with subset edos {{EDOs|2, 4, 103, and 206}}. [[1236edo]], which triples it, gives a good correction to harmonics 5, 7, and 11. | ||
== 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 | ||
| Line 24: | Line 33: | ||
| {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }} | | {{monzo| 32 -7 -9 }}, {{monzo| -5 31 -19 }} | ||
| {{mapping| 412 653 957 }} | | {{mapping| 412 653 957 }} | ||
| | | −0.1501 | ||
| 0.2182 | | 0.2182 | ||
| 7.49 | | 7.49 | ||
| Line 31: | Line 40: | ||
| 6144/6125, 2460375/2458624, 49009212/48828125 | | 6144/6125, 2460375/2458624, 49009212/48828125 | ||
| {{mapping| 412 653 957 1157 }} | | {{mapping| 412 653 957 1157 }} | ||
| | | −0.2085 | ||
| 0.2143 | | 0.2143 | ||
| 7.36 | | 7.36 | ||
|} | |||
=== 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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| 16875/16384 | | 16875/16384 | ||
| [[Septisuperfourth]] (7-limit) | | [[Septisuperfourth]] (7-limit) | ||
|} | |||
<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 | |||