402edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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* {{val| 402 637 '''934''' 1129 }} (402c) | * {{val| 402 637 '''934''' 1129 }} (402c) | ||
Using the patent val, it tempers out [[4375/4374]], 7381125/7340032 and 3200000/3176523 in the 7-limit | Using the patent val, it tempers out the [[semicomma]] in the 5-limit and [[4375/4374]], 7381125/7340032, and 3200000/3176523 in the 7-limit, [[support]]ing [[Abigail]]. | ||
Using the 402d val, it tempers out [[250047/250000]], 1500625/1492992 and 2460375/2458624 in the 7-limit | Using the 402d val, it tempers out [[250047/250000]], and 1500625/1492992 and 2460375/2458624 in the 7-limit. | ||
Using the 402c val, it tempers out [[3136/3125]], 321489/320000 and 13060694016/12867859375 in the 7-limit | Using the 402c val, it tempers out the [[schisma]] in the 5-limit; and [[3136/3125]], 321489/320000, and 13060694016/12867859375 in the 7-limit, supporting [[bischismic]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 402 factors into 2 × 3 × 67, 402edo has subset edos {{EDOs| 2, 3, 6, 67, 134, and 201 }}. [[804edo]], which doubles it, gives a good correction to the | Since 402 factors into {{nowrap|2 × 3 × 67}}, 402edo has subset edos {{EDOs| 2, 3, 6, 67, 134, and 201 }}. [[804edo]], which doubles it, gives a good correction to the harmonics 5 and 7. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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=== 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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| 271.64 | | 271.64 | ||
| 75/64 | | 75/64 | ||
| [[Orson]] | | [[Orson]] (402) | ||
|- | |- | ||
| 1 | | 1 | ||
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| 343.28 | | 343.28 | ||
| 8000/6561 | | 8000/6561 | ||
| [[Raider]] | | [[Raider]] (402) | ||
|- | |||
| 2 | |||
| 70\402 | |||
| 208.96 | |||
| 44/39 | |||
| [[Abigail]] (402) | |||
|- | |||
| 2 | |||
| 167\402<br />(34\402) | |||
| 498.51<br />(101.49) | |||
| 4/3<br />(200/189) | |||
| [[Bischismic]] (402c, 7-limit) | |||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||