392edo: 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 == | ||
392et is consistent to the [[7-odd-limit]] | 392et is [[consistent]] to the [[7-odd-limit]] with a flat tendency in the [[prime harmonic]]s. As an equal temperament, it [[tempering out|tempers out]] the [[parakleisma]] in the 5-limit; 321489/320000 ([[varunisma]]), 420175/419904 ([[wizma]]), 703125/702464 ([[meter]]), and 823543/819200 ([[quince comma]]) in the 7-limit; and [[441/440]], [[8019/8000]], [[9801/9800]], and [[41503/41472]] in the 11-limit. It [[support]]s [[qak]] and [[octowerck]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
392 factors into 2<sup>3</sup> × 7<sup>2</sup>, with subset edos {{EDOs|2, 4, 7, 8, 14, 28, 49, 56, 98, and 196}}. | 392 factors into 2<sup>3</sup> × 7<sup>2</sup>, with subset edos {{EDOs| 2, 4, 7, 8, 14, 28, 49, 56, 98, and 196 }}. | ||
== Regular temperament properties == | == 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|-621 392}} | ! rowspan="2" | [[Comma list]] | ||
|{{mapping|392 621}} | ! rowspan="2" | [[Mapping]] | ||
| 0.2948 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3 | |||
| {{monzo| -621 392 }} | |||
| {{mapping| 392 621 }} | |||
| +0.2948 | |||
| 0.2949 | | 0.2949 | ||
| 9.63 | | 9.63 | ||
|- | |- | ||
|2.3.5 | | 2.3.5 | ||
|{{monzo|8 14 -13}}, {{monzo|-49 28 2}} | | {{monzo| 8 14 -13 }}, {{monzo| -49 28 2 }} | ||
|{{mapping|392 621 910}} | | {{mapping| 392 621 910 }} | ||
| 0.2826 | | +0.2826 | ||
| 0.2414 | | 0.2414 | ||
| 7.89 | | 7.89 | ||
|- | |- | ||
|2.3.5.7 | | 2.3.5.7 | ||
|321489/320000, 420175/419904, 703125/702464 | | 321489/320000, 420175/419904, 703125/702464 | ||
|{{mapping|392 621 910 1100}} | | {{mapping| 392 621 910 1100 }} | ||
| 0.3437 | | +0.3437 | ||
| 0.2343 | | 0.2343 | ||
| 7.65 | | 7.65 | ||
|- | |- | ||
|2.3.5.7.11 | | 2.3.5.7.11 | ||
|441/440, 8019/8000, 41503/41472, | | 441/440, 8019/8000, 41503/41472, 703125/702464 | ||
|{{mapping|392 621 910 1100 1356}} | | {{mapping| 392 621 910 1100 1356 }} | ||
| 0.2922 | | +0.2922 | ||
| 0.2335 | | 0.2335 | ||
| 7.63 | | 7.63 | ||
| Line 53: | Line 54: | ||
=== 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> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|37\392 | | 37\392 | ||
|113.27 | | 113.27 | ||
|16/15 | | 16/15 | ||
|[[Misneb]] | | [[Misneb]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|103\392 | | 103\392 | ||
|315.31 | | 315.31 | ||
|6/5 | | 6/5 | ||
|[[Parakleismic]] | | [[Parakleismic]] (5-limit) | ||
|- | |- | ||
|1 | | 1 | ||
|149\392 | | 149\392 | ||
|456.12 | | 456.12 | ||
|125/96 | | 125/96 | ||
|[[Qak]] | | [[Qak]] | ||
|- | |- | ||
|8 | | 8 | ||
|185\392<br>(11\392) | | 185\392<br />(11\392) | ||
|566.33<br>(33.67) | | 566.33<br />(33.67) | ||
|104/75<br>(55/54) | | 104/75<br />(55/54) | ||
|[[Octowerck]] | | [[Octowerck]] (392f) | ||
|- | |- | ||
|28 | | 28 | ||
|163\392<br>(5\392) | | 163\392<br />(5\392) | ||
|498.98<br>(15.31) | | 498.98<br />(15.31) | ||
|4/3<br>(105/104) | | 4/3<br />(105/104) | ||
|[[Oquatonic]] | | [[Oquatonic]] (5-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 | |||
<nowiki>* | |||