3776edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | == Theory == | ||
3776edo is a good 2.3.11.13.19 subgroup system. It does not tune the [[15-odd-limit]] consistently, though a reasonable represenation exists through the 19-limit patent val. There, it provides the [[optimal patent val]] for the [[oganesson]] temperament in the 7-, 11-, 13-, 17-, and the 19-limit. It tempers out the [[quartisma]] in the 11-limit, and is a tuning for the rank-3 [[van gogh]] temperament. | 3776edo is a good 2.3.11.13.19 subgroup system. It does not tune the [[15-odd-limit]] consistently, though a reasonable represenation exists through the 19-limit patent val. There, it provides the [[optimal patent val]] for the [[oganesson]] temperament in the 7-, 11-, 13-, 17-, and the 19-limit. It tempers out the [[quartisma]] in the 11-limit, and is a tuning for the rank-3 [[van gogh]] temperament. | ||
In the 19-limit, and 2.3.5.17.19 subgroup, 3776edo tempers out the comma that associates [[171/170]] to 1 step of 118edo, hence enabling usage of this interval as microchroma. This is strengthened by 3776edo's strong and consistent approximations of [[19/17]] and [[10/9]], intervals that are one 171/170 apart. | In the 19-limit, and 2.3.5.17.19 subgroup, 3776edo tempers out the comma that associates [[171/170]] to 1 step of 118edo, hence enabling usage of this interval as microchroma. This is strengthened by 3776edo's strong and consistent approximations of [[19/17]] and [[10/9]], intervals that are one 171/170 apart. | ||
=== Odd harmonics === | === Odd harmonics === | ||
{{harmonics in equal|3776}} | {{harmonics in equal|3776}} | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Periods<br>per 8ve | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
!Generator | |- | ||
!Cents | ! Periods<br />per 8ve | ||
!Associated<br>ratio | ! Generator* | ||
!Temperaments | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
|118 | | 118 | ||
|1781\3776<br>(21\3776) | | 1781\3776<br>(21\3776) | ||
|565.995<br>(6.67) | | 565.995<br>(6.67) | ||
|165/119<br>(?) | | 165/119<br>(?) | ||
|[[Oganesson]] | | [[Oganesson]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
[[Category:Equal divisions of the octave|####]] | [[Category:Equal divisions of the octave|####]] | ||
<!-- 4-digit number --> | <!-- 4-digit number --> | ||