360edo: Difference between revisions
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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In the 5-limit, the [[patent val]] [[support]]s the [[misty]] temperament, and in the 7-limit 360edo supports the [[trimisty]] (name proposed by Eliora) 63 & 99 temperament with the comma basis {[[10976/10935]], 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave. | In the 5-limit, the [[patent val]] [[support]]s the [[misty]] temperament, and in the 7-limit 360edo supports the [[trimisty]] (name proposed by Eliora) 63 & 99 temperament with the comma basis {[[10976/10935]], 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave. | ||
360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the 140 & 220 temperament with period 1\20. Aside from that, it provides the optimal patent val for the 41 & 360 temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports 12 & 360 with the comma basis { [[390625/388962]], 67108864/66430125}. | 360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the {{nowrap|140 & 220}} temperament with period 1\20. Aside from that, it provides the optimal patent val for the {{nowrap|41 & 360}} temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports {{nowrap|12 & 360}} with the comma basis {[[390625/388962]], 67108864/66430125}. | ||
Aside from the patent val, there is a number of mappings to be considered. The 360d val, | Aside from the patent val, there is a number of mappings to be considered. The 360d val, {{val|360 571 836 '''1010'''}}, tempers out 3136/3125, 5120/5103, and extends the misty temperament in to the 7-limit. It is also a tuning for the 12th-octave [[magnesium]] temperament. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
360 is the 13th [[highly composite edo]], with many proper divisors: {{EDOs| 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 }}. One step of 360edo is known as '''the Dröbisch angle''', an [[interval size measure]] first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle". | 360 is the 13th [[highly composite edo]], with many proper divisors: {{EDOs| 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 }}. One step of 360edo is known as '''the Dröbisch angle''', an [[interval size measure]] first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle". | ||
== Table of intervals == | == Table of intervals == | ||
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=== 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 | ||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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|- | |- | ||
| 3 | | 3 | ||
| 149\360<br>(29\360) | | 149\360<br />(29\360) | ||
| 703.33<br>(303.33) | | 703.33<br />(303.33) | ||
| 4/3<br>(135/128) | | 4/3<br />(135/128) | ||
| [[Misty]] | | [[Misty]] | ||
|- | |- | ||
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|- | |- | ||
| 9 | | 9 | ||
| 149\360<br>(29\360) | | 149\360<br />(29\360) | ||
| 703.33<br>(36.67) | | 703.33<br />(36.67) | ||
| 4/3<br>(135/128) | | 4/3<br />(135/128) | ||
| [[Trimisty]] | | [[Trimisty]] | ||
|- | |- | ||
|12 | | 12 | ||
|73\360 | | 73\360<br />(13\360) | ||
(13\360) | | 243.333<br />(43.333) | ||
|243.333 | | 3145728/2734375<br />(?) | ||
(43.333) | | [[Magnesium]] (360d) | ||
|3145728/2734375 | |||
(?) | |||
|[[Magnesium]] (360d) | |||
|- | |- | ||
| 20 | | 20 | ||
| 149\360<br>(5\360) | | 149\360<br />(5\360) | ||
| 703.33<br>(43.33) | | 703.33<br />(43.33) | ||
| 4/3<br>(126/125) | | 4/3<br />(126/125) | ||
| [[Degrees]] | | [[Degrees]] | ||
|} | |} | ||
<nowiki>* | <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 | ||
== Music == | == Music == | ||