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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 7 × 13
{{ED intro}}
| Step size = 3.29670¢
| Fifth = 213\364 (702.20¢)
| Semitones = 35:27 (115.38¢ : 89.01¢)
| Consistency = 21
}}
The '''364 equal divisions of the octave''' ('''364edo'''), or the '''364(-tone) equal temperament''' ('''364tet''', '''364et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 364 parts of about 3.30 [[cent]]s each.


== Theory ==
== Theory ==
364edo is consistent through the [[21-odd-limit]], [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| -65 0 28 }}; (28-5-comma) in the [[5-limit]]; 65625/65536 (horwell), 390625/388962 ([[Dimcomp comma|dimcomp]]), and 420175/419904 (wizma) in the [[7-limit]] (supporting [[fifthplus]] and [[oquatonic]]); 1375/1372, [[6250/6237]], [[19712/19683]], and 41503/41472 in the [[11-limit]] (as well as [[9801/9800]]); [[625/624]], [[1716/1715]], [[2080/2079]], [[2200/2197]], and 14641/14625 in the [[13-limit]] (as well as [[4096/4095]], [[4225/4224]], and [[10985/10976]]); [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, 2025/2023, and 8624/8619 in the [[17-limit]] (as well as 2431/2430, 4914/4913, and [[5832/5831]]); [[1216/1215]], 1331/1330, 1540/1539, and [[1729/1728]] in the [[19-limit]].
364edo is [[consistent]] through the [[21-odd-limit]] with good average accuracy.  


364 is divisible by, and thus contains sub-edos {{EDOs|1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182.}}
As an equal temperament, it [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| -65 0 28 }} ([[oquatonic comma]]) in the [[5-limit]]; 65625/65536 ([[horwell comma]]), 390625/388962 ([[dimcomp comma]]), and 420175/419904 ([[wizma]]) in the [[7-limit]] ([[support]]ing [[fifthplus]] and [[oquatonic]]); [[1375/1372]], [[6250/6237]], [[9801/9800]], [[19712/19683]], and [[41503/41472]] in the [[11-limit]]; [[625/624]], [[1716/1715]], [[2080/2079]], [[2200/2197]], [[4096/4095]], [[4225/4224]], [[10985/10976]], and 14641/14625 in the [[13-limit]]; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], [[2431/2430]], [[4914/4913]], [[5832/5831]], and 8624/8619 in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|364}}
{{Harmonics in equal|364|columns=11}}
{{Harmonics in equal|364|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 364edo (continued)}}
 
=== Subsets and supersets ===
Since 364 factors into primes as {{nowrap| 2<sup>2</sup> × 7 × 13 }}, 364edo has subset edos {{EDOs| 2, 4, 7, 13, 14, 26, 28, 52, 91, 182 }}.
 
=== Miscellany ===
364edo can act as "pseudo-24024edo" in a sense that it can replicate being a multiple of [[11edo]], [[12edo]], [[13edo]] and [[14edo]]. It has 13 and 14 as its divisors, while at the same time supporting the Supermajor[11] scale from 91edo, which is a very precise temperament, and WorldCalendar[12] scale, which mimics 12edo. While it does not exactly replicate 11edo and 12edo, it comes close enough in harmonic parameters these edos are sought after.


== 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]]
Line 28: Line 30:
|-
|-
| 2.3
| 2.3
| {{monzo| 577 -364 }}
| {{Monzo| 577 -364 }}
| [{{val| 364 577 }}]
| {{Mapping| 364 577 }}
| -0.0766
| −0.0766
| 0.0766
| 0.0766
| 2.32
| 2.32
Line 36: Line 38:
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| -65 0 28 }}
| 1600000/1594323, {{monzo| -65 0 28 }}
| [{{val| 364 577 845 }}]
| {{Mapping| 364 577 845 }}
| +0.0350
| +0.0350
| 0.1698
| 0.1698
Line 43: Line 45:
| 2.3.5.7
| 2.3.5.7
| 65625/65536, 390625/388962, 420125/419904
| 65625/65536, 390625/388962, 420125/419904
| [{{val| 364 577 845 1022 }}]
| {{Mapping| 364 577 845 1022 }}
| -0.0098
| −0.0098
| 0.1662
| 0.1662
| 5.04
| 5.04
Line 50: Line 52:
| 2.3.5.7.11
| 2.3.5.7.11
| 1375/1372, 6250/6237, 19712/19683, 41503/41472
| 1375/1372, 6250/6237, 19712/19683, 41503/41472
| [{{val| 364 577 845 1022 1259 }}]
| {{Mapping| 364 577 845 1022 1259 }}
| +0.0366
| +0.0366
| 0.1753
| 0.1753
Line 57: Line 59:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 625/624, 1375/1372, 2080/2079, 2200/2197, 14641/14625
| 625/624, 1375/1372, 2080/2079, 2200/2197, 14641/14625
| [{{val| 364 577 845 1022 1259 1347 }}]
| {{Mapping| 364 577 845 1022 1259 1347 }}
| +0.0245
| +0.0245
| 0.1622
| 0.1622
Line 64: Line 66:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 625/624, 715/714, 1089/1088, 1225/1224, 2025/2023, 2200/2197
| 625/624, 715/714, 1089/1088, 1225/1224, 2025/2023, 2200/2197
| [{{val| 364 577 845 1022 1259 1347 1488 }}]
| {{Mapping| 364 577 845 1022 1259 1347 1488 }}
| +0.0022
| +0.0022
| 0.1599
| 0.1599
Line 71: Line 73:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 625/624, 715/714, 1089/1088, 1216/1215, 1225/1224, 1331/1330, 1729/1728
| 625/624, 715/714, 1089/1088, 1216/1215, 1225/1224, 1331/1330, 1729/1728
| [{{val| 364 577 845 1022 1259 1347 1488 1546 }}]
| {{Mapping| 364 577 845 1022 1259 1347 1488 1546 }}
| +0.0257
| +0.0257
| 0.1620
| 0.1620
Line 79: Line 81:
=== 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 octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 90: Line 93:
| 339.56
| 339.56
| 243/200
| 243/200
| [[Amity]] / [[paramity]]
| [[Paramity]]
|-
|-
| 1
| 1
Line 96: Line 99:
| 412.09
| 412.09
| 80/63
| 80/63
| [[Witch]]
| [[Witcher]]
|-
|-
| 1
| 1
Line 111: Line 114:
|-
|-
| 2
| 2
| 57\364
| 125\364<br>(57\364)
| 187.91
| 412.09<br>(187.91)
| 49/44
| 80/63<br>(49/44)
| [[Semiwitch]]
| [[Semiwitcher]]
|-
| 2
| 151\364<br>(31\364)
| 497.80<br>(102.20)
| 4/3<br>(35/33)
| [[Gariwizmic]]
|-
| 4
| 30\364
| 98.90
| 18/17
| [[World calendar]]
|-
| 13
| 151\364<br>(11\364)
| 497.80<br>(36.26)
| 4/3<br>(?)
| [[Aluminium]]
|-
| 26
| 151\364<br>(11\364)
| 497.80<br>(36.26)
| 4/3<br>(?)
| [[Iron]]
|-
|-
| 28
| 28
Line 121: Line 148:
| 4/3<br>(105/104)
| 4/3<br>(105/104)
| [[Oquatonic]]
| [[Oquatonic]]
|-
| 91
| 151\364<br>(3\364)
| 497.80<br>(3.30)
| 4/3<br>(176/175)
| [[Protactinium]]
|}
|}
<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


[[Category:Equal divisions of the octave]]
== Scales ==
* WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30
Retrieved from "https://en.xen.wiki/w/364edo"