364edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|364}}
{{ED intro}}


== 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 ===
{{Harmonics in equal|364|columns=11}}
{{Harmonics in equal|364|columns=11}}
{{Harmonics in equal|364|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 364edo (continued)}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
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.
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 25: 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 33: 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 40: 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 47: 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 54: 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 61: 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 68: 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 76: 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 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
Line 87: Line 93:
| 339.56
| 339.56
| 243/200
| 243/200
| [[Amity]] / [[paramity]]
| [[Paramity]]
|-
|-
| 1
| 1
Line 93: Line 99:
| 412.09
| 412.09
| 80/63
| 80/63
| [[Witch]]
| [[Witcher]]
|-
|-
| 1
| 1
Line 108: 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
| 4
Line 118: Line 130:
| 18/17
| 18/17
| [[World calendar]]
| [[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 125: Line 149:
| [[Oquatonic]]
| [[Oquatonic]]
|-
|-
|91
| 91
| 151\364<br>(3\364)
| 151\364<br>(3\364)
|497.80<br>(3.30)
| 497.80<br>(3.30)
|4/3<br>(176/175)
| 4/3<br>(176/175)
|[[Protactinium]]
| [[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 ==
== Scales ==
 
* WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30
* WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30<!-- 3-digit number -->

Latest revision as of 12:54, 15 May 2026

← 363edo 364edo 365edo →
Prime factorization 22 × 7 × 13
Step size 3.2967 ¢ 
Fifth 213\364 (702.198 ¢)
Semitones (A1:m2) 35:27 (115.4 ¢ : 89.01 ¢)
Consistency limit 21
Distinct consistency limit 21

364 equal divisions of the octave (abbreviated 364edo or 364ed2), also called 364-tone equal temperament (364tet) or 364 equal temperament (364et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 364 equal parts of about 3.3 ¢ each. Each step represents a frequency ratio of 21/364, or the 364th root of 2.

Theory

364edo is consistent through the 21-odd-limit with good average accuracy.

As an equal temperament, it tempers out 1600000/1594323 (amity comma) and [-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 (supporting 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

Approximation of prime harmonics in 364edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.24 -0.60 +0.40 -0.77 +0.13 +0.54 -0.81 +1.40 -1.01 -1.08
Relative (%) +0.0 +7.4 -18.2 +12.3 -23.3 +4.0 +16.4 -24.6 +42.3 -30.5 -32.7
Steps
(reduced) 		364
(0) 		577
(213) 		845
(117) 		1022
(294) 		1259
(167) 		1347
(255) 		1488
(32) 		1546
(90) 		1647
(191) 		1768
(312) 		1803
(347)
Approximation of prime harmonics in 364edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -0.79 -0.49 -0.53 +0.43 +0.12 -0.93 +0.70 -0.19 +1.62 -0.32 +1.40
Relative (%) -24.1 -14.9 -16.0 +13.0 +3.7 -28.2 +21.2 -5.6 +49.2 -9.6 +42.4
Steps
(reduced) 		1896
(76) 		1950
(130) 		1975
(155) 		2022
(202) 		2085
(265) 		2141
(321) 		2159
(339) 		2208
(24) 		2239
(55) 		2253
(69) 		2295
(111)

Subsets and supersets

Since 364 factors into primes as 22 × 7 × 13, 364edo has subset 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [577 -364⟩ [⟨364 577]] −0.0766 0.0766 2.32
2.3.5 1600000/1594323, [-65 0 28⟩ [⟨364 577 845]] +0.0350 0.1698 5.15
2.3.5.7 65625/65536, 390625/388962, 420125/419904 [⟨364 577 845 1022]] −0.0098 0.1662 5.04
2.3.5.7.11 1375/1372, 6250/6237, 19712/19683, 41503/41472 [⟨364 577 845 1022 1259]] +0.0366 0.1753 5.32
2.3.5.7.11.13 625/624, 1375/1372, 2080/2079, 2200/2197, 14641/14625 [⟨364 577 845 1022 1259 1347]] +0.0245 0.1622 4.92
2.3.5.7.11.13.17 625/624, 715/714, 1089/1088, 1225/1224, 2025/2023, 2200/2197 [⟨364 577 845 1022 1259 1347 1488]] +0.0022 0.1599 4.85
2.3.5.7.11.13.17.19 625/624, 715/714, 1089/1088, 1216/1215, 1225/1224, 1331/1330, 1729/1728 [⟨364 577 845 1022 1259 1347 1488 1546]] +0.0257 0.1620 4.91

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 103\364 339.56 243/200 Paramity
1 125\364 412.09 80/63 Witcher
1 149\364 491.21 3645/2744 Fifthplus
1 151\364 497.80 4/3 Gary
2 125\364
(57\364) 		412.09
(187.91) 		80/63
(49/44) 		Semiwitcher
2 151\364
(31\364) 		497.80
(102.20) 		4/3
(35/33) 		Gariwizmic
4 30\364 98.90 18/17 World calendar
13 151\364
(11\364) 		497.80
(36.26) 		4/3
(?) 		Aluminium
26 151\364
(11\364) 		497.80
(36.26) 		4/3
(?) 		Iron
28 151\364
(5\364) 		497.80
(16.48) 		4/3
(105/104) 		Oquatonic
91 151\364
(3\364) 		497.80
(3.30) 		4/3
(176/175) 		Protactinium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

  • WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30
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