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


== Theory ==
== Theory ==
364edo is [[consistent]] through the [[21-odd-limit]]. It [[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]] (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.  
 
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)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 364 factors into {{factorization|364}}, 364edo has subset edos {{EDOs| 2, 4, 7, 13, 14, 26, 28, 52, 91, 182 }}.
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 }}.


=== Miscellaneous properties ===
=== 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.
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.


Line 20: Line 23:
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 27: Line 30:
|-
|-
| 2.3
| 2.3
| {{monzo| 577 -364 }}
| {{Monzo| 577 -364 }}
| {{mapping| 364 577 }}
| {{Mapping| 364 577 }}
| -0.0766
| −0.0766
| 0.0766
| 0.0766
| 2.32
| 2.32
Line 35: Line 38:
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| -65 0 28 }}
| 1600000/1594323, {{monzo| -65 0 28 }}
| {{mapping| 364 577 845 }}
| {{Mapping| 364 577 845 }}
| +0.0350
| +0.0350
| 0.1698
| 0.1698
Line 42: Line 45:
| 2.3.5.7
| 2.3.5.7
| 65625/65536, 390625/388962, 420125/419904
| 65625/65536, 390625/388962, 420125/419904
| {{mapping| 364 577 845 1022 }}
| {{Mapping| 364 577 845 1022 }}
| -0.0098
| −0.0098
| 0.1662
| 0.1662
| 5.04
| 5.04
Line 49: 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
| {{mapping| 364 577 845 1022 1259 }}
| {{Mapping| 364 577 845 1022 1259 }}
| +0.0366
| +0.0366
| 0.1753
| 0.1753
Line 56: 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
| {{mapping| 364 577 845 1022 1259 1347 }}
| {{Mapping| 364 577 845 1022 1259 1347 }}
| +0.0245
| +0.0245
| 0.1622
| 0.1622
Line 63: 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
| {{mapping| 364 577 845 1022 1259 1347 1488 }}
| {{Mapping| 364 577 845 1022 1259 1347 1488 }}
| +0.0022
| +0.0022
| 0.1599
| 0.1599
Line 70: 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
| {{mapping| 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 80: Line 83:
|+ style="font-size: 105%;" | 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*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! 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
| 4
Line 123: Line 132:
|-
|-
| 13
| 13
| 151\364<br />(11\364)
| 151\364<br>(11\364)
| 497.80<br />(36.26)
| 497.80<br>(36.26)
| 4/3<br />(?)
| 4/3<br>(?)
| [[Aluminium]]
| [[Aluminium]]
|-
|-
| 26
| 26
| 151\364<br />(11\364)
| 151\364<br>(11\364)
| 497.80<br />(36.26)
| 497.80<br>(36.26)
| 4/3<br />(?)
| 4/3<br>(?)
| [[Iron]]
| [[Iron]]
|-
|-
| 28
| 28
| 151\364<br />(5\364)
| 151\364<br>(5\364)
| 497.80<br />(16.48)
| 497.80<br>(16.48)
| 4/3<br />(105/104)
| 4/3<br>(105/104)
| [[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 lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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


== 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

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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