364edo: Difference between revisions

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== Theory ==
== Theory ==
364edo is consistent through the [[21-odd-limit]], [[tempering out]] 1600000/1594323 ([[amity comma]]) and {{monzo| -65 0 28 }}; ([[oquatonic 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]]. The equal temperament [[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]] (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]].
 
364 is divisible by, and thus contains sub-edos {{EDOs|1, 2, 4, 7, 13, 14, 26, 28, 52, 91, 182}}.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|364|columns=11}}
{{Harmonics in equal|364|columns=11}}
=== Subsets and supersets ===
Since 364 factors into {{factorization|364}}, 364edo has subset edos {{EDOs| 2, 4, 7, 13, 14, 26, 28, 52, 91, 182 }}.


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


== 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 577 -364 }}
| {{monzo| 577 -364 }}
| [{{val| 364 577 }}]
| {{mapping| 364 577 }}
| -0.0766
| -0.0766
| 0.0766
| 0.0766
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| 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
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| 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
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| 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
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| 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
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| 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
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| 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
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|+Table of rank-2 temperaments by generator
|+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
|-
|-
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| [[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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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 -->

Revision as of 15:14, 9 November 2023

← 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

Template:EDO intro

Theory

364edo is consistent through the 21-odd-limit. The equal temperament 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, 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.

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)

Subsets and supersets

Since 364 factors into 22 × 7 × 13, 364edo has subset edos 2, 4, 7, 13, 14, 26, 28, 52, 91, 182.

Miscellaneous properties

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 Amity / paramity
1 125\364 412.09 80/63 Witch
1 149\364 491.21 3645/2744 Fifthplus
1 151\364 497.80 4/3 Gary
2 57\364 187.91 49/44 Semiwitch
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 it is distinct

Scales

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