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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|152}}
{{ED intro}}


== Theory ==
== Theory ==
152edo is a strong [[11-limit]] system, with the [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] slightly sharp. It [[tempers out]] 1600000/1594323 ([[amity comma]]) and {{monzo| 32 -7 -9 }} ([[escapade comma]]) in the 5-limit; [[4375/4374]], [[5120/5103]], [[6144/6125]] and [[16875/16807]] in the 7-limit; [[540/539]], [[1375/1372]], [[3025/3024]], [[4000/3993]], [[5632/5625]] and [[9801/9800]] in the 11-limit. It provides the [[optimal patent val]] for the 11-limit linear temperaments [[amity]], [[grendel]], and [[kwai]], and the 11-limit planar temperament [[laka]].  
364edo is [[consistent]] through the [[21-odd-limit]] with good average accuracy.  


It has two reasonable mappings for 13, with the 152f val scoring much better. The 152f val tempers out [[352/351]], [[625/624]], [[640/637]], [[729/728]], [[847/845]], [[1188/1183]], [[1575/1573]], [[1716/1715]] and [[2080/2079]], [[support]]ing and giving an excellent tuning for amity, kwai, and laka. The optimal tuning of this temperament is [[consistent]] in the 15-integer-limit. The [[patent val]] tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1001/1000]], [[1573/1568]], and [[4096/4095]], providing the optimal patent val for the 13-limit rank-5 temperament tempering out 169/168, as well as some further temperaments thereof, such as [[octopus]].
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]].
 
[[Paul Erlich]] has suggested that 152edo could be considered a sort of [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3038.html#3041 universal tuning].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|152}}
{{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 152 factors into {{factorisation}}, 152edo has subset edos {{EDOs| 2, 4, 8, 19, 38, 76 }}.  
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 ==
Line 21: 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 28: Line 30:
|-
|-
| 2.3
| 2.3
| {{monzo| 241 -152 }}
| {{Monzo| 577 -364 }}
| {{mapping| 152 241 }}
| {{Mapping| 364 577 }}
| −0.213
| −0.0766
| 0.213
| 0.0766
| 2.70
| 2.32
|-
|-
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| 32 -7 -9 }}
| 1600000/1594323, {{monzo| -65 0 28 }}
| {{mapping| 152 241 353 }}
| {{Mapping| 364 577 845 }}
| −0.218
| +0.0350
| 0.174
| 0.1698
| 2.21
| 5.15
|-
|-
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 5120/5103, 16875/16807
| 65625/65536, 390625/388962, 420125/419904
| {{mapping| 152 241 353 427 }}
| {{Mapping| 364 577 845 1022 }}
| −0.362
| −0.0098
| 0.291
| 0.1662
| 3.69
| 5.04
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 1375/1372, 4000/3993, 5120/5103
| 1375/1372, 6250/6237, 19712/19683, 41503/41472
| {{mapping| 152 241 353 427 526 }}
| {{Mapping| 364 577 845 1022 1259 }}
| −0.365
| +0.0366
| 0.260
| 0.1753
| 3.30
| 5.32
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 352/351, 540/539, 625/624, 729/728, 1575/1573
| 625/624, 1375/1372, 2080/2079, 2200/2197, 14641/14625
| {{mapping| 152 241 353 427 526 563 }} (152f)
| {{Mapping| 364 577 845 1022 1259 1347 }}
| −0.494
| +0.0245
| 0.373
| 0.1622
| 4.73
| 4.92
|-
| 2.3.5.7.11.13.17
| 625/624, 715/714, 1089/1088, 1225/1224, 2025/2023, 2200/2197
| {{Mapping| 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
| {{Mapping| 364 577 845 1022 1259 1347 1488 1546 }}
| +0.0257
| 0.1620
| 4.91
|}
|}
* 152et (152fg val) has lower absolute errors in the 11-, 19-, and 23-limit than any previous equal temperaments. In the 11-limit it is the first to beat [[130edo|130]] and is superseded by [[224edo|224]]. In the 19- and 23-limit it is the first to beat [[140edo|140]] and is superseded by [[159edo|159]].
* It is best at the no-17 19- and 23-limit, in which it has lower relative errors than any previous equal temperaments. Not until [[270edo|270]] do we find a better equal temperament that does better in either of those subgroups.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
Line 69: 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
|-
|-
| 1
| 1
| 7\152
| 103\364
| 55.26
| 339.56
| 33/32
| 243/200
| [[Escapade]] / [[alphaquarter]]
| [[Paramity]]
|-
|-
| 1
| 1
| 31\152
| 125\364
| 244.74
| 412.09
| 15/13
| 80/63
| [[Subsemifourth]]
| [[Witcher]]
|-
|-
| 1
| 1
| 39\152
| 149\364
| 307.89
| 491.21
| 3200/2673
| 3645/2744
| [[Familia]]
| [[Fifthplus]]
|-
|-
| 1
| 1
| 43\152
| 151\364
| 339.47
| 497.80
| 243/200
| [[Amity]]
|-
| 1
| 49\152
| 386.84
| 5/4
| [[Grendel]]
|-
| 1
| 63\152
| 497.37
| 4/3
| 4/3
| [[Kwai]]
| [[Gary]]
|-
| 1
| 71\152
| 560.53
| 242/175
| [[Whoops]]
|-
| 2
| 7\152
| 55.26
| 33/32
| [[Septisuperfourth]]
|-
| 2
| 9\152
| 71.05
| 25/24
| [[Vishnu]] / [[acyuta]] (152f) / [[ananta]] (152)
|-
|-
| 2
| 2
| 43\152<br />(33\152)
| 125\364<br>(57\364)
| 339.47<br />(260.53)
| 412.09<br>(187.91)
| 243/200<br />(64/55)
| 80/63<br>(49/44)
| [[Hemiamity]]
| [[Semiwitcher]]
|-
|-
| 2
| 2
| 55\152<br />(21\152)
| 151\364<br>(31\364)
| 434.21<br />(165.79)
| 497.80<br>(102.20)
| 9/7<br />(11/10)
| 4/3<br>(35/33)
| [[Supers]]
| [[Gariwizmic]]
|-
|-
| 4
| 4
| 63\152<br />(13\152)
| 30\364
| 497.37<br />(102.63)
| 98.90
| 4/3<br />(35/33)
| 18/17
| [[Undim]] / [[unlit]]
| [[World calendar]]
|-
|-
| 8
| 13
| 63\152<br />(6\152)
| 151\364<br>(11\364)
| 497.37<br />(47.37)
| 497.80<br>(36.26)
| 4/3<br />(36/35)
| 4/3<br>(?)
| [[Twilight]]
| [[Aluminium]]
|-
|-
| 8
| 26
| 74\152<br />(2\152)
| 151\364<br>(11\364)
| 584.21<br />(15.79)
| 497.80<br>(36.26)
| 7/5<br />(126/125)
| 4/3<br>(?)
| [[Octoid]] (152f) / [[octopus]] (152)
| [[Iron]]
|-
|-
| 19
| 28
| 63\152<br />(1\152)
| 151\364<br>(5\364)
| 497.37<br />(7.89)
| 497.80<br>(16.48)
| 4/3<br />(225/224)
| 4/3<br>(105/104)
| [[Enneadecal]]
| [[Oquatonic]]
|-
|-
| 38
| 91
| 63\152<br />(1\152)
| 151\364<br>(3\364)
| 497.37<br />(7.89)
| 497.80<br>(3.30)
| 4/3<br />(225/224)
| 4/3<br>(176/175)
| [[Hemienneadecal]]
| [[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
 
== Music ==
; [[birdshite stalactite]]
* "athlete's feet" from ''razorblade tiddlywinks'' (2023) – [https://open.spotify.com/track/32c34U3syZDMAJkBzgh2pd Spotify] | [https://birdshitestalactite.bandcamp.com/track/athletes-feet Bandcamp] | [https://www.youtube.com/watch?v=lXqVaVn3SrA YouTube]


[[Category:Amity]]
== Scales ==
[[Category:Grendel]]
* WorldCalendar[12]: 31 30 30 31 30 30 31 30 30 31 30 30
[[Category:Kwai]]
[[Category:Laka]]
[[Category:Listen]]

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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