331edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|331}} == Theory == 331et tempers out 78125000/78121827, 5120/5103 and 1959552/1953125 in the 7-limit; 806736/805255, 1835008/1830125, 101921..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|331}}
{{ED intro}}
 
== Theory ==
== Theory ==
331et tempers out [[78125000/78121827]], [[5120/5103]] and 1959552/1953125 in the 7-limit; 806736/805255, 1835008/1830125, 1019215872/1019046875, 12005/11979, [[16384/16335]], 2359296/2358125, 42875/42768, 180224/180075, 1684375/1679616, 968000/964467, [[3025/3024]], 78408/78125, 1362944/1361367, 4108797/4096000 and 43923/43750 in the 11-limit.
331edo is only [[consistent]] to the [[5-odd-limit]] and the errors of both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large, commending itself as a temperament of the 2.9.15.7.11.13.17.19 [[subgroup]].
===Odd harmonics===
 
Using the [[patent val]] nonetheless, the equal temperament [[tempering out|tempers out]] [[5120/5103]], 1959552/1953125 and [[78125000/78121827]] in the 7-limit; [[3025/3024]], 12005/11979, [[16384/16335]], 42875/42768, 43923/43750, 78408/78125, and 180224/180075 in the 11-limit.
 
=== Odd harmonics ===
{{Harmonics in equal|331}}
{{Harmonics in equal|331}}
===Subsets and supersets===
 
=== Subsets and supersets ===
331edo is the 67th [[prime edo]]. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.  
331edo is the 67th [[prime edo]]. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.  
==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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.9
! [[TE error|Absolute]] (¢)
|{{monzo|-1049 331}}
! [[TE simple badness|Relative]] (%)
|{{val|331 1049}}
|-
| 0.1402
| 2.9
| {{monzo| -1049 331 }}
| {{mapping| 331 1049 }}
| +0.1402
| 0.1402
| 0.1402
| 3.87
| 3.87
|-
|-
|2.9.15
| 2.9.15
|{{monzo|-7 17 -12}}, {{monzo|-74 -5 23}}
| {{monzo| -7 17 -12 }}, {{monzo| -81 12 11 }}
|{{val|331 1049 1293}}
| {{mapping| 331 1049 1293 }}
| 0.1494
| +0.1238
| 0.1152
| 0.1168
| 3.18
| 3.22
|-
| 2.9.15.7
| 65625/65536, 420175/419904, 80387359983/80000000000
| {{mapping| 331 1049 1293 929 }}
| +0.1685
| 0.1275
| 3.52
|-
|-
|2.9.15.7
| 2.9.15.7.11
|65625/65536, 420175/419904, 80387359983/80000000000
| 9801/9800, 41503/41472, 137781/137500, 759375/758912
|{{val|331 1049 1293 929}}
| {{mapping| 331 1049 1293 929 1145 }}
| 0.1878
| +0.1499
| 0.1199
| 0.1200
| 3.31
| 3.31
|-
|-
|2.9.15.7.11
| 2.9.15.7.11.13
|9801/9800, 41503/41472, 137781/137500, 759375/758912
| 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875
|{{val|331 1049 1293 929 1145}}
| {{mapping| 331 1049 1293 929 1145 1225 }}
| 0.1653
| +0.0997
| 0.1163
| 3.21
|-
|2.9.15.7.11.13
|729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000
|{{val|331 1049 1293 929 1145 1225}}
| 0.1125
| 0.1587
| 4.38
|-
|2.9.15.7.11.13.17
|729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 14161/14157, 14175/14144, 43904/43875, 18816/18785, 92823/95744
|{{val|331 1049 1293 929 1145 1225 1353}}
| 0.0901
| 0.1568
| 0.1568
| 4.33
| 4.33
|-
| 2.9.15.7.11.13.17
| 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 18816/18785
| {{mapping| 331 1049 1293 929 1145 1225 1353 }}
| +0.0791
| 0.1537
| 4.24
|}
|}


=== 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
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|89\331
| 107\331
|322.66
| 387.92
|6/5
| 5/4
|Magicaltet
| [[Würschmidt]] (331, 5-limit)
|-
|1
|137\331
|496.68
|5457/4096
|Edson
|}
|}
<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 ==
* [[Magicaltet7]]
* [[Magicaltet11]]
* [[Magicaltet15]]


==Scales==
== Music ==
*[[Magicaltet7]]
; [[User:Francium|Francium]]
*[[Magicaltet11]]
* "Silent Silence" from ''Edson EP'' (2023) – [https://open.spotify.com/track/6q3xr4E4QIL9BMaZyf6LXd Spotify] | [https://francium223.bandcamp.com/track/silent-silence Bandcamp] | [https://www.youtube.com/watch?v=g3FF6oqnilk YouTube] – in Edson, 331edo tuning
*[[Magicaltet15]]
* "Moth Mustard" from ''Unsuspecting Tyrant Double-Decker Beef Fort'' (2026) – [https://open.spotify.com/track/6M1I3YWmWvHM5bnMUz4bYg Spotify] | [https://francium223.bandcamp.com/track/moth-mustard Bandcamp] | [https://www.youtube.com/watch?v=arg4hjvpeuY YouTube]


==Music==
[[Category:Listen]]
*[https://www.youtube.com/watch?v=g3FF6oqnilk Silent Silence] by [[User:Francium|Francium]]

Latest revision as of 13:32, 13 March 2026

← 330edo 331edo 332edo →
Prime factorization 331 (prime)
Step size 3.62538 ¢ 
Fifth 194\331 (703.323 ¢)
Semitones (A1:m2) 34:23 (123.3 ¢ : 83.38 ¢)
Dual sharp fifth 194\331 (703.323 ¢)
Dual flat fifth 193\331 (699.698 ¢)
Dual major 2nd 56\331 (203.021 ¢)
Consistency limit 5
Distinct consistency limit 5

331 equal divisions of the octave (abbreviated 331edo or 331ed2), also called 331-tone equal temperament (331tet) or 331 equal temperament (331et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 331 equal parts of about 3.63 ¢ each. Each step represents a frequency ratio of 21/331, or the 331st root of 2.

Theory

331edo is only consistent to the 5-odd-limit and the errors of both harmonics 3 and 5 are quite large, commending itself as a temperament of the 2.9.15.7.11.13.17.19 subgroup.

Using the patent val nonetheless, the equal temperament tempers out 5120/5103, 1959552/1953125 and 78125000/78121827 in the 7-limit; 3025/3024, 12005/11979, 16384/16335, 42875/42768, 43923/43750, 78408/78125, and 180224/180075 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 331edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.37 +1.60 -0.85 -0.89 -0.26 +0.56 -0.66 +0.18 -0.23 +0.52 -1.08
Relative (%) +37.7 +44.2 -23.4 -24.5 -7.2 +15.4 -18.1 +5.0 -6.4 +14.3 -29.9
Steps
(reduced) 		525
(194) 		769
(107) 		929
(267) 		1049
(56) 		1145
(152) 		1225
(232) 		1293
(300) 		1353
(29) 		1406
(82) 		1454
(130) 		1497
(173)

Subsets and supersets

331edo is the 67th prime edo. 662edo, which doubles it, gives a good correction to the harmonics 3 and 5.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [-1049 331⟩ [⟨331 1049]] +0.1402 0.1402 3.87
2.9.15 [-7 17 -12⟩, [-81 12 11⟩ [⟨331 1049 1293]] +0.1238 0.1168 3.22
2.9.15.7 65625/65536, 420175/419904, 80387359983/80000000000 [⟨331 1049 1293 929]] +0.1685 0.1275 3.52
2.9.15.7.11 9801/9800, 41503/41472, 137781/137500, 759375/758912 [⟨331 1049 1293 929 1145]] +0.1499 0.1200 3.31
2.9.15.7.11.13 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875 [⟨331 1049 1293 929 1145 1225]] +0.0997 0.1568 4.33
2.9.15.7.11.13.17 729/728, 833/832, 1089/1088, 2025/2023, 10648/10647, 18816/18785 [⟨331 1049 1293 929 1145 1225 1353]] +0.0791 0.1537 4.24

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 107\331 387.92 5/4 Würschmidt (331, 5-limit)

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

Scales

Music

Francium
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