331edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|331}}
{{EDO intro|331}}
== 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" | [[Subgroup]]
! rowspan="2" |[[Comma list|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]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.9
| 2.9
|{{monzo|-1049 331}}
| {{monzo| -1049 331 }}
|{{val|331 1049}}
| {{mapping| 331 1049 }}
| 0.1402
| 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| -74 -5 23 }}
|{{val|331 1049 1293}}
| {{mapping| 331 1049 1293 }}
| 0.1494
| 0.1494
| 0.1152
| 0.1152
| 3.18
| 3.18
|-
|-
|2.9.15.7
| 2.9.15.7
|65625/65536, 420175/419904, 80387359983/80000000000
| 65625/65536, 420175/419904, 80387359983/80000000000
|{{val|331 1049 1293 929}}
| {{mapping| 331 1049 1293 929 }}
| 0.1878
| 0.1878
| 0.1199
| 0.1199
| 3.31
| 3.31
|-
|-
|2.9.15.7.11
| 2.9.15.7.11
|9801/9800, 41503/41472, 137781/137500, 759375/758912
| 9801/9800, 41503/41472, 137781/137500, 759375/758912
|{{val|331 1049 1293 929 1145}}
| {{mapping| 331 1049 1293 929 1145 }}
| 0.1653
| 0.1653
| 0.1163
| 0.1163
| 3.21
| 3.21
|-
|-
|2.9.15.7.11.13
| 2.9.15.7.11.13
|729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000
| 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000
|{{val|331 1049 1293 929 1145 1225}}
| {{mapping| 331 1049 1293 929 1145 1225 }}
| 0.1125
| 0.1125
| 0.1587
| 0.1587
| 4.38
| 4.38
|-
|-
|2.9.15.7.11.13.17
| 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
| 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}}
| {{mapping| 331 1049 1293 929 1145 1225 1353 }}
| 0.0901
| 0.0901
| 0.1568
| 0.1568
Line 65: Line 71:
|+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
|-
|-
|1
| 1
|89\331
| 89\331
|322.66
| 322.66
|6/5
| 6/5
|[[Magicaltet]]
| [[Magicaltet]]
|-
|-
|1
| 1
|107\331
| 107\331
|387.92
| 387.92
|5/4
| 5/4
|[[Würschmidt]]
| [[Würschmidt]]
|-
|-
|1
| 1
|137\331
| 137\331
|496.68
| 496.68
|5457/4096
| 5457/4096
|[[Edson]]
| [[Edson]]
|}
|}


==Scales==
== Scales ==
*[[Magicaltet7]]
* [[Magicaltet7]]
*[[Magicaltet11]]
* [[Magicaltet11]]
*[[Magicaltet15]]
* [[Magicaltet15]]


==Music==
== Music ==
*[https://www.youtube.com/watch?v=g3FF6oqnilk Silent Silence] by [[User:Francium|Francium]]
; [[User:Francium|Francium]]
* "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] – edson in 331edo tuning

Revision as of 09:14, 15 December 2023

← 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

Template:EDO intro

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⟩, [-74 -5 23⟩ [⟨331 1049 1293]] 0.1494 0.1152 3.18
2.9.15.7 65625/65536, 420175/419904, 80387359983/80000000000 [⟨331 1049 1293 929]] 0.1878 0.1199 3.31
2.9.15.7.11 9801/9800, 41503/41472, 137781/137500, 759375/758912 [⟨331 1049 1293 929 1145]] 0.1653 0.1163 3.21
2.9.15.7.11.13 729/728, 1575/1573, 10648/10647, 41503/41472, 43904/43875, 53361/53248, 20336647/2028000 [⟨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 [⟨331 1049 1293 929 1145 1225 1353]] 0.0901 0.1568 4.33

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 89\331 322.66 6/5 Magicaltet
1 107\331 387.92 5/4 Würschmidt
1 137\331 496.68 5457/4096 Edson

Scales

Music

Francium
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