320edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 72: Line 64:
| 0.1899
| 0.1899
| 5.06
| 5.06
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 114: Line 100:
|-
|-
| 5
| 5
| 133\320<br>(5\320)
| 133\320<br />(5\320)
| 498.75<br>(18.75)
| 498.75<br />(18.75)
| 4/3<br>(81/80)
| 4/3<br />(81/80)
| [[Pental (temperament)|Pental]]
| [[Pental (temperament)|Pental]]
|-
|-
| 8
| 8
| 133\320<br>(9\320)
| 133\320<br />(9\320)
| 566.25<br>(33.75)
| 566.25<br />(33.75)
| 104/75<br>(55/54)
| 104/75<br />(55/54)
| [[Octowerck]]
| [[Octowerck]]
|-
|-
| 10
| 10
| 19\320<br>(13\320)
| 19\320<br />(13\320)
| 71.25<br>(48.75)
| 71.25<br />(48.75)
| 25/24<br>(36/35)
| 25/24<br />(36/35)
| [[Decavish]]
| [[Decavish]]
|-
|-
| 10
| 10
| 133\320<br>(5\320)
| 133\320<br />(5\320)
| 498.75<br>(18.75)
| 498.75<br />(18.75)
| 4/3<br>(81/80)
| 4/3<br />(81/80)
| [[Decal]]
| [[Decal]]
|-
|-
| 20
| 20
| 151\320<br>(7\320)
| 151\320<br />(7\320)
| 566.25<br>(26.25)
| 566.25<br />(26.25)
| 165/119<br>(?)
| 165/119<br />(?)
| [[Soviet ferris wheel]]
| [[Soviet ferris wheel]]
|-
|-
| 32
| 32
| 133\320<br>(3\320)
| 133\320<br />(3\320)
| 498.75<br>(11.25)
| 498.75<br />(11.25)
| 4/3<br>(?)
| 4/3<br />(?)
| [[Bezique]]
| [[Bezique]]
|-
|-
| 80
| 80
| 99\320<br>(3\320)
| 99\320<br />(3\320)
| 371.25<br>(11.25)
| 371.25<br />(11.25)
| 2275/1836<br>(?)
| 2275/1836<br />(?)
| [[Mercury]]
| [[Mercury]]
|-
|-
| 80
| 80
| 133\320<br>(1\320)
| 133\320<br />(1\320)
| 498.75<br>(3.75)
| 498.75<br />(3.75)
| 4/3<br>(245/243)
| 4/3<br />(245/243)
| [[Octogintic]]
| [[Octogintic]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Varuna]]
[[Category:Varuna]]
[[Category:Werckismic]]
[[Category:Werckismic]]

Revision as of 05:12, 16 November 2024

← 319edo 320edo 321edo →
Prime factorization 26 × 5
Step size 3.75 ¢ 
Fifth 187\320 (701.25 ¢)
Semitones (A1:m2) 29:25 (108.8 ¢ : 93.75 ¢)
Consistency limit 19
Distinct consistency limit 19

Template:EDO intro

Theory

320edo is consistent in the 19-odd-limit and a fairly good tuning for the 19-limit. It has a flat tendency for most prime harmonics from 3 to 19, with the sole exception of 17.

The equal temperament tempers out 65625/65536 (horwell comma) and 420175/419904 (wizma) in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports the varuna temperament, the rank-3 temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the optimal patent val. It also provides the optimal patent val for the rank-4 werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to an extension of varuna.

Prime harmonics

Approximation of prime harmonics in 320edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.71 -0.06 -1.33 -0.07 -0.53 +0.04 -1.26 +1.73 +1.67 -1.29
Relative (%) +0.0 -18.8 -1.7 -35.4 -1.8 -14.1 +1.2 -33.7 +46.0 +44.6 -34.3
Steps
(reduced) 		320
(0) 		507
(187) 		743
(103) 		898
(258) 		1107
(147) 		1184
(224) 		1308
(28) 		1359
(79) 		1448
(168) 		1555
(275) 		1585
(305)

Subsets and supersets

Since 320 factors into 26 × 5, 320edo has subset edos 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160.

Regular temperament properties

Template:Comma basis begin |- | 2.3 | [-507 320⟩ | [⟨320 507]] | +0.2224 | 0.2224 | 5.93 |- | 2.3.5 | [23 6 -14⟩, [-28 25 -5⟩ | [⟨320 507 743]] | +0.1574 | 0.2036 | 5.43 |- | 2.3.5.7 | 65625/65536, 235298/234375, 321489/320000 | [⟨320 507 743 898]] | +0.2361 | 0.2229 | 5.94 |- | 2.3.5.7.11 | 441/440, 8019/8000, 41503/41472, 65625/65536 | [⟨320 507 743 898 1107]] | +0.1928 | 0.2173 | 5.80 |- | 2.3.5.7.11.13 | 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655 | [⟨320 507 743 898 1107 1184]] | +0.1845 | 0.1993 | 5.31 |- | 2.3.5.7.11.13.17 | 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224 | [⟨320 507 743 898 1107 1184 1308]] | +0.1565 | 0.1968 | 5.25 |- | 2.3.5.7.11.13.17.19 | 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520 | [⟨320 507 743 898 1107 1184 1308 1359]] | +0.1741 | 0.1899 | 5.06 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 7\320 | 26.25 | [-2 13 -8⟩ | Sfourth (5-limit) |- | 1 | 131\320 | 491.25 | 3645/2744 | Fifthplus |- | 1 | 157\320 | 588.75 | 45/32 | Untriton (5-limit) |- | 1 | 93\320 | 348.75 | 6144/3757 | Hectosaros leap week |- | 2 | 19\320 | 71.25 | 25/24 | Narayana |- | 5 | 133\320
(5\320) | 498.75
(18.75) | 4/3
(81/80) | Pental |- | 8 | 133\320
(9\320) | 566.25
(33.75) | 104/75
(55/54) | Octowerck |- | 10 | 19\320
(13\320) | 71.25
(48.75) | 25/24
(36/35) | Decavish |- | 10 | 133\320
(5\320) | 498.75
(18.75) | 4/3
(81/80) | Decal |- | 20 | 151\320
(7\320) | 566.25
(26.25) | 165/119
(?) | Soviet ferris wheel |- | 32 | 133\320
(3\320) | 498.75
(11.25) | 4/3
(?) | Bezique |- | 80 | 99\320
(3\320) | 371.25
(11.25) | 2275/1836
(?) | Mercury |- | 80 | 133\320
(1\320) | 498.75
(3.75) | 4/3
(245/243) | Octogintic Template:Rank-2 end Template:Orf

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