320edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|320}}
{{ED intro}}


== Theory ==
== Theory ==
320et tempers out 65625/65536 (horwell) and 420175/419904 (wizma) in the 7-limit and [[441/440]], [[8019/8000]] and [[9801/9800]] in the 11-limit, and so [[support]]s 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 a rank-3 temperament.  
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 harmonic]]s from 3 to 19, with the sole exception of [[17/1|17]].
 
As an equal temperament, it [[tempering out|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 [[support]]s 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 ===
=== Prime harmonics ===
{{Harmonics in equal|320|intervals=prime|columns=11}}
{{Harmonics in equal|320|intervals=prime|columns=11}}


=== Divisors ===
=== Subsets and supersets ===
Since 320 factors into 2<sup>6</sup> × 5, 320edo has subset edos {{EDOs| 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160 }}.
Since 320 factors into 2<sup>6</sup> × 5, 320edo has subset edos {{EDOs| 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160 }}.


== 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]]
! 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| -507 320 }}
| {{Monzo| -507 320 }}
| [{{val| 320 507 }}]
| {{Mapping| 320 507 }}
| +0.2224
| +0.2224
| 0.2224
| 0.2224
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 23 6 -14 }}, {{monzo| -28 25 -5 }}
| {{Monzo| 23 6 -14 }}, {{monzo| -28 25 -5 }}
| [{{val| 320 507 743 }}]
| {{Mapping| 320 507 743 }}
| +0.1574
| +0.1574
| 0.2036
| 0.2036
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| 2.3.5.7
| 2.3.5.7
| 65625/65536, 235298/234375, 321489/320000
| 65625/65536, 235298/234375, 321489/320000
| [{{val| 320 507 743 898 }}]
| {{Mapping| 320 507 743 898 }}
| +0.2361
| +0.2361
| 0.2229
| 0.2229
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| 2.3.5.7.11
| 2.3.5.7.11
| 441/440, 8019/8000, 41503/41472, 65625/65536
| 441/440, 8019/8000, 41503/41472, 65625/65536
| [{{val| 320 507 743 898 1107 }}]
| {{Mapping| 320 507 743 898 1107 }}
| +0.1928
| +0.1928
| 0.2173
| 0.2173
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655
| 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655
| [{{val| 320 507 743 898 1107 1184 }}]
| {{Mapping| 320 507 743 898 1107 1184 }}
| +0.1845
| +0.1845
| 0.1993
| 0.1993
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224
| 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224
| [{{val| 320 507 743 898 1107 1184 1308 }}]
| {{Mapping| 320 507 743 898 1107 1184 1308 }}
| +0.1565
| +0.1565
| 0.1968
| 0.1968
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520
| 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520
| [{{val| 320 507 743 898 1107 1184 1308 1359 }}]
| {{Mapping| 320 507 743 898 1107 1184 1308 1359 }}
| +0.1741
| +0.1741
| 0.1899
| 0.1899
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=== 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
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 7\320
| 7\320
| 26.25
| 26.25
| {{monzo| -2 13 -8 }}
| {{Monzo| -2 13 -8 }}
| [[Sfourth]] (5-limit)
| [[Sfourth]] (5-limit)
|-
|-
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| 498.75<br>(18.75)
| 498.75<br>(18.75)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Pental]]
| [[Quintile]]
|-
|-
| 8
| 8
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| 498.75<br>(18.75)
| 498.75<br>(18.75)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Decal]]
| [[Decile]]
|-
| 20
| 151\320<br>(7\320)
| 566.25<br>(26.25)
| 165/119<br>(?)
| [[Soviet ferris wheel]]
|-
| 32
| 133\320<br>(3\320)
| 498.75<br>(11.25)
| 4/3<br>(?)
| [[Bezique]]
|-
|-
| 80
| 80
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| [[Octogintic]]
| [[Octogintic]]
|}
|}
<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


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

Latest revision as of 13:33, 13 March 2026

← 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

320 equal divisions of the octave (abbreviated 320edo or 320ed2), also called 320-tone equal temperament (320tet) or 320 equal temperament (320et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 320 equal parts of exactly 3.75 ¢ each. Each step represents a frequency ratio of 21/320, or the 320th root of 2.

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.

As an equal temperament, it 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
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

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
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) 		Quintile
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) 		Decile
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

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

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