320edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''320 equal divisions of the octave''' ('''320edo'''), or the '''320(-tone) equal temperament''' ('''320tet''', '''320et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 320 [[equal]] parts of precisely 3.75 [[cent]]s each.
{{EDO intro|320}}


== Theory ==
== Theory ==
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=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|320|intervals=prime|columns=11}}
{{Harmonics in equal|320|intervals=prime|columns=11}}
=== Divisors ===
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 ==
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! 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]] (¢)
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Cents<br>(Reduced)
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| [[Untriton]] (5-limit)
| [[Untriton]] (5-limit)
|-
|-
|1
| 1
|93\320
| 93\320
|348.75
| 348.75
|6144/3757
| 6144/3757
|[[Hectosaros leap week]] (no-5, no-11)
| [[Hectosaros leap week]]
|-
|-
| 2
| 2
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| [[Decal]]
| [[Decal]]
|-
|-
|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
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|}
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Varuna]]
[[Category:Varuna]]
[[Category:Werckismic]]
[[Category:Werckismic]]

Revision as of 08:00, 6 January 2023

← 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

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 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 a rank-3 temperament.

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)

Divisors

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
(Reduced) 		Cents
(Reduced) 		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) 		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
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
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