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320edo: Difference between revisions

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The '''320 equal division''' divides the [[octave]] into 320 [[equal]] parts of precisely 3.75 [[cent]]s each.
{{Infobox ET}}
{{ED intro}}


It 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.  
== 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 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 ===
{{Primes in edo|320}}
{{Harmonics in equal|320|intervals=prime|columns=11}}
 
=== 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 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -507 320 }}
| {{Mapping| 320 507 }}
| +0.2224
| 0.2224
| 5.93
|-
| 2.3.5
| {{Monzo| 23 6 -14 }}, {{monzo| -28 25 -5 }}
| {{Mapping| 320 507 743 }}
| +0.1574
| 0.2036
| 5.43
|-
| 2.3.5.7
| 65625/65536, 235298/234375, 321489/320000
| {{Mapping| 320 507 743 898 }}
| +0.2361
| 0.2229
| 5.94
|-
| 2.3.5.7.11
| 441/440, 8019/8000, 41503/41472, 65625/65536
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 320 507 743 898 1107 1184 1308 1359 }}
| +0.1741
| 0.1899
| 5.06
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 7\320
| 26.25
| {{Monzo| -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<br>(5\320)
| 498.75<br>(18.75)
| 4/3<br>(81/80)
| [[Quintile]]
|-
| 8
| 133\320<br>(9\320)
| 566.25<br>(33.75)
| 104/75<br>(55/54)
| [[Octowerck]]
|-
| 10
| 19\320<br>(13\320)
| 71.25<br>(48.75)
| 25/24<br>(36/35)
| [[Decavish]]
|-
| 10
| 133\320<br>(5\320)
| 498.75<br>(18.75)
| 4/3<br>(81/80)
| [[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
| 99\320<br>(3\320)
| 371.25<br>(11.25)
| 2275/1836<br>(?)
| [[Mercury]]
|-
| 80
| 133\320<br>(1\320)
| 498.75<br>(3.75)
| 4/3<br>(245/243)
| [[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:Equal divisions of the octave]]
[[Category:Varuna]]
[[Category:Varuna]]
[[Category:Werckismic]]
[[Category:Werckismic]]
Retrieved from "https://en.xen.wiki/w/320edo"