320edo: Difference between revisions

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== 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 ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

Since 320 factors into 26 × 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~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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|-

| 2.3

| {{~~monzo~~| -507 320 }}

| {{Monzo| -507 320 }}

| [{{~~val~~| 320 507 }}]

| {{Mapping| 320 507 }}

| +0.2224

| 0.2224

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|-

| 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

| 0.2036

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| 2.3.5.7

| 65625/65536, 235298/234375, 321489/320000

| [{{~~val~~| 320 507 743 898 }}]

| {{Mapping| 320 507 743 898 }}

| +0.2361

| 0.2229

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| 2.3.5.7.11

| 441/440, 8019/8000, 41503/41472, 65625/65536

| [{{~~val~~| 320 507 743 898 1107 }}]

| {{Mapping| 320 507 743 898 1107 }}

| +0.1928

| 0.2173

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| 2.3.5.7.11.13

| 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

| 0.1993

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| 2.3.5.7.11.13.17

| 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

| 0.1968

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| 2.3.5.7.11.13.17.19

| 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

| 0.1899

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=== Rank-2 temperaments ===

{| 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 per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| 7\320

| 26.25

| {{~~monzo~~| -2 13 -8 }}

| {{Monzo| -2 13 -8 }}

| [[Sfourth]] (5-limit)

|-

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| 498.75 (18.75)

| 4/3 (81/80)

| [[~~Pental~~]]

| [[Quintile]]

|-

| 8

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| 498.75 (18.75)

| 4/3 (81/80)

| [[~~Decal~~]]

| [[Decile]]

|-

| 20

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| [[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:Werckismic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|320}}
+{{ED intro}}
 == Theory ==
-et 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.
+edo 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 ===
 {{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 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 23: / Line 26: @@
 |-
 | 2.3
-| {{monzo| -507 320 }}
+| {{Monzo| -507 320 }}
-| [{{val| 320 507 }}]
+| {{Mapping| 320 507 }}
 | +0.2224
 | 0.2224
@@ Line 30: / Line 33: @@
 |-
 | 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
 | 0.2036
@@ Line 38: / Line 41: @@
 | 2.3.5.7
 | 65625/65536, 235298/234375, 321489/320000
-| [{{val| 320 507 743 898 }}]
+| {{Mapping| 320 507 743 898 }}
 | +0.2361
 | 0.2229
@@ Line 45: / Line 48: @@
 | 2.3.5.7.11
 | 441/440, 8019/8000, 41503/41472, 65625/65536
-| [{{val| 320 507 743 898 1107 }}]
+| {{Mapping| 320 507 743 898 1107 }}
 | +0.1928
 | 0.2173
@@ Line 52: / Line 55: @@
 | 2.3.5.7.11.13
 | 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
 | 0.1993
@@ Line 59: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 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
 | 0.1968
@@ Line 66: / Line 69: @@
 | 2.3.5.7.11.13.17.19
 | 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
 | 0.1899
@@ Line 74: / Line 77: @@
 === Rank-2 temperaments ===
 {| 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
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 84: / Line 88: @@
 | 7\320
 | 26.25
-| {{monzo| -2 13 -8 }}
+| {{Monzo| -2 13 -8 }}
 | [[Sfourth]] (5-limit)
 |-
@@ Line 115: / Line 119: @@
 | 498.75<br>(18.75)
 | 4/3<br>(81/80)
-| [[Pental]]
+| [[Quintile]]
 |-
 | 8
@@ Line 133: / Line 137: @@
 | 498.75<br>(18.75)
 | 4/3<br>(81/80)
-| [[Decal]]
+| [[Decile]]
 |-
 | 20
@@ Line 159: / Line 163: @@
 | [[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:Werckismic]]