323edo: Difference between revisions

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== Theory ==

323edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

323edo is a strong [[5-limit]] system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].

As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], ~~supporting~~ 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the 5-limit; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], [[support]]ing 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].

In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.

In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. [[1375/1372]], [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], [[12005/11979]], and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament tempering out 14641/14580.

~~Since 323 factors into~~ {{~~factorisation~~|323}}~~, 323edo shares the excellent approximations of [[25/24]]~~ in ~~[[17edo]] and~~ of ~~[[28/27]] and the [[6/5]]~~ in ~~[[19edo]].~~

=== Prime harmonics ===

{{Harmonics in equal|323|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 323edo (continued)}}

=== ~~Prime harmonics~~ ===

=== Subsets and supersets ===

Since 323 factors into primes as {{nowrap| 17 × 19 }}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[6/5]] and [[28/27]] in [[19edo]].

== Regular temperament properties ==

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

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

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

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

! colspan="2" | Tuning error

|-

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|-

| 2.3

| {{~~monzo~~| 512 -323 }}

| {{Monzo| 512 -323 }}

| {{~~mapping~~| 323 512 }}

| {{Mapping| 323 512 }}

| −0.0669

| 0.0669

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|-

| 2.3.5

| {{~~monzo~~| 24 -21 4 }}, {{monzo| 38 -2 -15 }}

| {{Monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}

| {{~~mapping~~| 323 512 750 }}

| {{Mapping| 323 512 750 }}

| −0.0538

| 0.0577

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| 2.3.5.7

| 4375/4374, 589824/588245, 703125/702464

| {{~~mapping~~| 323 512 750 907 }}

| {{Mapping| 323 512 750 907 }}

| −0.1146

| 0.1165

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| 2.3.5.7.13

| 676/675, 4096/4095, 4375/4374, 16848/16807

| {{~~mapping~~| 323 512 750 907 1195 }}

| {{Mapping| 323 512 750 907 1195 }}

| −0.0431

| 0.1770

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| 2.3.5.7.13.17

| 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374

| {{~~mapping~~| 323 512 750 907 1195 1320 }}

| {{Mapping| 323 512 750 907 1195 1320 }}

| +0.0020

| 0.1905

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| 2.3.5.7.11

| 1375/1372, 4375/4374, 5632/5625, 14641/14580

| {{~~mapping~~| 323 512 750 907 1117 }} (323)

| {{Mapping| 323 512 750 907 1117 }} (323)

| −0.0066

| 0.2399

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| 2.3.5.7.11.13

| 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374

| {{~~mapping~~| 323 512 750 907 1117 1195 }} (323)

| {{Mapping| 323 512 750 907 1117 1195 }} (323)

| +0.0350

| 0.2380

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| 2.3.5.7.11

| 540/539, 4375/4374, 12005/11979, 16384/16335

| {{~~mapping~~| 323 512 750 907 1118 }} (323e)

| {{Mapping| 323 512 750 907 1118 }} (323e)

| −0.2213

| 0.2375

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| 2.3.5.7.11.13

| 364/363, 540/539, 676/675, 4096/4095, 4375/4374

| {{~~mapping~~| 323 512 750 907 1118 1195 }} (323e)

| {{Mapping| 323 512 750 907 1118 1195 }} (323e)

| −0.1440

| 0.2773

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

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| 200/189

| [[Hemiluna]] (323)

|-

| 1

| 27\323

| 100.31

| 675/637

| [[Heptacot]] (323)

|-

| 1

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| 128\323

| 475.54

| ~~320~~/~~243~~

| 25/19

| [[Vulture]]

|-

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| [[Enneadecal]]

|}

<nowiki />* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<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:Deuteromere]]

[[Category:Lambeth]]

[[Category:Stockhausenic]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a strong 5-limit system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].
+edo is a strong [[5-limit]] system and an excellent tuning when considered in the no-11 [[subgroup]], with errors of 25% or less all the way into the [[31-limit]].
-As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the [[5-limit]]; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], supporting 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].
+As an equal temperament, it [[tempering out|tempers out]] the [[vulture comma]], {{monzo| 24 -21 4 }} and the [[luna comma]], {{monzo| 38 -2 -15 }}, in the 5-limit; [[4375/4374]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]], [[support]]ing 7-limit [[vulture]], [[lunatic]], [[enneadecal]], and [[gamera]].
-In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. 1375/1372, [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], 12005/11979, and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the 2.3.5.11 subgroup temperament tempering out 14641/14580.
+In the 11-limit, the 323e val and the [[patent val]] are comparable in errors. [[1375/1372]], [[5632/5625]], [[14641/14580]], and [[19712/19683]] are tempered out in the patent val; [[540/539]], [[6250/6237]], [[12005/11979]], and [[16384/16335]] are tempered out in the 323e val. It provides the [[optimal patent val]] for the rank-5 temperament tempering out [[1573/1568]], the lambeth comma, as well as 13-limit [[stockhausenic]], and [[deuteromere]], the [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament tempering out 14641/14580.
-Since 323 factors into {{factorisation|323}}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[28/27]] and the [[6/5]] in [[19edo]].
+=== Prime harmonics ===
+{{Harmonics in equal|323|columns=11}}
+{{Harmonics in equal|323|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 323edo (continued)}}
-=== Prime harmonics ===
+=== Subsets and supersets ===
-{{Harmonics in equal|323}}
+Since 323 factors into primes as {{nowrap| 17 × 19 }}, 323edo shares the excellent approximations of [[25/24]] in [[17edo]] and of [[6/5]] and [[28/27]] in [[19edo]].
 == Regular temperament properties ==
@@ Line 20: / Line 22: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 27: / Line 29: @@
 |-
 | 2.3
-| {{monzo| 512 -323 }}
+| {{Monzo| 512 -323 }}
-| {{mapping| 323 512 }}
+| {{Mapping| 323 512 }}
 | −0.0669
 | 0.0669
@@ Line 34: / Line 36: @@
 |-
 | 2.3.5
-| {{monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
+| {{Monzo| 24 -21 4 }}, {{monzo| 38 -2 -15 }}
-| {{mapping| 323 512 750 }}
+| {{Mapping| 323 512 750 }}
 | −0.0538
 | 0.0577
@@ Line 42: / Line 44: @@
 | 2.3.5.7
 | 4375/4374, 589824/588245, 703125/702464
-| {{mapping| 323 512 750 907 }}
+| {{Mapping| 323 512 750 907 }}
 | −0.1146
 | 0.1165
@@ Line 49: / Line 51: @@
 | 2.3.5.7.13
 | 676/675, 4096/4095, 4375/4374, 16848/16807
-| {{mapping| 323 512 750 907 1195 }}
+| {{Mapping| 323 512 750 907 1195 }}
 | −0.0431
 | 0.1770
@@ Line 56: / Line 58: @@
 | 2.3.5.7.13.17
 | 442/441, 676/675, 2500/2499, 4096/4095, 4375/4374
-| {{mapping| 323 512 750 907 1195 1320 }}
+| {{Mapping| 323 512 750 907 1195 1320 }}
 | +0.0020
 | 0.1905
@@ Line 63: / Line 65: @@
 | 2.3.5.7.11
 | 1375/1372, 4375/4374, 5632/5625, 14641/14580
-| {{mapping| 323 512 750 907 1117 }} (323)
+| {{Mapping| 323 512 750 907 1117 }} (323)
 | −0.0066
 | 0.2399
@@ Line 70: / Line 72: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 1375/1372, 4096/4095, 4375/4374
-| {{mapping| 323 512 750 907 1117 1195 }} (323)
+| {{Mapping| 323 512 750 907 1117 1195 }} (323)
 | +0.0350
 | 0.2380
@@ Line 77: / Line 79: @@
 | 2.3.5.7.11
 | 540/539, 4375/4374, 12005/11979, 16384/16335
-| {{mapping| 323 512 750 907 1118 }} (323e)
+| {{Mapping| 323 512 750 907 1118 }} (323e)
 | −0.2213
 | 0.2375
@@ Line 84: / Line 86: @@
 | 2.3.5.7.11.13
 | 364/363, 540/539, 676/675, 4096/4095, 4375/4374
-| {{mapping| 323 512 750 907 1118 1195 }} (323e)
+| {{Mapping| 323 512 750 907 1118 1195 }} (323e)
 | −0.1440
 | 0.2773
@@ Line 95: / Line 97: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 106: / Line 108: @@
 | 200/189
 | [[Hemiluna]] (323)
+|-
+| 1
+| 27\323
+| 100.31
+| 675/637
+| [[Heptacot]] (323)
 |-
 | 1
@@ Line 134: / Line 142: @@
 | 128\323
 | 475.54
-| 320/243
+| 25/19
 | [[Vulture]]
 |-
@@ Line 149: / Line 157: @@
 | [[Enneadecal]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<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:Deuteromere]]
 [[Category:Lambeth]]
 [[Category:Stockhausenic]]