328edo: Difference between revisions

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== Theory ==

328edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[164edo]]. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].

328edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].

=== Prime harmonics ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

Since 328 factors into ~~23 × 41~~, it has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.

Since 328 factors into {{Factorisation|328}}, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

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

! 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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| 2.3.5.7

| 2401/2400, 3136/3125, 589824/588245

| [{{~~val~~| 328 520 762 921 }}]

| {{mapping| 328 520 762 921 }}

| -0.298

| −0.298

| 0.229

| 6.27

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| 2.3.5.7.11

| 2401/2400, 3136/3125, 9801/9800, 19712/19683

| [{{~~val~~| 328 520 762 921 1135 }}]

| {{mapping| 328 520 762 921 1135 }}

| -0.303

| −0.303

| 0.205

| 5.61

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| 2.3.5.7.11.13

| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647

| [{{~~val~~| 328 520 762 921 1135 1214 }}]

| {{mapping| 328 520 762 921 1135 1214 }}

| -0.295

| −0.295

| 0.188

| 5.15

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| 2.3.5.7.11.13.17

| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125

| [{{~~val~~| 328 520 762 921 1135 1214 1341 }}]

| {{mapping| 328 520 762 921 1135 1214 1341 }}

| -0.293

| −0.293

| 0.174

| 4.77

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{| 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)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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|-

| 2

| 111\328 (53\328)

| 111\328 (53\328)

| 406.10 (193.90)

| 406.10 (193.90)

| 495/392 (28/25)

| 495/392 (28/25)

| [[Semihemiwürschmidt]]

|-

| 8

| 136\328 (13\328)

| 136\328 (13\328)

| 497.56 (47.56)

| 497.56 (47.56)

| 4/3 (36/35)

| 4/3 (36/35)

| [[Twilight]]

|-

| 41

| 49\328 (1\328)

| 49\328 (1\328)

| 179.27 (3.66)

| 179.27 (3.66)

| 567/512 (352/351)

| 567/512 (352/351)

| [[Hemicountercomp]]

|}

<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:Hemiwürschmidt]]

[[Category:Semiporwell]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|328}}
+{{ED intro}}
 == Theory ==
-edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[164edo]]. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].
+edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].
 === Prime harmonics ===
-{{Harmonics in equal|328|columns=11}}
+{{Harmonics in equal|328|intervals=prime|columns=11}}
-=== Divisors ===
+=== Subsets and supersets ===
-Since 328 factors into 2<sup>3</sup> × 41, it has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.
+Since 328 factors into {{Factorisation|328}}, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.
 == 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 (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 24: / Line 25: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 589824/588245
-| [{{val| 328 520 762 921 }}]
+| {{mapping| 328 520 762 921 }}
-| -0.298
+| −0.298
 | 0.229
 | 6.27
@@ Line 31: / Line 32: @@
 | 2.3.5.7.11
 | 2401/2400, 3136/3125, 9801/9800, 19712/19683
-| [{{val| 328 520 762 921 1135 }}]
+| {{mapping| 328 520 762 921 1135 }}
-| -0.303
+| −0.303
 | 0.205
 | 5.61
@@ Line 38: / Line 39: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
-| [{{val| 328 520 762 921 1135 1214 }}]
+| {{mapping| 328 520 762 921 1135 1214 }}
-| -0.295
+| −0.295
 | 0.188
 | 5.15
@@ Line 45: / Line 46: @@
 | 2.3.5.7.11.13.17
 | 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
-| [{{val| 328 520 762 921 1135 1214 1341 }}]
+| {{mapping| 328 520 762 921 1135 1214 1341 }}
-| -0.293
+| −0.293
 | 0.174
 | 4.77
@@ Line 55: / Line 56: @@
 {| 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)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 81: / Line 83: @@
 |-
 | 2
-| 111\328<br>(53\328)
+| 111\328<br />(53\328)
-| 406.10<br>(193.90)
+| 406.10<br />(193.90)
-| 495/392<br>(28/25)
+| 495/392<br />(28/25)
 | [[Semihemiwürschmidt]]
 |-
 | 8
-| 136\328<br>(13\328)
+| 136\328<br />(13\328)
-| 497.56<br>(47.56)
+| 497.56<br />(47.56)
-| 4/3<br>(36/35)
+| 4/3<br />(36/35)
 | [[Twilight]]
 |-
 | 41
-| 49\328<br>(1\328)
+| 49\328<br />(1\328)
-| 179.27<br>(3.66)
+| 179.27<br />(3.66)
-| 567/512<br>(352/351)
+| 567/512<br />(352/351)
 | [[Hemicountercomp]]
 |}
+<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|###]] <!-- 3-digit number -->
 [[Category:Hemiwürschmidt]]
 [[Category:Semiporwell]]