328edo: Difference between revisions

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== Theory ==

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]].

328edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. Like 164edo, it inherits the [[3/2|perfect fifth]] from [[41edo]]. 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 ===

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=== Subsets and supersets ===

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

== 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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| 2.3.5.7

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

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

| {{Mapping| 328 520 762 921 }}

| −0.298

| 0.229

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| 2.3.5.7.11

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

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

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

| −0.303

| 0.205

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| 2.3.5.7.11.13

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

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

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

| −0.295

| 0.188

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| 2.3.5.7.11.13.17

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

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

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

| −0.293

| 0.174

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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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|-

| 2

| ~~111~~\328~~<br~~ />(53\328)

| 47\328

| ~~406.10 (~~193.90)

| 171.95

| ~~495/392 (~~28/25)

| 243/220

| [[Semiosiris]]

|-

| 2

| 53\328

| 193.90

| 28/25

| [[Semihemiwürschmidt]]

|-

| 8

| ~~136\328 (~~13\328)

| 13\328

| ~~497.56 (~~47.56)

| 47.56

| ~~4/3 (~~36/35)

| 36/35

| [[Twilight]]

|-

| 41

| ~~49\328 (~~1\328)

| 1\328

| ~~179.27 (~~3.66)

| 3.66

| ~~567/512 (~~352/351)

| 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~~

<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]

[[Category:Hemiwürschmidt]]

[[Category:Semiporwell]]

@@ Line 3: / Line 3: @@
 == Theory ==
-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]].
+edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. Like 164edo, it inherits the [[3/2|perfect fifth]] from [[41edo]]. 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 ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
 Since 328 factors into {{Factorisation|328}}, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.
 == Regular temperament properties ==
@@ Line 17: / Line 17: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 25: / Line 25: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 589824/588245
-| {{mapping| 328 520 762 921 }}
+| {{Mapping| 328 520 762 921 }}
 | −0.298
 | 0.229
@@ Line 32: / Line 32: @@
 | 2.3.5.7.11
 | 2401/2400, 3136/3125, 9801/9800, 19712/19683
-| {{mapping| 328 520 762 921 1135 }}
+| {{Mapping| 328 520 762 921 1135 }}
 | −0.303
 | 0.205
@@ Line 39: / Line 39: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
-| {{mapping| 328 520 762 921 1135 1214 }}
+| {{Mapping| 328 520 762 921 1135 1214 }}
 | −0.295
 | 0.188
@@ Line 46: / Line 46: @@
 | 2.3.5.7.11.13.17
 | 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
-| {{mapping| 328 520 762 921 1135 1214 1341 }}
+| {{Mapping| 328 520 762 921 1135 1214 1341 }}
 | −0.293
 | 0.174
@@ Line 58: / Line 58: @@
 |+ 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 83: / Line 83: @@
 |-
 | 2
-| 111\328<br />(53\328)
+| 47\328
-| 406.10<br />(193.90)
+| 171.95
-| 495/392<br />(28/25)
+| 243/220
+| [[Semiosiris]]
+|-
+| 2
+| 53\328
+| 193.90
+| 28/25
 | [[Semihemiwürschmidt]]
 |-
 | 8
-| 136\328<br />(13\328)
+| 13\328
-| 497.56<br />(47.56)
+| 47.56
-| 4/3<br />(36/35)
+| 36/35
 | [[Twilight]]
 |-
 | 41
-| 49\328<br />(1\328)
+| 1\328
-| 179.27<br />(3.66)
+| 3.66
-| 567/512<br />(352/351)
+| 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
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 [[Category:Hemiwürschmidt]]
 [[Category:Semiporwell]]