328edo: Difference between revisions

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The '''328 equal divisions of the octave''' ('''328edo'''), or the '''328(-tone) equal temperament''' ('''328tet''', '''328et''') when viewed from a [[regular temperament]] perspective, divides the octave into 328 [[equal]] parts of 3.659 [[cent]]s each.

== Theory ==

328edo is [[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 ~~supports~~ [[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]].

328 ~~factors into 23 × 41, with subset edos 2, 4, 8, 41, 82, and 164.~~

=== Prime harmonics ===

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

=== Subsets and supersets ===

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" | [[Subgroup]]

! 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

| [{{~~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

Line 36:

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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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=== Rank-2 temperaments ===

Note: 5-limit temperaments supported by 164et are not listed.

{| 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 ~~octave~~

|-

! 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)

| [[~~Hemicounterpyth~~]]

| [[Hemicountercomp]]

|}

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

~~[[Category:Equal divisions of the octave]]~~

[[Category:Hemiwürschmidt]]

[[Category:Semiporwell]]

@@ Line 1: / Line 1: @@
-The '''328 equal divisions of the octave''' ('''328edo'''), or the '''328(-tone) equal temperament''' ('''328tet''', '''328et''') when viewed from a [[regular temperament]] perspective, divides the octave into 328 [[equal]] parts of 3.659 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
-edo is [[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 supports [[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]].
-factors into 2<sup>3</sup> × 41, with subset edos 2, 4, 8, 41, 82, and 164.
+=== Prime harmonics ===
+{{Harmonics in equal|328|intervals=prime|columns=11}}
-=== Prime harmonics ===
+=== Subsets and supersets ===
-{{Primes in edo|328}}
+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" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 22: / 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 29: / 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 36: / 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 43: / 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 50: / Line 53: @@
 === Rank-2 temperaments ===
+Note: 5-limit temperaments supported by 164et are not listed.
 {| 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 octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 77: / 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)
-| [[Hemicounterpyth]]
+| [[Hemicountercomp]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave]]
 [[Category:Hemiwürschmidt]]
 [[Category:Semiporwell]]