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 ~~harmonics~~ are much improved. It has a sharp tendency, with ~~[[harmonic]]s~~ 3 through 17 all tuned sharp. 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 23 × 41, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.

== Regular temperament properties ==

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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

| 0.229

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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

| 0.205

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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

| 0.188

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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

| 0.174

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|+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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| [[Hemicountercomp]]

|}

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

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

[[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 harmonics are much improved. It has a sharp tendency, with [[harmonic]]s 3 through 17 all tuned sharp. 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|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 2<sup>3</sup> × 41, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.
 == Regular temperament properties ==
@@ Line 24: / Line 24: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 589824/588245
-| [{{val| 328 520 762 921 }}]
+| {{mapping| 328 520 762 921 }}
 | -0.298
 | 0.229
@@ Line 31: / Line 31: @@
 | 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
 | 0.205
@@ Line 38: / Line 38: @@
 | 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
 | 0.188
@@ Line 45: / Line 45: @@
 | 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
 | 0.174
@@ Line 57: / Line 57: @@
 |+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 98: / Line 98: @@
 | [[Hemicountercomp]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Hemiwürschmidt]]
 [[Category:Semiporwell]]