328edo: Difference between revisions

← Older edit

VisualWikitext

@@ 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
 |}
-[[Category:Equal divisions of the octave]]
+=== Rank-2 temperaments ===
+Note: 5-limit temperaments supported by 164et are not listed.
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
+|-
+| 1
+| 53\328
+| 193.90
+| 28/25
+| [[Hemiwürschmidt]]
+|-
+| 1
+| 117\328
+| 428.05
+| 2800/2187
+| [[Osiris]]
+|-
+| 2
+| 17\328
+| 62.20
+| 28/27
+| [[Eagle]]
+|-
+| 2
+| 111\328<br />(53\328)
+| 406.10<br />(193.90)
+| 495/392<br />(28/25)
+| [[Semihemiwürschmidt]]
+|-
+| 8
+| 136\328<br />(13\328)
+| 497.56<br />(47.56)
+| 4/3<br />(36/35)
+| [[Twilight]]
+|-
+| 41
+| 49\328<br />(1\328)
+| 179.27<br />(3.66)
+| 567/512<br />(352/351)
+| [[Hemicountercomp]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Hemiwürschmidt]]
 [[Category:Semiporwell]]