330edo: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|330}}
+{{ED intro}}
-edo has a flat tendency, with its 3rd, 5th, and 7th harmonics tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.
+== Theory ==
+edo has a flat tendency, with its [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] tuned progressively flatter. In the 11-limit, the 330e [[val]] {{val| 330 523 766 926 '''1141''' }} scores significantly better in [[TE error]] than its [[patent val]] {{val| 330 523 766 926 '''1142''' }} and allows an extension to the 13-limit.
-It tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[meter comma|meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].
+The equal temperament tempers out 32805/32768 ([[schisma]]) in the [[5-limit]]; 250047/250000 ([[landscape comma]]), 703125/702464 ([[meter]]) and 4802000/4782969 ([[canousma]]) in the [[7-limit]]. Using the 330e val, it tempers out 385/384 ([[keenanisma]]), 9801/9800 ([[kalisma]]), and 14641/14580 ([[semicanousma]]) in the [[11-limit]]; 847/845 ([[847/845|cuthbert]]) and 1001/1000 ([[1001/1000|sinbadma]]) in the [[13-limit]].
 It provides a nice tuning for [[keenanismic family|keenanismic]], the rank-4 temperament that tempers out 385/384 (even better than its [[optimal patent val]] [[284edo]]), and actually a close-to-optimal tuning for 11-limit [[Canou family #semicanou|semicanou]], the rank-3 temperament that tempers out 9801/9800 and 14641/14580.
@@ Line 12: / Line 13: @@
 === Subsets and supersets ===
-Since 330 factors into 2 × 3 × 5 × 11, it has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.
+Since 330 factors into {{factorisation|330}}, 330edo has subset edos {{EDOs| 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, and 165 }}.
+== 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 (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| -523 330 }}
+| {{mapping| 330 523 }}
+| +0.0432
+| 0.0432
+| 1.19
+|-
+| 2.3.5
+| 32805/32768, {{monzo| -2 -50 35 }}
+| {{mapping| 330 523 766 }}
+| +0.1521
+| 0.1581
+| 4.35
+|-
+| 2.3.5.7
+| 32805/32768, 250047/250000, 823543/819200
+| {{mapping| 330 523 766 926 }}
+| +0.2524
+| 0.2212
+| 6.08
+|}
+=== Rank-2 temperaments ===
+{| 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
+| 137\330
+| 498.182
+| 4/3
+| [[Helmholtz (temperament)|Helmholtz]]
+|-
+| 3
+| 137\330<br />(27\330)
+| 498.182<br />(98.182)
+| 4/3<br />(200/189)
+| [[Term (temperament)|Term]]
+|-
+| 6
+| 137\330<br />(27\330)
+| 498.182<br />(98.182)
+| 4/3<br />(200/189)
+| [[Semiterm]] (330eff)
+|-
+| 22
+| 137\330<br />(2\330)
+| 498.182<br />(7.273)
+| 4/3<br />({{monzo| 16 -13 2 }})
+| [[Major arcana]] (5-limit)
+|}
+<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:Canou]]
 [[Category:Semicanousmic]]
 [[Category:Keenanismic]]