292edo: Difference between revisions

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-The '''292 equal divisions of the octave''' ('''292edo''') is the [[EDO|equal division of the octave]] into 292 parts of 4.1095 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
-edo is closely related to [[146edo]], but the patent vals differ on the mapping for 3. It tempers out {{monzo| 3 -18 11 }} (quartonic comma) and {{monzo| 38 -2 -15 }} (luna/hemithirds comma) in the [[5-limit]]; 5120/5103 ([[5120/5103|hemifamity]]), 390625/388962 ([[dimcomp comma|dimcomp]]), 420175/419904 (wizma), and 4802000/4782969 ([[canousma]]) in the [[7-limit]]; 1375/1372, 5632/5625, [[6250/6237]], [[9801/9800]] and [[14641/14580]] in the [[11-limit]]; [[352/351]], [[625/624]], [[847/845]], [[1716/1715]], and [[2080/2079]] in the [[13-limit]].
+== Theory ==
+edo is closely related to [[146edo]], but the [[patent val]]s differ on the mapping for [[3/1|3]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 3 -18 11 }} ([[quartonic comma]]) and {{monzo| 38 -2 -15 }} ([[luna comma|luna/hemithirds comma]]) in the [[5-limit]]; 5120/5103 ([[5120/5103|hemifamity]]), 390625/388962 ([[dimcomp comma|dimcomp]]), 420175/419904 ([[wizma]]), and 4802000/4782969 ([[canousma]]) in the [[7-limit]]; 1375/1372, [[5632/5625]], [[6250/6237]], [[9801/9800]] and [[14641/14580]] in the [[11-limit]]; [[352/351]], [[625/624]], [[847/845]], [[1716/1715]], and [[2080/2079]] in the [[13-limit]].
-It provides the [[optimal patent val]] for the [[undim]] temperament in the 7-, 11-, and 13-limit, and notably [[support]]s [[Hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[Luna family #Semiluna|semiluna]].
+It provides the [[optimal patent val]] for the [[undim]] temperament in the 7-, 11-, and 13-limit, and notably [[support]]s [[hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[semiluna]].
-[[Category:Equal divisions of the octave]]
+=== Prime harmonics ===
-[[Category:Hemifamity]]
+{{Harmonics in equal|292}}
+=== Subsets and supersets ===
+Since 292 factors into 2<sup>2</sup> × 73, 292edo has subset edos {{EDOs| 2, 4, 73, and 146 }}.
+== 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| 463 -292 }}
+| {{mapping| 292 463 }}
+| −0.2476
+| 0.2475
+| 6.02
+|-
+| 2.3.5
+| {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}
+| {{mapping| 292 463 678 }}
+| −0.1633
+| 0.2346
+| 5.71
+|-
+| 2.3.5.7
+| 5120/5103, 390625/388962, 420175/419904
+| {{mapping| 292 463 678 820 }}
+| −0.2148
+| 0.2219
+| 5.40
+|-
+| 2.3.5.7.11
+| 1375/1372, 5120/5103, 5632/5625, 14641/14580
+| {{mapping| 292 463 678 820 1010 }}
+| −0.1353
+| 0.2544
+| 6.19
+|-
+| 2.3.5.7.11.13
+| 352/351, 625/624, 847/845, 1716/1715, 14641/14580
+| {{mapping| 292 463 678 820 1010 1081 }}
+| −0.3480
+| 0.2736
+| 6.66
+|-
+| 2.3.5.7.11.13.17
+| 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023
+| {{mapping| 292 463 678 820 1010 1081 1194 }}
+| −0.2376
+| 0.2696
+| 6.56
+|}
+=== 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
+| 11\292
+| 45.21
+| 250/243
+| [[Quartonic]] (5-limit)
+|-
+| 1
+| 47\292
+| 193.15
+| 262144/234375
+| [[Luna]]
+|-
+| 1
+| 59\292
+| 242.47
+| 147/128
+| [[Septiquarter]]
+|-
+| 1
+| 111\292
+| 456.16
+| 125/96
+| [[Qak]]
+|-
+| 2
+| 47\292
+| 193.15
+| 121/108
+| [[Semiluna]]
+|-
+| 2
+| 59\292
+| 242.47
+| 121/105
+| [[Semiseptiquarter]]
+|-
+| 4
+| 121\292
+| 497.26
+| 4/3
+| [[Undim]]
+|}
+<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:Aberschismic]]
 [[Category:Septiquarter]]
 [[Category:Semiluna]]
 [[Category:Undim]]