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'''292edo''' is the [[EDO|equal division of the octave]] into 292 parts of 4.1095 cents each. It is closely related to [[146edo]], but the patent vals differ on the mapping for 3. It tempers out 390625000/387420489 (quartonic comma) and 274877906944/274658203125 (hemithirds comma) in the 5-limit; 5120/5103, 390625/388962, and 420175/419904 in the 7-limit; 1375/1372, 5632/5625, 6250/6237, and 14641/14580 in the 11-limit. It supports the [[Hemifamity temperaments|septiquarter temperament]] and [[Dimcomp family|dimcomp rank three temperament]].
{{Infobox ET}}
{{ED intro}}


[[Category:EDO]]
== Theory ==
292edo 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 [[semiluna]].
 
=== Prime harmonics ===
{{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]]
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