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'''292edo''' is the [[EDO|equal division of the octave]] into 292 parts of 4.1095 cents each.
{{Infobox ET}}
{{ED intro}}


292edo 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 (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 ==
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 notably supports [[Hemifamity temperaments #Semiseptiquarter|semiseptiquarter]] and [[Luna family #Semiluna|semiluna]]. It also provides a nice tuning for the rank-3 [[Dimcomp family|dimcomp]] and [[Canou family #Semicanou|semicanou]].  
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:Theory]]
=== Prime harmonics ===
[[Category:Equal divisions of the octave]]
{{Harmonics in equal|292}}
[[Category:Hemifamity]]
 
=== 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:Septiquarter]]
[[Category:Semiluna]]
[[Category:Semiluna]]
[[Category:Canou]]
[[Category:Undim]]
[[Category:Semicanousmic]]

Latest revision as of 12:39, 6 June 2026

← 291edo 292edo 293edo →
Prime factorization 22 × 73
Step size 4.10959 ¢ 
Fifth 171\292 (702.74 ¢)
Semitones (A1:m2) 29:21 (119.2 ¢ : 86.3 ¢)
Consistency limit 9
Distinct consistency limit 9

292 equal divisions of the octave (abbreviated 292edo or 292ed2), also called 292-tone equal temperament (292tet) or 292 equal temperament (292et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 292 equal parts of about 4.11 ¢ each. Each step represents a frequency ratio of 21/292, or the 292nd root of 2.

Theory

292edo is closely related to 146edo, but the patent vals differ on the mapping for 3. As an equal temperament, it tempers out [3 -18 11⟩ (quartonic comma) and [38 -2 -15⟩ (luna/hemithirds comma) in the 5-limit; 5120/5103 (hemifamity), 390625/388962 (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 supports semiseptiquarter and semiluna.

Prime harmonics

Approximation of prime harmonics in 292edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.78 -0.01 +1.04 -0.63 +1.94 +1.89 -1.62 +0.49 +1.93 +1.54
Relative (%) +0.0 +19.1 -0.3 +25.2 -15.4 +47.2 +46.1 -39.5 +12.0 +47.0 +37.5
Steps
(reduced) 		292
(0) 		463
(171) 		678
(94) 		820
(236) 		1010
(134) 		1081
(205) 		1194
(26) 		1240
(72) 		1321
(153) 		1419
(251) 		1447
(279)

Subsets and supersets

Since 292 factors into 22 × 73, 292edo has subset edos 2, 4, 73, and 146.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [463 -292⟩ [⟨292 463]] −0.2476 0.2475 6.02
2.3.5 [3 -18 11⟩, [38 -2 -15⟩ [⟨292 463 678]] −0.1633 0.2346 5.71
2.3.5.7 5120/5103, 390625/388962, 420175/419904 [⟨292 463 678 820]] −0.2148 0.2219 5.40
2.3.5.7.11 1375/1372, 5120/5103, 5632/5625, 14641/14580 [⟨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 [⟨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 [⟨292 463 678 820 1010 1081 1194]] −0.2376 0.2696 6.56

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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