292edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>2</sup> × 73
{{ED intro}}
| Step size = 4.10959¢
| Fifth = 171\292 (702.74¢)
| Semitones = 29:21 (119.18¢ : 86.30¢)
| Consistency = 9
}}
{{EDO intro|292}}


== Theory ==
== Theory ==
292edo 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]].  
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]].  
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 ===
=== Prime harmonics ===
{{Harmonics in equal|292}}
{{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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3
| 2.3
| {{monzo| 463 -292 }}
| {{monzo| 463 -292 }}
| [{{val| 292 463 }}]
| {{mapping| 292 463 }}
| -0.2476
| −0.2476
| 0.2475
| 0.2475
| 6.02
| 6.02
Line 36: Line 34:
| 2.3.5
| 2.3.5
| {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}
| {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}
| [{{val| 292 463 678 }}]
| {{mapping| 292 463 678 }}
| -0.1633
| −0.1633
| 0.2346
| 0.2346
| 5.71
| 5.71
Line 43: Line 41:
| 2.3.5.7
| 2.3.5.7
| 5120/5103, 390625/388962, 420175/419904
| 5120/5103, 390625/388962, 420175/419904
| [{{val| 292 463 678 820 }}]
| {{mapping| 292 463 678 820 }}
| -0.2148
| −0.2148
| 0.2219
| 0.2219
| 5.40
| 5.40
Line 50: Line 48:
| 2.3.5.7.11
| 2.3.5.7.11
| 1375/1372, 5120/5103, 5632/5625, 14641/14580
| 1375/1372, 5120/5103, 5632/5625, 14641/14580
| [{{val| 292 463 678 820 1010 }}]
| {{mapping| 292 463 678 820 1010 }}
| -0.1353
| −0.1353
| 0.2544
| 0.2544
| 6.19
| 6.19
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 352/351, 625/624, 847/845, 1716/1715, 14641/14580
| 352/351, 625/624, 847/845, 1716/1715, 14641/14580
| [{{val| 292 463 678 820 1010 1081 }}]
| {{mapping| 292 463 678 820 1010 1081 }}
| -0.3480
| −0.3480
| 0.2736
| 0.2736
| 6.66
| 6.66
Line 64: Line 62:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023
| 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023
| [{{val| 292 463 678 820 1010 1081 1194 }}]
| {{mapping| 292 463 678 820 1010 1081 1194 }}
| -0.2376
| −0.2376
| 0.2696
| 0.2696
| 6.56
| 6.56
Line 72: Line 70:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 121: Line 120:
| [[Undim]]
| [[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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Aberschismic]]
[[Category:Hemifamity]]
[[Category:Septiquarter]]
[[Category:Septiquarter]]
[[Category:Semiluna]]
[[Category:Semiluna]]
[[Category:Undim]]
[[Category:Undim]]

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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