292edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 22 × 73~~

~~| Step size = 4.10959¢~~

~~| Fifth = 171\292 (702.74¢)~~

~~| Semitones = 29:21 (119.18¢ : 86.30¢)~~

~~| Consistency = 9~~

}}

== 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 ===

=== Subsets and supersets ===

Since 292 factors into 22 × 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" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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| 2.3

| {{monzo| 463 -292 }}

| [{{~~val~~| 292 463 }}]

| {{mapping| 292 463 }}

| -0.2476

| −0.2476

| 0.2475

| 6.02

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| 2.3.5

| {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}

| [{{~~val~~| 292 463 678 }}]

| {{mapping| 292 463 678 }}

| -0.1633

| −0.1633

| 0.2346

| 5.71

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| 2.3.5.7

| 5120/5103, 390625/388962, 420175/419904

| [{{~~val~~| 292 463 678 820 }}]

| {{mapping| 292 463 678 820 }}

| -0.2148

| −0.2148

| 0.2219

| 5.40

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| 2.3.5.7.11

| 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

| 6.19

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| 2.3.5.7.11.13

| 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

| 6.66

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| 2.3.5.7.11.13.17

| 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

| 6.56

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=== Rank-2 temperaments ===

{| 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 per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

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| [[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|###]] ~~

[[Category:Aberschismic]]

~~[[Category:Hemifamity~~]]

[[Category:Septiquarter]]

[[Category:Semiluna]]

[[Category:Undim]]

@@ Line 1: / Line 1: @@
-{{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 ==
-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]].
+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 [[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 ===
 {{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" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 29: / Line 27: @@
 | 2.3
 | {{monzo| 463 -292 }}
-| [{{val| 292 463 }}]
+| {{mapping| 292 463 }}
-| -0.2476
+| −0.2476
 | 0.2475
 | 6.02
@@ Line 36: / Line 34: @@
 | 2.3.5
 | {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}
-| [{{val| 292 463 678 }}]
+| {{mapping| 292 463 678 }}
-| -0.1633
+| −0.1633
 | 0.2346
 | 5.71
@@ Line 43: / Line 41: @@
 | 2.3.5.7
 | 5120/5103, 390625/388962, 420175/419904
-| [{{val| 292 463 678 820 }}]
+| {{mapping| 292 463 678 820 }}
-| -0.2148
+| −0.2148
 | 0.2219
 | 5.40
@@ Line 50: / Line 48: @@
 | 2.3.5.7.11
 | 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
 | 6.19
@@ Line 57: / Line 55: @@
 | 2.3.5.7.11.13
 | 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
 | 6.66
@@ Line 64: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 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
 | 6.56
@@ Line 72: / Line 70: @@
 === Rank-2 temperaments ===
 {| 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
 |-
@@ Line 121: / Line 120: @@
 | [[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:Semiluna]]
 [[Category:Undim]]