292edo: Difference between revisions

(5 intermediate revisions by 2 users not shown)

Line 1:

== Theory ==

292edo is closely related to [[146edo]], but the [[patent val]]s differ on the mapping for [[3/1|3]]. ~~The~~ equal temperament [[tempering out|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 ===

Line 11:

=== Subsets and supersets ===

Since 292 factors into ~~{{factorization|292}}~~, 292edo has subset edos {{EDOs| 2, 4, 73, and 146 }}.

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

== Regular temperament properties ==

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{monzo| 463 -292 }}

| {{mapping| 292 463 }}

| ~~−0~~.2476

| −0.2476

| 0.2475

| 6.02

Line 26:

Line 35:

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

| {{mapping| 292 463 678 }}

| ~~−0~~.1633

| −0.1633

| 0.2346

| 5.71

Line 33:

Line 42:

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

| {{mapping| 292 463 678 820 }}

| ~~−0~~.2148

| −0.2148

| 0.2219

| 5.40

Line 40:

Line 49:

| 1375/1372, 5120/5103, 5632/5625, 14641/14580

| {{mapping| 292 463 678 820 1010 }}

| ~~−0~~.1353

| −0.1353

| 0.2544

| 6.19

Line 47:

Line 56:

| 352/351, 625/624, 847/845, 1716/1715, 14641/14580

| {{mapping| 292 463 678 820 1010 1081 }}

| ~~−0~~.3480

| −0.3480

| 0.2736

| 6.66

Line 54:

Line 63:

| 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023

| {{mapping| 292 463 678 820 1010 1081 1194 }}

| ~~−0~~.2376

| −0.2376

| 0.2696

| 6.56

Line 60:

Line 69:

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

Line 103:

Line 119:

| 4/3

| [[Undim]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<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:~~Hemifamity~~]]

[[Category:Aberschismic]]

[[Category:Septiquarter]]

[[Category:Semiluna]]

[[Category:Undim]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
-edo is closely related to [[146edo]], but the [[patent val]]s differ on the mapping for [[3/1|3]]. The equal temperament [[tempering out|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 ===
@@ Line 11: / Line 11: @@
 === Subsets and supersets ===
-Since 292 factors into {{factorization|292}}, 292edo has subset edos {{EDOs| 2, 4, 73, and 146 }}.
+Since 292 factors into 2<sup>2</sup> × 73, 292edo has subset edos {{EDOs| 2, 4, 73, and 146 }}.
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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 }}
-| &minus;0.2476
+| −0.2476
 | 0.2475
 | 6.02
@@ Line 26: / Line 35: @@
 | {{monzo| 3 -18 11 }}, {{monzo| 38 -2 -15 }}
 | {{mapping| 292 463 678 }}
-| &minus;0.1633
+| −0.1633
 | 0.2346
 | 5.71
@@ Line 33: / Line 42: @@
 | 5120/5103, 390625/388962, 420175/419904
 | {{mapping| 292 463 678 820 }}
-| &minus;0.2148
+| −0.2148
 | 0.2219
 | 5.40
@@ Line 40: / Line 49: @@
 | 1375/1372, 5120/5103, 5632/5625, 14641/14580
 | {{mapping| 292 463 678 820 1010 }}
-| &minus;0.1353
+| −0.1353
 | 0.2544
 | 6.19
@@ Line 47: / Line 56: @@
 | 352/351, 625/624, 847/845, 1716/1715, 14641/14580
 | {{mapping| 292 463 678 820 1010 1081 }}
-| &minus;0.3480
+| −0.3480
 | 0.2736
 | 6.66
@@ Line 54: / Line 63: @@
 | 352/351, 625/624, 715/714, 847/845, 1225/1224, 2025/2023
 | {{mapping| 292 463 678 820 1010 1081 1194 }}
-| &minus;0.2376
+| −0.2376
 | 0.2696
 | 6.56
@@ Line 60: / Line 69: @@
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 103: / Line 119: @@
 | 4/3
 | [[Undim]]
-{{rank-2 end}}
+|}
-{{orf}}
+<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:Hemifamity]]
+[[Category:Aberschismic]]
 [[Category:Septiquarter]]
 [[Category:Semiluna]]
 [[Category:Undim]]