140edo: Difference between revisions

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== Theory ==

In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the kwazy comma, {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53 & 140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].

If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].

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=== Prime harmonics ===

{{Harmonics in equal|140|columns=10|start=11|collapsed=true|title=Approximation of prime harmonics in 140edo (continued)}}

=== ~~Divisors~~ ===

=== Subsets and supersets ===

Since 140 factors into 22 × 5 × 7, it has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70 }}.

Since 140 factors into 22 × 5 × 7, 140edo has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70 }}.

== Regular temperament properties ==

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| 2.3.5

| 15625/15552, {{monzo| 35 -25 2 }}

| [{{~~val~~| 140 222 325 }}]

| {{mapping| 140 222 325 }}

| -0.104

| 0.346

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| 2.3.5.7

| 2401/2400, 5120/5103, 15625/15552

| [{{~~val~~| 140 222 325 393 }}]

| {{mapping| 140 222 325 393 }}

| -0.055

| 0.311

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| 2.3.5.7.11

| 385/384, 1331/1323, 1375/1372, 2200/2187

| [{{~~val~~| 140 222 325 393 484 }}]

| {{mapping| 140 222 325 393 484 }}

| +0.115

| 0.439

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| 2.3.5.7.11.13

| 325/324, 352/351, 385/384, 625/624, 1331/1323

| [{{~~val~~| 140 222 325 393 484 518 }}]

| {{mapping| 140 222 325 393 484 518 }}

| +0.119

| 0.401

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Oquatonic]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Countercata]]

@@ Line 3: / Line 3: @@
 == Theory ==
-In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the kwazy comma, {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53&amp;140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
+In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It [[support]]s the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53 &amp; 140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
 If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
@@ Line 9: / Line 9: @@
 === Prime harmonics ===
 {{Harmonics in equal|140|columns=10}}
-{{Harmonics in equal|140|start=11|columns=10}}
+{{Harmonics in equal|140|columns=10|start=11|collapsed=true|title=Approximation of prime harmonics in 140edo (continued)}}
-=== Divisors ===
+=== Subsets and supersets ===
-Since 140 factors into 2<sup>2</sup> × 5 × 7, it has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70 }}.
+Since 140 factors into 2<sup>2</sup> × 5 × 7, 140edo has subset edos {{EDOs| 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70 }}.
 == Regular temperament properties ==
@@ Line 27: / Line 27: @@
 | 2.3.5
 | 15625/15552, {{monzo| 35 -25 2 }}
-| [{{val| 140 222 325 }}]
+| {{mapping| 140 222 325 }}
 | -0.104
 | 0.346
@@ Line 34: / Line 34: @@
 | 2.3.5.7
 | 2401/2400, 5120/5103, 15625/15552
-| [{{val| 140 222 325 393 }}]
+| {{mapping| 140 222 325 393 }}
 | -0.055
 | 0.311
@@ Line 41: / Line 41: @@
 | 2.3.5.7.11
 | 385/384, 1331/1323, 1375/1372, 2200/2187
-| [{{val| 140 222 325 393 484 }}]
+| {{mapping| 140 222 325 393 484 }}
 | +0.115
 | 0.439
@@ Line 48: / Line 48: @@
 | 2.3.5.7.11.13
 | 325/324, 352/351, 385/384, 625/624, 1331/1323
-| [{{val| 140 222 325 393 484 518 }}]
+| {{mapping| 140 222 325 393 484 518 }}
 | +0.119
 | 0.401
@@ Line 58: / Line 58: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 153: / Line 153: @@
 | [[Oquatonic]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 [[Category:Countercata]]