Wesley family: Difference between revisions

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This tempers out the wesley comma, 78125/73728. The wesley comma is unchanged in Wesley Woolhouse's 7/26-comma meantone - it is an eigenmonzo, or unchanged-interval, in other words. Here is [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html a talk by Gene Ward Smith].  
The '''wesley family''' of [[Rank-2 temperament|rank-2]] [[temperament]]s [[Tempering out|tempers out]] the [[wesley comma]], 78125/73728. The wesley comma is unchanged in [[Wesley Woolhouse]]'s 7/26-comma meantone – it is an [[Eigenmonzo|eigenmonzo (i.e. unchanged-interval)]]. Here is [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html a talk by Gene Ward Smith].  


== Wesley ==
== Wesley ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma]]: 78125/73728
[[Comma list]]: 78125/73728


[[Mapping]]: [{{val| 1 4 3 }}, {{val| 0 -7 -2 }}]
{{Mapping|legend=1| 1 4 3 | 0 -7 -2 }}


[[POTE generator]]: ~125/96 = 414.509
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 414.509


{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
Line 15: Line 15:


== Septimal wesley ==
== Septimal wesley ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 405/392, 875/864
[[Comma list]]: 405/392, 875/864


[[Mapping]]: [{{val| 1 4 3 8 }}, {{val| 0 -7 -2 -15 }}]
{{Mapping|legend=1| 1 4 3 8 | 0 -7 -2 -15 }}


{{Multival|legend=1|7 2 15 -13 4 29}}
{{Multival|legend=1| 7 2 15 -13 4 29 }}


[[POTE generator]]: ~9/7 = 415.519
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 415.519


{{Optimal ET sequence|legend=1| 3d, 23d, 26 }}
{{Optimal ET sequence|legend=1| 3d, 23d, 26 }}
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Comma list: 45/44, 99/98, 875/864
Comma list: 45/44, 99/98, 875/864


Mapping: [{{val| 1 4 3 8 9 }}, {{val| 0 -7 -2 -15 -16 }}]
Mapping: {{mapping| 1 4 3 8 9 | 0 -7 -2 -15 -16 }}


POTE generator: ~9/7 = 415.769
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.769


{{Optimal ET sequence|legend=1| 3de, 23de, 26, 75bcd }}
{{Optimal ET sequence|legend=1| 3de, 23de, 26, 75bcd }}
Line 47: Line 47:
Comma list: 45/44, 78/77, 99/98, 325/324
Comma list: 45/44, 78/77, 99/98, 325/324


Mapping: [{{val| 1 4 3 8 9 12 }}, {{val| 0 -7 -2 -15 -16 -24 }}]
Mapping: {{mapping| 1 4 3 8 9 12 | 0 -7 -2 -15 -16 -24 }}


POTE generator: ~9/7 = 415.645
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.645


{{Optimal ET sequence|legend=1| 3def, 26 }}
{{Optimal ET sequence|legend=1| 3def, 26 }}
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== Snipes ==
== Snipes ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 6125/5832
[[Comma list]]: 225/224, 6125/5832


[[Mapping]]: [{{val| 1 4 3 9 }}, {{val| 0 -7 -2 -18 }}]
{{Mapping|legend=1| 1 4 3 9 | 0 -7 -2 -18 }}


{{Multival|legend=1| 7 2 18 -13 9 36 }}
{{Multival|legend=1| 7 2 18 -13 9 36 }}


[[POTE generator]]: ~35/27 = 413.513
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 413.513


{{Optimal ET sequence|legend=1| 29, 90cd, 119ccdd }}
{{Optimal ET sequence|legend=1| 29, 90cd, 119ccdd }}
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Comma list: 55/54, 225/224, 245/242
Comma list: 55/54, 225/224, 245/242


Mapping: [{{val| 1 4 3 9 10 }}, {{val| 0 -7 -2 -18 -19 }}]
Mapping: {{mapping| 1 4 3 9 10 | 0 -7 -2 -18 -19 }}


POTE generator: ~14/11 = 413.490
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 413.490


{{Optimal ET sequence|legend=1| 29, 90cd, 119ccdde }}
{{Optimal ET sequence|legend=1| 29, 90cd, 119ccdde }}
Line 84: Line 84:


== Roman ==
== Roman ==
{{see also|Avicennmic temperaments #Roman}}
{{See also| Avicennmic temperaments #Roman }}


=== 7-limit (Crusher) ===
=== 7-limit a.k.a. crusher ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 525/512, 3125/3024
[[Comma list]]: 525/512, 3125/3024


[[Mapping]]: [{{val| 1 4 3 -1 }}, {{val| 0 -7 -2 11 }}]
{{Mapping|legend=1| 1 4 3 -1 | 0 -7 -2 11 }}


{{Multival|legend=1|7 2 -11 -13 -37 -31}}
{{Multival|legend=1| 7 2 -11 -13 -37 -31 }}


[[POTE generator]]: ~63/50 = 414.552
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~63/50 = 414.552


{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
Line 106: Line 106:
Comma list: 100/99, 245/242, 525/512
Comma list: 100/99, 245/242, 525/512


Mapping: [{{val| 1 4 3 -1 0 }}, {{val| 0 -7 -2 11 10 }}]
Mapping: {{mapping| 1 4 3 -1 0 | 0 -7 -2 11 10 }}


POTE generator: ~14/11 = 414.471
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 414.471


{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
Line 119: Line 119:
Comma list: 65/64, 100/99, 105/104, 245/242
Comma list: 65/64, 100/99, 105/104, 245/242


Mapping: [{{val| 1 4 3 -1 0 3 }}, {{val| 0 -7 -2 11 10 2 }}]
Mapping: {{mapping| 1 4 3 -1 0 3 | 0 -7 -2 11 10 2 }}


POTE generator: ~14/11 = 414.472
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 414.472


{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55cf }}
{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55cf }}
Line 128: Line 128:


== Dubbla ==
== Dubbla ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 78125/73728
[[Comma list]]: 50/49, 78125/73728


[[Mapping]]: [{{val| 2 1 4 5 }}, {{val| 0 7 2 2 }}]
{{Mapping|legend=1| 2 1 4 5 | 0 7 2 2 }}


{{Multival|legend=1|14 4 4 -26 -33 -2}}
{{Multival|legend=1| 14 4 4 -26 -33 -2 }}


[[POTE generator]]: ~192/175 = 185.738
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 185.738


{{Optimal ET sequence|legend=1| 6, 26, 58c, 84c }}
{{Optimal ET sequence|legend=1| 6, 26, 58c, 84c }}
Line 147: Line 147:
Comma list: 50/49, 125/121, 1344/1331
Comma list: 50/49, 125/121, 1344/1331


Mapping: [{{val| 2 1 4 5 6 }}, {{val| 0 7 2 2 3 }}]
Mapping: {{mapping| 2 1 4 5 6 | 0 7 2 2 3 }}


POTE generator: ~11/10 = 185.879
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 185.879


{{Optimal ET sequence|legend=1| 6, 26, 58c, 84c }}
{{Optimal ET sequence|legend=1| 6, 26, 58c, 84c }}
Line 160: Line 160:
Comma list: 50/49, 105/104, 125/121, 144/143
Comma list: 50/49, 105/104, 125/121, 144/143


Mapping: [{{val| 2 1 4 5 6 4 }}, {{val| 0 7 2 2 3 11 }}]
Mapping: {{mapping| 2 1 4 5 6 4 | 0 7 2 2 3 11 }}


POTE generator: ~11/10 = 185.702
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 185.702


{{Optimal ET sequence|legend=1| 26, 58c, 84c }}
{{Optimal ET sequence|legend=1| 26, 58c, 84c }}
Line 170: Line 170:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Wesley family| ]] <!-- main article -->
[[Category:Wesley family| ]] <!-- main article -->
[[Category:Wesley| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 08:29, 21 September 2023

The wesley family of rank-2 temperaments tempers out the wesley comma, 78125/73728. The wesley comma is unchanged in Wesley Woolhouse's 7/26-comma meantone – it is an eigenmonzo (i.e. unchanged-interval). Here is a talk by Gene Ward Smith.

Wesley

Subgroup: 2.3.5

Comma list: 78125/73728

Mapping: [⟨1 4 3], ⟨0 -7 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~125/96 = 414.509

Optimal ET sequence: 3, 23, 26, 29, 55c

Badness: 0.247718

Septimal wesley

Subgroup: 2.3.5.7

Comma list: 405/392, 875/864

Mapping: [⟨1 4 3 8], ⟨0 -7 -2 -15]]

Wedgie: ⟨⟨ 7 2 15 -13 4 29 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.519

Optimal ET sequence: 3d, 23d, 26

Badness: 0.095344

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 875/864

Mapping: [⟨1 4 3 8 9], ⟨0 -7 -2 -15 -16]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.769

Optimal ET sequence: 3de, 23de, 26, 75bcd

Badness: 0.049066

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 99/98, 325/324

Mapping: [⟨1 4 3 8 9 12], ⟨0 -7 -2 -15 -16 -24]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.645

Optimal ET sequence: 3def, 26

Badness: 0.038402

Snipes

Subgroup: 2.3.5.7

Comma list: 225/224, 6125/5832

Mapping: [⟨1 4 3 9], ⟨0 -7 -2 -18]]

Wedgie: ⟨⟨ 7 2 18 -13 9 36 ]]

Optimal tuning (POTE): ~2 = 1\1, ~35/27 = 413.513

Optimal ET sequence: 29, 90cd, 119ccdd

Badness: 0.117943

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 225/224, 245/242

Mapping: [⟨1 4 3 9 10], ⟨0 -7 -2 -18 -19]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 413.490

Optimal ET sequence: 29, 90cd, 119ccdde

Badness: 0.054296

Roman

See also: Avicennmic temperaments § Roman

7-limit a.k.a. crusher

Subgroup: 2.3.5.7

Comma list: 525/512, 3125/3024

Mapping: [⟨1 4 3 -1], ⟨0 -7 -2 11]]

Wedgie: ⟨⟨ 7 2 -11 -13 -37 -31 ]]

Optimal tuning (POTE): ~2 = 1\1, ~63/50 = 414.552

Optimal ET sequence: 3, 23, 26, 29, 55c

Badness: 0.113386

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 525/512

Mapping: [⟨1 4 3 -1 0], ⟨0 -7 -2 11 10]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 414.471

Optimal ET sequence: 3, 23, 26, 29, 55c

Badness: 0.052841

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 100/99, 105/104, 245/242

Mapping: [⟨1 4 3 -1 0 3], ⟨0 -7 -2 11 10 2]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 414.472

Optimal ET sequence: 3, 23, 26, 29, 55cf

Badness: 0.030043

Dubbla

Subgroup: 2.3.5.7

Comma list: 50/49, 78125/73728

Mapping: [⟨2 1 4 5], ⟨0 7 2 2]]

Wedgie: ⟨⟨ 14 4 4 -26 -33 -2 ]]

Optimal tuning (POTE): ~2 = 1\1, ~192/175 = 185.738

Optimal ET sequence: 6, 26, 58c, 84c

Badness: 0.181726

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 125/121, 1344/1331

Mapping: [⟨2 1 4 5 6], ⟨0 7 2 2 3]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 185.879

Optimal ET sequence: 6, 26, 58c, 84c

Badness: 0.075842

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 105/104, 125/121, 144/143

Mapping: [⟨2 1 4 5 6 4], ⟨0 7 2 2 3 11]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 185.702

Optimal ET sequence: 26, 58c, 84c

Badness: 0.049603

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