Wesley family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
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].  
The '''wesley family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] 78125/73728, the [[wesley comma]]. The wesley comma is unchanged in [[Wesley Woolhouse]]'s 7/26-comma meantone – it is an [[eigenmonzo|eigenmonzo (i.e. unchanged-interval)]], which [[Gene Ward Smith]] has talked about<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html Yahoo! Tuning Group | ''26et and meantone projections'']</ref>.  


== Wesley ==
== Wesley ==
Line 8: Line 8:


{{Mapping|legend=1| 1 4 3 | 0 -7 -2 }}
{{Mapping|legend=1| 1 4 3 | 0 -7 -2 }}
: mapping generators: ~2, ~125/96
: mapping generators: ~2, ~125/96


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~125/96 = 414.509
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/96 = 414.509{{c}}


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


[[Badness]]: 0.247718
[[Badness]] (Smith): 0.247718


== Septimal wesley ==
== Septimal wesley ==
Line 24: Line 23:
{{Mapping|legend=1| 1 4 3 8 | 0 -7 -2 -15 }}
{{Mapping|legend=1| 1 4 3 8 | 0 -7 -2 -15 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 415.519
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 415.519{{c}}


{{Optimal ET sequence|legend=1| 3d, …, 23d, 26 }}
{{Optimal ET sequence|legend=1| 3d, …, 23d, 26 }}


[[Badness]]: 0.095344
[[Badness]] (Smith): 0.095344


=== 11-limit ===
=== 11-limit ===
Line 37: Line 36:
Mapping: {{mapping| 1 4 3 8 9 | 0 -7 -2 -15 -16 }}
Mapping: {{mapping| 1 4 3 8 9 | 0 -7 -2 -15 -16 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.769
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 415.769{{c}}


{{Optimal ET sequence|legend=1| 3de, …, 23de, 26 }}
{{Optimal ET sequence|legend=0| 3de, …, 23de, 26 }}


Badness: 0.049066
Badness (Smith): 0.049066


=== 13-limit ===
=== 13-limit ===
Line 50: Line 49:
Mapping: {{mapping| 1 4 3 8 9 12 | 0 -7 -2 -15 -16 -24 }}
Mapping: {{mapping| 1 4 3 8 9 12 | 0 -7 -2 -15 -16 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 415.645
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 415.645{{c}}


{{Optimal ET sequence|legend=1| 3def, 23deff, 26 }}
{{Optimal ET sequence|legend=0| 3def, 23deff, 26 }}


Badness: 0.038402
Badness (Smith): 0.038402


== Snipes ==
== Snipes ==
Line 63: Line 62:
{{Mapping|legend=1| 1 4 3 9 | 0 -7 -2 -18 }}
{{Mapping|legend=1| 1 4 3 9 | 0 -7 -2 -18 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~35/27 = 413.513
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/27 = 413.513{{c}}


{{Optimal ET sequence|legend=1| 3d, …, 26d, 29 }}
{{Optimal ET sequence|legend=1| 3d, …, 26d, 29 }}


[[Badness]]: 0.117943
[[Badness]] (Smith): 0.117943


=== 11-limit ===
=== 11-limit ===
Line 76: Line 75:
Mapping: {{mapping| 1 4 3 9 10 | 0 -7 -2 -18 -19 }}
Mapping: {{mapping| 1 4 3 9 10 | 0 -7 -2 -18 -19 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 413.490
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 413.490{{c}}


{{Optimal ET sequence|legend=1| 3de, …, 26de, 29 }}
{{Optimal ET sequence|legend=0| 3de, …, 26de, 29 }}


Badness: 0.054296
Badness (Smith): 0.054296


== Roman ==
== Roman ==
Line 93: Line 92:
{{Mapping|legend=1| 1 4 3 -1 | 0 -7 -2 11 }}
{{Mapping|legend=1| 1 4 3 -1 | 0 -7 -2 11 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~63/50 = 414.552
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~63/50 = 414.552{{c}}


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


[[Badness]]: 0.113386
[[Badness]] (Smith): 0.113386


=== 11-limit ===
=== 11-limit ===
Line 106: Line 105:
Mapping: {{mapping| 1 4 3 -1 0 | 0 -7 -2 11 10 }}
Mapping: {{mapping| 1 4 3 -1 0 | 0 -7 -2 11 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 414.471
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 414.471{{c}}


{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}
{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55c }}


Badness: 0.052841
Badness (Smith): 0.052841


=== 13-limit ===
=== 13-limit ===
Line 119: Line 118:
Mapping: {{mapping| 1 4 3 -1 0 3 | 0 -7 -2 11 10 2 }}
Mapping: {{mapping| 1 4 3 -1 0 3 | 0 -7 -2 11 10 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 414.472
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 414.472{{c}}


{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55cf }}
{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55cf }}


Badness: 0.030043
Badness (Smith): 0.030043


== Dubbla ==
== Dubbla ==
Line 132: Line 131:
{{Mapping|legend=1| 2 1 4 5 | 0 7 2 2 }}
{{Mapping|legend=1| 2 1 4 5 | 0 7 2 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~192/175 = 185.738
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~192/175 = 185.738{{c}}


{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}
{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}


[[Badness]]: 0.181726
[[Badness]] (Smith): 0.181726


=== 11-limit ===
=== 11-limit ===
Line 145: Line 144:
Mapping: {{mapping| 2 1 4 5 6 | 0 7 2 2 3 }}
Mapping: {{mapping| 2 1 4 5 6 | 0 7 2 2 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 185.879
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 185.879{{c}}


{{Optimal ET sequence|legend=1| 6, 20b, 26 }}
{{Optimal ET sequence|legend=0| 6, 20b, 26 }}


Badness: 0.075842
Badness (Smith): 0.075842


=== 13-limit ===
=== 13-limit ===
Line 158: Line 157:
Mapping: {{mapping| 2 1 4 5 6 4 | 0 7 2 2 3 11 }}
Mapping: {{mapping| 2 1 4 5 6 4 | 0 7 2 2 3 11 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 185.702
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 185.702{{c}}
 
{{Optimal ET sequence|legend=0| 6f, 20bff, 26 }}


{{Optimal ET sequence|legend=1| 6f, 20bff, 26 }}
Badness (Smith): 0.049603


Badness: 0.049603
== References ==


[[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 15:25, 2 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The wesley family of temperaments tempers out 78125/73728, the wesley comma. The wesley comma is unchanged in Wesley Woolhouse's 7/26-comma meantone – it is an eigenmonzo (i.e. unchanged-interval), which Gene Ward Smith has talked about[1].

Wesley

Subgroup: 2.3.5

Comma list: 78125/73728

Mapping: [⟨1 4 3], ⟨0 -7 -2]]

mapping generators: ~2, ~125/96

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~125/96 = 414.509 ¢

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

Badness (Smith): 0.247718

Septimal wesley

Subgroup: 2.3.5.7

Comma list: 405/392, 875/864

Mapping: [⟨1 4 3 8], ⟨0 -7 -2 -15]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~9/7 = 415.519 ¢

Optimal ET sequence: 3d, …, 23d, 26

Badness (Smith): 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 = 1200.000 ¢, ~9/7 = 415.769 ¢

Optimal ET sequence: 3de, …, 23de, 26

Badness (Smith): 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 = 1200.000 ¢, ~9/7 = 415.645 ¢

Optimal ET sequence: 3def, 23deff, 26

Badness (Smith): 0.038402

Snipes

Subgroup: 2.3.5.7

Comma list: 225/224, 6125/5832

Mapping: [⟨1 4 3 9], ⟨0 -7 -2 -18]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/27 = 413.513 ¢

Optimal ET sequence: 3d, …, 26d, 29

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 413.490 ¢

Optimal ET sequence: 3de, …, 26de, 29

Badness (Smith): 0.054296

Roman

See also: Avicennmic temperaments

7-limit Roman is also known as "crusher".

Subgroup: 2.3.5.7

Comma list: 525/512, 3125/3024

Mapping: [⟨1 4 3 -1], ⟨0 -7 -2 11]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~63/50 = 414.552 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 414.471 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~14/11 = 414.472 ¢

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

Badness (Smith): 0.030043

Dubbla

Subgroup: 2.3.5.7

Comma list: 50/49, 78125/73728

Mapping: [⟨2 1 4 5], ⟨0 7 2 2]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~192/175 = 185.738 ¢

Optimal ET sequence: 6, 20b, 26, 58c

Badness (Smith): 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 = 1200.000 ¢, ~11/10 = 185.879 ¢

Optimal ET sequence: 6, 20b, 26

Badness (Smith): 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 = 1200.000 ¢, ~11/10 = 185.702 ¢

Optimal ET sequence: 6f, 20bff, 26

Badness (Smith): 0.049603

References

  1. ↑ Yahoo! Tuning Group | 26et and meantone projections
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