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, in other words. Here is [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html a talk by Gene Ward Smith].  
{{Technical data page}}
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 ==
Subgroup: 2.3.5
The generator of wesley is one half of a [[~]][[5/2]], around 786 [[cent]]s, seven of which after [[octave reduction]] reach the [[3/2|perfect fifth]]. The [[ploidacot]] signature of this temperament is delta-heptacot. A good tuning for the generator may be found between [[26edo|17\26]] and [[29edo|19\29]].  


[[Comma]]: 78125/73728
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val| 1 4 3 }}, {{val| 0 -7 -2 }}]
[[Comma list]]: 78125/73728


[[POTE generator]]: ~125/96 = 414.509
{{Mapping|legend=1| 1 -3 1 | 0 7 2 }}
: mapping generators: ~2, ~192/125


{{Val list|legend=1| 3, 23, 26, 29, 55c }}
[[Optimal tuning]]s:
* [[WE]] ~2 = 1202.9225{{c}}, ~192/125 = 787.4042{{c}}
: [[error map]]: {{val| +2.922 +1.107 -8.583 }}
* [[CWE]] ~2 = 1200.0000{{c}}, ~192/125 = 785.7808{{c}}
: error map: {{val| 0.000 -1.490 -14.752 }}


[[Badness]]: 0.247718
{{Optimal ET sequence|legend=1| 3, …, 23, 26, 29, 55c }}
 
[[Badness]] (Sintel): 5.81


== 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 -3 1 -7 | 0 7 2 15 }}
 
{{Multival|legend=1|7 2 15 -13 4 29}}


[[POTE generator]]: ~9/7 = 415.519
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.6434{{c}}, ~14/9 = 786.8631{{c}}
: [[error map]]: {{val| +3.634 -4.844 -8.944 +8.617 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 784.7754{{c}}
: error map: {{val| 0.000 -8.527 -16.763 +2.805 }}


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


[[Badness]]: 0.095344
[[Badness]] (Sintel): 2.41


=== 11-limit ===
=== 11-limit ===
Line 34: Line 44:
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 -3 1 -7 -7 | 0 7 2 15 16 }}


POTE generator: ~9/7 = 415.769
Optimal tunings:  
* WE: ~2 = 1203.6405{{c}}, ~11/7 = 786.6101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 784.5242{{c}}


Vals: {{Val list| 3de, 23de, 26, 75bcd }}
{{Optimal ET sequence|legend=0| 3de, …, 23de, 26 }}


Badness: 0.049066
Badness (Sintel): 1.62


=== 13-limit ===
=== 13-limit ===
Line 47: Line 59:
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 -3 1 -7 -7 -12 | 0 7 2 15 16 24 }}


POTE generator: ~9/7 = 415.645
Optimal tunings:  
* WE: ~2 = 1200.0000{{c}}, ~11/7 = 786.5973{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 784.5923{{c}}


Vals: {{Val list| 3def, 26 }}
{{Optimal ET sequence|legend=0| 3def, 23deff, 26 }}


Badness: 0.038402
Badness (Sintel): 1.59


== 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 -3 1 -9 | 0 7 2 18 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.9268{{c}}, ~54/35 = 787.7496{{c}}
: [[error map]]: {{val| +1.927 +6.511 -8.888 -6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/35 = 786.6189{{c}}
: error map: {{val| 0.000 +4.377 -13.076 -9.686 }}
 
{{Optimal ET sequence|legend=1| 3d, …, 26d, 29 }}
 
[[Badness]] (Sintel): 2.98
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 225/224, 245/242


{{Multival|legend=1|7 2 18 -13 9 36}}
Mapping: {{mapping| 1 -3 1 -9 -9 | 0 7 2 18 19 }}


[[POTE generator]]: ~35/27 = 413.513
Optimal tunings:  
* WE: ~2 = 1201.9201{{c}}, ~11/7 = 787.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 786.6402{{c}}


{{Val list|legend=1| 29, 90cd, 119cd }}
{{Optimal ET sequence|legend=0| 3de, …, 26de, 29 }}


[[Badness]]: 0.117943
Badness (Sintel): 1.79


== Roman ==
== Roman ==
{{see also|Avicennmic temperaments #Roman}}
Roman tempers out 525/512, the [[avicennma]]. The 7-limit version has also been proposed as ''crusher''.


=== 7-limit (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 -3 1 10 | 0 7 2 -11 }}
 
{{Multival|legend=1|7 2 -11 -13 -37 -31}}


[[POTE generator]]: ~63/50 = 414.552
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1203.0258{{c}}, ~63/40 = 787.4289{{c}}
: [[error map]]: {{val| +3.026 +0.970 -8.430 -0.286 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 785.4717{{c}}
: error map: {{val| 0.000 -3.653 -15.370 -9.0145 }}


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


[[Badness]]: 0.113386
[[Badness]] (Sintel): 2.87


=== 11-limit ===
=== 11-limit ===
Line 93: Line 125:
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 -3 1 10 10 | 0 7 2 -11 -10 }}


POTE generator: ~14/11 = 414.471
Optimal tunings:  
* WE: ~2 = 1202.7762{{c}}, ~11/7 = 787.3465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 785.5130{{c}}


Vals: {{Val list| 3, 23, 26, 29, 55c }}
{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55c }}


Badness: 0.052841
Badness (Sintel): 1.75


=== 13-limit ===
=== 13-limit ===
Line 106: Line 140:
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 -3 1 10 10 5 | 0 7 2 -11 -10 -2 }}


POTE generator: ~14/11 = 414.472
Optimal tunings:  
* WE: ~2 = 1202.8327{{c}}, ~11/7 = 787.3820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 785.5085{{c}}


Vals: {{Val list| 3, 23, 26, 29, 55cf }}
{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55cf }}


Badness: 0.030043
Badness (Sintel): 1.24


== 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 }}
: mapping generators: ~7/5, ~192/175


{{Multival|legend=1|14 4 4 -26 -33 -2}}
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.8055{{c}}, ~192/175 = 185.9882{{c}}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~192/175 = 185.8808{{c}}


[[POTE generator]]: ~192/175 = 185.738
{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}


{{Val list|legend=1| 6, 26, 58c, 84c }}
[[Badness]] (Sintel): 4.60
 
[[Badness]]: 0.181726


=== 11-limit ===
=== 11-limit ===
Line 134: Line 171:
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 tunings:  
* WE: ~7/5 = 600.3702{{c}}, ~11/10 = 185.9936{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.9408{{c}}


Vals: {{Val list| 6, 26, 58c, 84c }}
{{Optimal ET sequence|legend=0| 6, 20b, 26 }}


Badness: 0.075842
Badness (Sintel): 2.51


=== 13-limit ===
=== 13-limit ===
Line 147: Line 186:
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 tunings:  
* WE: ~7/5 = 600.3563{{c}}, ~11/10 = 185.8121{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.7623{{c}}


Vals: {{Val list| 26, 58c, 84c }}
{{Optimal ET sequence|legend=0| 6f, 20bff, 26 }}


Badness: 0.049603
Badness (Sintel): 2.05


== References ==
[[Category:Wesley family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Wesley family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 12:25, 14 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

The generator of wesley is one half of a ~5/2, around 786 cents, seven of which after octave reduction reach the perfect fifth. The ploidacot signature of this temperament is delta-heptacot. A good tuning for the generator may be found between 17\26 and 19\29.

Subgroup: 2.3.5

Comma list: 78125/73728

Mapping: [⟨1 -3 1], ⟨0 7 2]]

mapping generators: ~2, ~192/125

Optimal tunings:

  • WE ~2 = 1202.9225 ¢, ~192/125 = 787.4042 ¢
error map: ⟨+2.922 +1.107 -8.583]
  • CWE ~2 = 1200.0000 ¢, ~192/125 = 785.7808 ¢
error map: ⟨0.000 -1.490 -14.752]

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

Badness (Sintel): 5.81

Septimal wesley

Subgroup: 2.3.5.7

Comma list: 405/392, 875/864

Mapping: [⟨1 -3 1 -7], ⟨0 7 2 15]]

Optimal tunings:

  • WE: ~2 = 1203.6434 ¢, ~14/9 = 786.8631 ¢
error map: ⟨+3.634 -4.844 -8.944 +8.617]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 784.7754 ¢
error map: ⟨0.000 -8.527 -16.763 +2.805]

Optimal ET sequence: 3d, …, 23d, 26

Badness (Sintel): 2.41

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1203.6405 ¢, ~11/7 = 786.6101 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 784.5242 ¢

Optimal ET sequence: 3de, …, 23de, 26

Badness (Sintel): 1.62

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~11/7 = 786.5973 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 784.5923 ¢

Optimal ET sequence: 3def, 23deff, 26

Badness (Sintel): 1.59

Snipes

Subgroup: 2.3.5.7

Comma list: 225/224, 6125/5832

Mapping: [⟨1 -3 1 -9], ⟨0 7 2 18]]

Optimal tunings:

  • WE: ~2 = 1201.9268 ¢, ~54/35 = 787.7496 ¢
error map: ⟨+1.927 +6.511 -8.888 -6.675]
  • CWE: ~2 = 1200.0000 ¢, ~54/35 = 786.6189 ¢
error map: ⟨0.000 +4.377 -13.076 -9.686]

Optimal ET sequence: 3d, …, 26d, 29

Badness (Sintel): 2.98

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1201.9201 ¢, ~11/7 = 787.7690 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 786.6402 ¢

Optimal ET sequence: 3de, …, 26de, 29

Badness (Sintel): 1.79

Roman

Roman tempers out 525/512, the avicennma. The 7-limit version has also been proposed as crusher.

Subgroup: 2.3.5.7

Comma list: 525/512, 3125/3024

Mapping: [⟨1 -3 1 10], ⟨0 7 2 -11]]

Optimal tunings:

  • WE: ~2 = 1203.0258 ¢, ~63/40 = 787.4289 ¢
error map: ⟨+3.026 +0.970 -8.430 -0.286]
  • CWE: ~2 = 1200.0000 ¢, ~63/40 = 785.4717 ¢
error map: ⟨0.000 -3.653 -15.370 -9.0145]

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

Badness (Sintel): 2.87

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1202.7762 ¢, ~11/7 = 787.3465 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 785.5130 ¢

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

Badness (Sintel): 1.75

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -3 1 10 10 5], ⟨0 7 2 -11 -10 -2]]

Optimal tunings:

  • WE: ~2 = 1202.8327 ¢, ~11/7 = 787.3820 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/7 = 785.5085 ¢

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

Badness (Sintel): 1.24

Dubbla

Subgroup: 2.3.5.7

Comma list: 50/49, 78125/73728

Mapping: [⟨2 1 4 5], ⟨0 7 2 2]]

mapping generators: ~7/5, ~192/175

Optimal tunings:

  • WE: ~7/5 = 600.8055 ¢, ~192/175 = 185.9882 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~192/175 = 185.8808 ¢

Optimal ET sequence: 6, 20b, 26, 58c

Badness (Sintel): 4.60

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 tunings:

  • WE: ~7/5 = 600.3702 ¢, ~11/10 = 185.9936 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~11/10 = 185.9408 ¢

Optimal ET sequence: 6, 20b, 26

Badness (Sintel): 2.51

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 tunings:

  • WE: ~7/5 = 600.3563 ¢, ~11/10 = 185.8121 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~11/10 = 185.7623 ¢

Optimal ET sequence: 6f, 20bff, 26

Badness (Sintel): 2.05

References

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