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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}}


Optimal GPV sequence: {{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}}


Optimal GPV sequence: {{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 }}
 
{{Multival|legend=1| 7 2 18 -13 9 36 }}


[[POTE generator]]: ~35/27 = 413.513
[[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 }}


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


[[Badness]]: 0.117943
[[Badness]] (Sintel): 2.98


=== 11-limit ===
=== 11-limit ===
Line 75: Line 91:
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 -3 1 -9 -9 | 0 7 2 18 19 }}


POTE generator: ~14/11 = 413.490
Optimal tunings:  
* WE: ~2 = 1201.9201{{c}}, ~11/7 = 787.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 786.6402{{c}}


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


Badness: 0.054296
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 106: 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}}


Optimal GPV sequence: {{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 119: 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}}


Optimal GPV sequence: {{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}}


[[POTE generator]]: ~192/175 = 185.738
[[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}}


{{Val list|legend=1| 6, 26, 58c, 84c }}
{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}


[[Badness]]: 0.181726
[[Badness]] (Sintel): 4.60


=== 11-limit ===
=== 11-limit ===
Line 147: 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}}


Optimal GPV sequence: {{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 160: 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 }}
 
Optimal tunings:
* WE: ~7/5 = 600.3563{{c}}, ~11/10 = 185.8121{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.7623{{c}}


POTE generator: ~11/10 = 185.702
{{Optimal ET sequence|legend=0| 6f, 20bff, 26 }}


Optimal GPV sequence: {{Val list| 26, 58c, 84c }}
Badness (Sintel): 2.05


Badness: 0.049603
== 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]]
Retrieved from "https://en.xen.wiki/w/Wesley_family"