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Horwell temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''horwell temperaments''', which temper out the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536).
This page collects [[rank-2 temperament|rank-2]] '''horwell temperaments''', which [[tempering out|temper out]] the [[horwell comma]] ({{monzo|legend=1| -16 1 5 1 }}, [[ratio]]: 65625/65536). For the family of [[rank-3 temperament]]s, see [[Horwell family]].  


Temperaments discussed elsewhere are  
Temperaments discussed elsewhere are  
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* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[Eris]]'' (+16875/16807) → [[Canopic clan #Eris|Canopic clan]]
* ''[[Eris]]'' (+16875/16807) → [[Canopic clan #Eris|Canopic clan]]
* ''[[Soviet ferris wheel]]'' (+{{monzo| -5 -9 -5 11 }}) → [[20th-octave temperaments #Soviet ferris wheel|20th-octave temperaments]]


Considered below are fifthplus, mutt, oquatonic, emkay, kastro, and bezique, in the order of increasing [[badness]].  
Considered below are fifthplus, mutt, oquatonic, emkay, kastro, soviet ferris wheel, and bezique, in the order of increasing [[badness]].  


== Fifthplus ==
== Fifthplus ==
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=== Prime archagall ===
=== Prime archagall ===
{{ See also | MVP archagall }}
{{ See also | 24576/24565 }}
Subgroup: 2.3.5.7.17
Subgroup: 2.3.5.7.17


Comma list: 1225/1224, 24576/24565, 57375/57344
Comma list: 1225/1224, 24576/24565, 57375/57344


Subgroup-val mapping: {{mapping| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}
{{Mapping|legend=2| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 441/440, 4375/4356, 16384/16335
Comma list: 441/440, 4375/4356, 16384/16335


Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}
{{Mapping|legend=0| 3 -2 6 20 21 | 0 7 1 -12 -11 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 364/363, 441/440, 625/624, 2200/2197
Comma list: 364/363, 441/440, 625/624, 2200/2197


Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}
{{Mapping|legend=0| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 1375/1372, 6250/6237, 65625/65536
Comma list: 1375/1372, 6250/6237, 65625/65536


Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
{{Mapping|legend=0| 28 0 65 123 230 | 0 1 0 -1 -3 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197


Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
{{Mapping|legend=0| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 3025/3024, 4000/3993, 65625/65536
Comma list: 3025/3024, 4000/3993, 65625/65536


Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}
{{Mapping|legend=0| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197


Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}
{{Mapping|legend=0| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 385/384, 3388/3375, 12005/11979
Comma list: 385/384, 3388/3375, 12005/11979


Mapping: {{mapping| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}
{{Mapping|legend=0| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 169/168, 364/363, 385/384, 3388/3375
Comma list: 169/168, 364/363, 385/384, 3388/3375


Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}
{{Mapping|legend=0| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}


Optimal tunings:  
Optimal tunings:  
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Badness (Sintel): 1.93
Badness (Sintel): 1.93
== Soviet ferris wheel ==
The soviet-ferris-wheel temperament may be described as 320 & 460. It was named by [[Eliora]] in 2022 because it has a period of 1/20 octave, and there are 20 cabins on a standard ferris wheel found throughout most of Eastern Europe and Central Asia (as in abandoned Pripyat wheel, for example).
Restricting the temperament to the 5-limit tempers out the comma which is produced by closing twenty [[375/256]]'s at 11 octaves, tempering this interval to 11\20.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 65625/65536, 1977326743/1968300000
{{Mapping|legend=1| 20 0 57 35 | 0 3 -1 2 }}
: mapping generators: ~16807/16200, ~3456/2401
[[Optimal tuning]]s:
* [[WE]]: ~16807/16200 = 60.0061{{c}}, ~3456/2401 = 634.0252{{c}} (~686/675 = 26.0415{{c}})
: [[error map]]: {{val| +0.121 +0.121 +0.007 -0.563 }}
* [[CWE]]: ~16807/16200 = 60.0000{{c}}, ~3456/2401 = 633.9770{{c}} (~686/675 = 26.0230{{c}})
: error map: {{val| 0.000 -0.024 -0.291 -0.872 }}
{{Optimal ET sequence|legend=1| 140, 320, 460, 600, 1060d }}
[[Badness]] (Sintel): 5.25
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 65625/65536, 184877/184320
{{Mapping|legend=0| 20 0 57 35 122 | 0 3 -1 2 5 }}
Optimal tunings:
* WE: ~512/495 = 60.0085{{c}}, ~231/160 = 633.9990{{c}} (~625/616 = 26.0943{{c}})
* CWE: ~512/495 = 60.0000{{c}}, ~231/160 = 633.9127{{c}} (~625/616 = 26.0873{{c}})
{{Optimal ET sequence|legend=0| 140, 320, 460 }}
Badness (Sintel): 2.92
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1001/1000, 4225/4224, 4459/4455, 26411/26364
{{Mapping|legend=0| 20 0 57 35 122 74 | 0 3 -1 2 5 0 }}
Optimal tunings:
* WE: ~512/495 = 60.0082{{c}}, ~75/52 = 633.9971{{c}} (~65/64 = 26.2937{{c}})
* CWE: ~512/495 = 60.0000{{c}}, ~75/52 = 633.9127{{c}} (~65/64 = 26.0873{{c}})
{{Optimal ET sequence|legend=0| 140, 320, 460 }}
Badness (Sintel): 1.48
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 833/832, 1001/1000, 1225/1224, 4225/4224, 4459/4455
{{Mapping|legend=0| 20 0 57 35 122 74 124 | 0 3 -1 2 5 0 -4 }}
Optimal tunings:
* WE: ~88/85 = 60.0081{{c}}, ~75/52 = 633.9982{{c}} (~65/64 = 26.0911{{c}})
* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.9117{{c}} (~65/64 = 26.0883{{c}})
{{Optimal ET sequence|legend=0| 140, 320, 460 }}
Badness (Sintel): 1.14
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 833/832, 1001/1000, 1225/1224, 1729/1728, 2926/2925, 4459/4455
{{Mapping|legend=0| 20 0 57 35 122 74 124 11 | 0 3 -1 2 5 0 -4 7 }}
Optimal tunings:
* WE: ~88/85 = 60.0075{{c}}, ~75/52 = 633.9713{{c}} (~65/64 = 26.1108{{c}})
* CWE: ~88/85 = 60.0000{{c}}, ~75/52 = 633.8948{{c}} (~65/64 = 26.1052{{c}})
{{Optimal ET sequence|legend=0| 140, 320, 460 }}
Badness (Sintel): 1.16


== Bezique ==
== Bezique ==
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Comma list: 9801/9800, 46656/46585, 65625/65536
Comma list: 9801/9800, 46656/46585, 65625/65536


Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}
{{Mapping|legend=0| 32 0 125 -113 60 | 0 1 -1 4 1 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655


Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
{{Mapping|legend=0| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}


Optimal tunings:  
Optimal tunings:  
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[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell| ]] <!-- key article -->
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Horwell_temperaments"