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Horwell temperaments: Difference between revisions

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Temperaments discussed elsewhere are  
Temperaments discussed elsewhere are  
* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
* ''[[Counterpara]]'' (+100442349/100000000) → [[Parakleismic family #Counterpara|Parakleismic family]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* [[Pontiac]] (+4375/4374) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Countercata]]'' (+5120/5103) → [[Kleismic family #Countercata|Kleismic family]]
* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
* ''[[Eris]]'' (+16875/16807) → [[Mirkwai clan #Eris|Mirkwai clan]]
* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
* ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
* [[Tertiaseptal]] (+2401/2400) → [[Breedsmic temperaments #Tertiaseptal|Breedsmic temperaments]]
* ''[[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]]
* ''[[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]].
== Fifthplus ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''
Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.
Fifthplus may be extended to the 2.3.5.7.17 subgroup, called prime archagall, derived from the fact that it is an extension of [[archagall]] to a prime subgroup. In either the 7-limit or the add-17 subgroup, [[171edo]] is exceptionally efficient and accurate, but in the latter case 171edo is the tuning where [[2401/2400]] (S49), [[2500/2499]] (S50), and [[1701/1700]] (S18/S20) all vanish, which is natural because this temperament tempers out [[1225/1224]] (S35, S49⋅S50) and [[5832/5831]] ((S18/S20)/S49) while not tempering out any of above individually.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 65625/65536, 420175/419904
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
: mapping generators: ~2, ~5488/3645
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}
: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}
: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}
[[Badness]] (Sintel): 0.654
=== Prime archagall ===
Subgroup: 2.3.5.7.17
Comma list: 1225/1224, 24576/24565, 57375/57344
Subgroup-val mapping: {{mapping| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}
Optimal tunings:
* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}
{{Optimal ET sequence|legend=0| 22, 105d, 127d, 149, 171, 1219, 1390 }}
Badness (Sintel): 0.421


== Mutt ==
== Mutt ==
Line 48: Line 90:
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:  
Line 63: Line 105:
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:  
Line 73: Line 115:
Badness (Sintel): 1.20
Badness (Sintel): 1.20


== Fifthplus ==
== Oquatonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sesesix]].''
: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
 
Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the [[dimcomp comma]] (390625/388962). In this temperament, the [[5/4]] major third is mapped to 9\28.


Fifthplus tempers out the [[wizma]] in addition to the horwell comma, and may be described as the {{nowrap| 22 & 171 }}. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 420175/419904
[[Comma list]]: 65625/65536, 390625/388962


{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
: mapping generators: ~2, ~5488/3645
: mapping generators: ~128/125, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0934{{c}}, ~5488/3645 = 708.8291{{c}}
* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}
: [[error map]]: {{val| +0.093 -0.007 -0.158 -0.059 }}
: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5488/3645 = 708.7752{{c}}
* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}
: error map: {{val| 0.000 -0.126 -0.391 -0.268 }}
: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}
 
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
 
[[Badness]] (Sintel): 2.23
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 1375/1372, 6250/6237, 65625/65536
 
{{Mapping|legend=0| 28 0 65 123 230 | 0 1 0 -1 -3 }}
 
Optimal tunings:
* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}
* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}
 
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
 
Badness (Sintel): 1.58
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
 
{{Mapping|legend=0| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
 
Optimal tunings:
* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}


{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


[[Badness]] (Sintel): 0.654
Badness (Sintel): 0.908


== Emkay ==
== Emkay ==
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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:  
Line 139: Line 213:
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:  
Line 176: Line 250:
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:  
Line 191: Line 265:
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:  
Line 200: Line 274:


Badness (Sintel): 1.93
Badness (Sintel): 1.93
== Oquatonic ==
: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
Oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the [[dimcomp comma]] (390625/388962). In this temperament, the [[5/4]] major third is mapped to 9\28.
The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref name="petr's long post"/>.
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 65625/65536, 390625/388962
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
: mapping generators: ~128/125, ~3
[[Optimal tuning]]s:
* [[WE]]: ~128/125 = 42.8570{{c}}, ~3/2 = 702.1112{{c}}
: [[error map]]: {{val| -0.004 +0.152 -0.609 +0.477 }}
* [[CWE]]: ~128/125 = 42.8571{{c}}, ~3/2 = 702.1132{{c}}
: error map: {{val| 0.000 +0.158 -0.599 +0.489 }}
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
[[Badness]] (Sintel): 2.23
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 6250/6237, 65625/65536
Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
Optimal tunings:
* WE: ~128/125 = 42.8577{{c}}, ~3/2 = 702.0275{{c}}
* CWE: ~128/125 = 42.8571{{c}}, ~3/2 = 702.0174{{c}}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
Badness (Sintel): 1.58
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
Optimal tunings:
* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}
Badness (Sintel): 0.908


== Bezique ==
== Bezique ==
Line 280: Line 300:
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:  
Line 295: Line 315:
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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== References ==
== References ==


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