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Horwell temperaments temper out the horwell comma, {{monzo|-16 1 5 1}} = 65625/65536.
{{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).


Discussed elsewhere are [[Hemimage temperaments|bisupermajor]], [[Kleismic family|countercata]], [[Mirkwai clan|eris]], [[Escapade family|escaped]], [[Hemimean clan|hemithirds]], [[Diaschismic family|keen]], [[Mabila family|mabila]], [[Maquila family|maquiloid]], [[Vishnuzmic family|narayana]], [[Semicomma family|orwell]], [[Amity family|paramity]], [[Schismatic family|pontiac]], [[Breedsmic temperaments|tertiaseptal]], and [[Würschmidt family|worschmidt]].
Temperaments discussed elsewhere are  
* [[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]]
* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
* [[Hemithirds]] (+1029/1024) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila 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]]


== Mutt ==
Considered below are fifthplus, mutt, oquatonic, emkay, kastro, and bezique, in the order of increasing [[badness]].
{{Main|Mutt temperament}}
 
== 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 }}. 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]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 420175/419904


Subgroup: 2.3.5
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
: mapping generators: ~2, ~5488/3645


[[Comma list]]: {{monzo| -44 -3 21 }}
[[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 }}


[[Mapping]]: [{{val|3 5 7}}, {{val|0 -7 -1}}]
{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}


[[POTE generator]]: ~5/4 = 385.980
[[Badness]] (Sintel): 0.654


{{Val list|legend=1| 3, 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
== Mutt ==
{{Main| Mutt }}
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Mutt (5-limit)]].''


[[Badness]]: 0.162467
Mutt tempers out the [[landscape comma]] in addition to the horwell comma, and may be described as the {{nowrap| 84 & 87 }} temperament.  


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 65625/65536, 250047/250000
[[Comma list]]: 65625/65536, 250047/250000


[[Mapping]]: [{{val|3 5 7 8}}, {{val|0 -7 -1 12}}]
{{Mapping|legend=1| 3 -2 6 20 | 0 7 1 -12 }}
: mapping generators: ~63/50, ~5/4


{{Multival|legend=1|21 3 -36 -44 -116 -92}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0351{{c}}, ~5/4 = 385.9974{{c}} (~126/125 = 14.0377{{c}})
: [[error map]]: {{val| +0.105 -0.043 -0.105 -0.092 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~5/4 = 385.9638{{c}} (~126/125 = 14.0362{{c}})
: error map: {{val| 0.000 -0.208 -0.350 -0.392 }}


[[POTE generator]]: ~5/4 = 385.964
{{Optimal ET sequence|legend=1| 84, 87, 171 }}


{{Val list|legend=1| 3, 84, 87, 171 }}
[[Badness]] (Sintel): 0.719
 
[[Badness]]: 0.028406


=== 11-limit ===
=== 11-limit ===
Line 38: Line 72:
Comma list: 441/440, 4375/4356, 16384/16335
Comma list: 441/440, 4375/4356, 16384/16335


Mapping: [{{val|3 5 7 8 10}}, {{val|0 -7 -1 12 11}}]
Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}


POTE generator: ~5/4 = 386.020
Optimal tunings:  
* WE: ~44/35 = 399.9783{{c}}, ~5/4 = 385.9993{{c}} (~126/125 = 13.9790{{c}})
* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0208{{c}} (~126/125 = 13.9792{{c}})


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429e }}
{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}


Badness: 0.058344
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
Line 51: Line 87:
Comma list: 364/363, 441/440, 625/624, 2200/2197
Comma list: 364/363, 441/440, 625/624, 2200/2197


Mapping: [{{val|3 5 7 8 10 11}}, {{val|0 -7 -1 12 11 3}}]
Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}
 
Optimal tunings:
* WE: ~44/35 = 399.9610{{c}}, ~5/4 = 385.9842{{c}} (~126/125 = 13.9768{{c}})
* CWE: ~44/35 = 400.0000{{c}}, ~5/4 = 386.0231{{c}} (~126/125 = 13.9769{{c}})
 
{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}
 
Badness (Sintel): 1.20
 
== 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.


POTE generator: ~5/4 = 386.022
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"/>.  


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429ef }}
[[Subgroup]]: 2.3.5.7


Badness: 0.029089
[[Comma list]]: 65625/65536, 390625/388962


== Fifthplus ==
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
Fifthplus (22&amp;171) tempers out the sesesix comma, {{monzo|-74 13 23}} in the 5-limit. The name "fifthplus" means using a sharp fifth interval (such as [[superpyth]] fifth) as a generator.
: 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 }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.58


[[Comma list]]: 65625/65536, 420175/419904
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val|1 11 -3 20}}, {{val|0 -23 13 -42}}]
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197


{{Multival|legend=1|23 -13 42 -74 2 134}}
Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}


[[POTE generator]]: ~5488/3645 = 708.774
Optimal tunings:  
* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}


{{Val list|legend=1| 22, 149, 171, 1903c, 2074c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


[[Badness]]: 0.025840
Badness (Sintel): 0.908


== Emkay ==
== Emkay ==
[[Emkay]] (87&amp;224) tempers out the same 5-limit comma as the [[Hemimean clan #Emka|emka temperament]] (37&amp;50), but with the horwell (65625/65536) rather than the hemimean (3136/3125) tempered out.
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''


Subgroup: 2.3.5.7
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
 
Emkay may be described as the {{nowrap| 87 & 224 }} temperament. It tempers out the same 5-limit comma as the [[emka]] (37 & 50), but with the horwell comma (65625/65536) rather than the hemimean comma (3136/3125) tempered out.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 244140625/243045684
[[Comma list]]: 65625/65536, 244140625/243045684


[[Mapping]]: [{{val|1 14 6 -28}}, {{val|0 -27 -8 67}}]
{{Mapping|legend=1| 1 -13 -2 39 | 0 27 8 -67 }}
: mapping generators: ~2, ~4536/3125


{{Multival|legend=1|27 8 -67 -50 -182 -178}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0279{{c}}, ~4536/3125 = 648.2405{{c}}
: [[error map]]: {{val| +0.028 +0.177 -0.445 +0.146 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4536/3125 = 648.2254{{c}}
: error map: {{val| 0.000 +0.133 -0.510 +0.069 }}


[[POTE generator]]: ~3125/2268 = 551.7745
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}


{{Val list|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
[[Badness]] (Sintel): 3.43
 
[[Badness]]: 0.135696


=== 11-limit ===
=== 11-limit ===
Line 98: Line 180:
Comma list: 3025/3024, 4000/3993, 65625/65536
Comma list: 3025/3024, 4000/3993, 65625/65536


Mapping: [{{val|1 14 6 -28 3}}, {{val|0 -27 -8 67 1}}]
Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}


POTE generator: ~11/8 = 551.7746
Optimal tunings:  
* WE: ~2 = 1199.9958{{c}}, ~16/11 = 648.2231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2254{{c}}


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness: 0.035586
Badness (Sintel): 1.18


=== 13-limit ===
=== 13-limit ===
Line 111: Line 195:
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197


Mapping: [{{val|1 14 6 -28 3 6}}, {{val|0 -27 -8 67 1 -5}}]
Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}


POTE generator: ~11/8 = 551.7749
Optimal tunings:  
* WE: ~2 = 1199.9694{{c}}, ~16/11 = 648.2085{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2251{{c}}


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness: 0.017853
Badness (Sintel): 0.738


== Kastro ==
== Kastro ==
{{see also| Very high accuracy temperaments #Astro }}
: ''For the 5-limit version, see [[Very high accuracy temperaments #Astro]].''


Subgroup: 2.3.5.7
Kastro may be described as the {{nowrap| 109 & 118 }} temperament, named by [[Petr Pařízek]] in 2011 as a variation of ''astro''<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 117649/116640
[[Comma list]]: 65625/65536, 117649/116640


[[Mapping]]: [{{val| 1 5 1 6 }}, {{val| 0 -31 12 -29 }}]
{{Mapping|legend=1| 1 -26 13 -23 | 0 31 -12 29 }}
: mapping generators: ~2, ~6272/3375


[[POTE generator]]: ~3375/3136 = 132.1845
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1529{{c}}, ~6272/3375 = 1067.9515{{c}}
: [[error map]]: {{val| +0.153 +0.567 +0.256 -1.749 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6272/3375 = 1067.8174{{c}}
: error map: {{val| 0.000 +0.384 -0.122 -2.122 }}


{{Val list|legend=1| 109, 118, 345d }}
{{Optimal ET sequence|legend=1| 109, 118, 345d, 463d, 581dd }}


[[Badness]]: 0.183435
[[Badness]] (Sintel): 4.64


=== 11-limit ===
=== 11-limit ===
Line 139: Line 232:
Comma list: 385/384, 3388/3375, 12005/11979
Comma list: 385/384, 3388/3375, 12005/11979


Mapping: [{{val| 1 5 1 6 5 }}, {{val| 0 -31 12 -29 -14 }}]
Mapping: {{mapping| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}


POTE generator: ~121/112 = 132.1864
Optimal tunings:  
* WE: ~2 = 1200.2427{{c}}, ~224/121 = 1068.0296{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/121 = 1067.8166{{c}}


Optimal GPV sequence: {{Val list| 109, 118, 345de, 463de, 581dde }}
{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}


Badness: 0.052693
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Line 152: Line 247:
Comma list: 169/168, 364/363, 385/384, 3388/3375
Comma list: 169/168, 364/363, 385/384, 3388/3375


Mapping: [{{val| 1 5 1 6 5 7 }}, {{val| 0 -31 12 -29 -14 -30 }}]
Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}
 
POTE generator: ~13/12 = 132.1789
 
Optimal GPV sequence: {{Val list| 109, 118f, 227f }}


Badness: 0.046695
Optimal tunings:  
* WE: ~2 = 1200.4303{{c}}, ~13/7 = 1068.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1067.8267{{c}}


== Oquatonic ==
{{Optimal ET sequence|legend=0| 109, 118f, 227f }}
The oquatonic has a period of 1/28 octave and tempers out the horwell (65625/65536) and the dimcomp (390625/388962), as well as the [[Hemfiness temperaments|hemfiness]] (4096000/4084101, saquinru-atriyo). In this temperament, major third of [[5/4]] is mapped into 9\28.


Subgroup: 2.3.5.7
Badness (Sintel): 1.93


[[Comma list]]: 65625/65536, 390625/388962
== Bezique ==
Bezique splits the octave into 32 equal parts and reaches 3/2, 8/5 and 11/8 in just one generator with the 64-tone mos. A notable edo tuning overshadowed by [[224edo]] is [[320edo]]. Bezique was named by [[Eliora]] in 2023 for the fact that the card game of bezique is played with two packs of 32 cards.


[[Mapping]]: [{{val| 28 0 65 123 }}, {{val| 0 1 0 -1 }}]
[[Subgroup]]: 2.3.5.7


Mapping generators: ~128/125, ~3
[[Comma list]]: 65625/65536, 847288609443/843308032000


{{Multival|legend=1| 28 0 -28 -65 -123 -65 }}
{{Mapping|legend=1| 32 0 125 -113 | 0 1 -1 4 }}
: mapping generators: ~100352/98415, ~3


[[POTE generator]]: ~3/2 = 702.1137
[[Optimal tuning]]s:
* [[WE]]: ~100352/98415 = 37.5038{{c}}, ~3/2 = 701.6058{{c}}
: [[error map]]: {{val| +0.120 -0.229 -0.071 +0.154 }}
* [[CWE]]: ~100352/98415 = 37.5000{{c}}, ~3/2 = 701.5544{{c}}
: error map: {{val| 0.000 -0.401 -0.368 -0.108 }}


{{Val list|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
{{Optimal ET sequence|legend=1| 96d, 224, 544, 768, 1312, 2080bc }}


[[Badness]]: 0.088286
[[Badness]] (Sintel): 6.82


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 1375/1372, 6250/6237, 65625/65536
Comma list: 9801/9800, 46656/46585, 65625/65536


Mapping: [{{val| 28 0 65 123 230 }}, {{val| 0 1 0 -1 -3 }}]
Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}


POTE generator: ~3/2 = 702.0186
Optimal tunings:  
* WE: ~45/44 = 37.5025{{c}}, ~3/2 = 701.5912{{c}}
* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5566{{c}}


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768 }}


Badness: 0.047853
Badness (Sintel): 2.25


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
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: [{{val| 28 0 65 123 230 148 }}, {{val| 0 1 0 -1 -3 -1 }}]
Optimal tunings:  
* WE: ~45/44 = 37.5021{{c}}, ~3/2 = 701.5769{{c}}
* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5490{{c}}


POTE generator: ~3/2 = 702.0288
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588 }}
Badness (Sintel): 1.23


Badness: 0.021968
== References ==


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