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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]], [[Würschmidt family|worschmidt]], and [[soviet ferris wheel]].
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 }} temperament. The name ''fifthplus'' means using a sharp fifth interval (such as a [[superpyth]] fifth) as a generator.


Subgroup: 2.3.5
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.


[[Comma list]]: {{monzo| -44 -3 21 }}
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|3 5 7}}, {{val|0 -7 -1}}]
[[Comma list]]: 65625/65536, 420175/419904


[[POTE generator]]: ~5/4 = 385.980
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
: mapping generators: ~2, ~5488/3645


{{Val list|legend=1| 3, 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
[[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 }}


[[Badness]]: 0.162467
{{Optimal ET sequence|legend=1| 22, 105d, 127d, 149, 171, 1903c, 2074c, …, 3613ccd }}


=== 7-limit ===
[[Badness]] (Sintel): 0.654
Subgroup: 2.3.5.7


[[Comma list]]: 65625/65536, 250047/250000
=== Prime archagall ===
Subgroup: 2.3.5.7.17


[[Mapping]]: [{{val|3 5 7 8}}, {{val|0 -7 -1 12}}]
Comma list: 1225/1224, 24576/24565, 57375/57344


{{Multival|legend=1|21 3 -36 -44 -116 -92}}
Subgroup-val mapping: {{mapping| 1 -12 10 -22 -3 | 0 23 -13 42 12 }}


[[POTE generator]]: ~5/4 = 385.964
Optimal tunings:  
* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}


{{Val list|legend=1| 3, 84, 87, 171 }}
{{Optimal ET sequence|legend=0| 22, 105d, 127d, 149, 171, 1219, 1390 }}


[[Badness]]: 0.028406
Badness (Sintel): 0.421


=== 11-limit ===
== Mutt ==
Subgroup: 2.3.5.7.11
{{Main| Mutt }}
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Mutt (5-limit)]].''
 
Mutt tempers out the [[landscape comma]] in addition to the horwell comma, and may be described as the {{nowrap| 84 & 87 }} temperament.  


Comma list: 441/440, 4375/4356, 16384/16335
[[Subgroup]]: 2.3.5.7


Mapping: [{{val|3 5 7 8 10}}, {{val|0 -7 -1 12 11}}]
[[Comma list]]: 65625/65536, 250047/250000


POTE generator: ~5/4 = 386.020
{{Mapping|legend=1| 3 -2 6 20 | 0 7 1 -12 }}
: mapping generators: ~63/50, ~5/4


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429e }}
[[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 }}


Badness: 0.058344
{{Optimal ET sequence|legend=1| 84, 87, 171 }}


=== 13-limit ===
[[Badness]] (Sintel): 0.719
Subgroup: 2.3.5.7.11.13


Comma list: 364/363, 441/440, 625/624, 2200/2197
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping: [{{val|3 5 7 8 10 11}}, {{val|0 -7 -1 12 11 3}}]
Comma list: 441/440, 4375/4356, 16384/16335


POTE generator: ~5/4 = 386.022
Mapping: {{mapping| 3 -2 6 20 21 | 0 7 1 -12 -11 }}


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429ef }}
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}})


Badness: 0.029089
{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}


== Fifthplus ==
Badness (Sintel): 1.93
Fifthplus (22&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.


Subgroup: 2.3.5.7
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 65625/65536, 420175/419904
Comma list: 364/363, 441/440, 625/624, 2200/2197


[[Mapping]]: [{{val|1 11 -3 20}}, {{val|0 -23 13 -42}}]
Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}


{{Multival|legend=1|23 -13 42 -74 2 134}}
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}})


[[POTE generator]]: ~5488/3645 = 708.774
{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}


{{Val list|legend=1| 22, 149, 171, 1903c, 2074c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd }}
Badness (Sintel): 1.20


[[Badness]]: 0.025840
== Oquatonic ==
: ''For the 5-limit version, see [[28th-octave temperaments #Oquatonic (5-limit)]].''


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


Subgroup: 2.3.5.7
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"/>.  


[[Comma list]]: 65625/65536, 244140625/243045684
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val|1 14 6 -28}}, {{val|0 -27 -8 67}}]
[[Comma list]]: 65625/65536, 390625/388962


{{Multival|legend=1|27 8 -67 -50 -182 -178}}
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
: mapping generators: ~128/125, ~3


[[POTE generator]]: ~3125/2268 = 551.7745
[[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 }}


{{Val list|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}


[[Badness]]: 0.135696
[[Badness]] (Sintel): 2.23


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


Comma list: 3025/3024, 4000/3993, 65625/65536
Comma list: 1375/1372, 6250/6237, 65625/65536


Mapping: [{{val|1 14 6 -28 3}}, {{val|0 -27 -8 67 1}}]
Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}


POTE generator: ~11/8 = 551.7746
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 GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


Badness: 0.035586
Badness (Sintel): 1.58


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


Comma list: 625/624, 1575/1573, 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: [{{val|1 14 6 -28 3 6}}, {{val|0 -27 -8 67 1 -5}}]
Optimal tunings:  
* WE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0289{{c}}
* CWE: ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}


POTE generator: ~11/8 = 551.7749
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}
Badness (Sintel): 0.908


Badness: 0.017853
== Emkay ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''


== Kastro ==
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
{{see also| Very high accuracy temperaments #Astro }}


Subgroup: 2.3.5.7
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.


[[Comma list]]: 65625/65536, 117649/116640
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 244140625/243045684


[[Mapping]]: [{{val| 1 5 1 6 }}, {{val| 0 -31 12 -29 }}]
{{Mapping|legend=1| 1 -13 -2 39 | 0 27 8 -67 }}
: mapping generators: ~2, ~4536/3125


[[POTE generator]]: ~3375/3136 = 132.1845
[[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 }}


{{Val list|legend=1| 109, 118, 345d }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}


[[Badness]]: 0.183435
[[Badness]] (Sintel): 3.43


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


Comma list: 385/384, 3388/3375, 12005/11979
Comma list: 3025/3024, 4000/3993, 65625/65536


Mapping: [{{val| 1 5 1 6 5 }}, {{val| 0 -31 12 -29 -14 }}]
Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}


POTE generator: ~121/112 = 132.1864
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| 109, 118, 345de, 463de, 581dde }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness: 0.052693
Badness (Sintel): 1.18


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


Comma list: 169/168, 364/363, 385/384, 3388/3375
Comma list: 625/624, 1575/1573, 2080/2079, 2200/2197


Mapping: [{{val| 1 5 1 6 5 7 }}, {{val| 0 -31 12 -29 -14 -30 }}]
Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}


POTE generator: ~13/12 = 132.1789
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| 109, 118f, 227f }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness: 0.046695
Badness (Sintel): 0.738


== Oquatonic ==
== Kastro ==
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.
: ''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>.  


[[Comma list]]: 65625/65536, 390625/388962
[[Subgroup]]: 2.3.5.7


[[Mapping]]: [{{val| 28 0 65 123 }}, {{val| 0 1 0 -1 }}]
[[Comma list]]: 65625/65536, 117649/116640
 
Mapping generators: ~128/125, ~3


{{Multival|legend=1| 28 0 -28 -65 -123 -65 }}
{{Mapping|legend=1| 1 -26 13 -23 | 0 31 -12 29 }}
: mapping generators: ~2, ~6272/3375


[[POTE generator]]: ~3/2 = 702.1137
[[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| 28, 56, 84, 140, 224, 364, 588, 952 }}
{{Optimal ET sequence|legend=1| 109, 118, 345d, 463d, 581dd }}


[[Badness]]: 0.088286
[[Badness]] (Sintel): 4.64


=== 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: 385/384, 3388/3375, 12005/11979


Mapping: [{{val| 28 0 65 123 230 }}, {{val| 0 1 0 -1 -3 }}]
Mapping: {{mapping| 1 -26 13 -23 -9 | 0 31 -12 29 14 }}


POTE generator: ~3/2 = 702.0186
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| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}
{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}


Badness: 0.047853
Badness (Sintel): 1.74


=== 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: 169/168, 364/363, 385/384, 3388/3375


Mapping: [{{val| 28 0 65 123 230 148 }}, {{val| 0 1 0 -1 -3 -1 }}]
Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}


POTE generator: ~3/2 = 702.0288
Optimal tunings:
* WE: ~2 = 1200.4303{{c}}, ~13/7 = 1068.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1067.8267{{c}}


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588 }}
{{Optimal ET sequence|legend=0| 109, 118f, 227f }}


Badness: 0.021968
Badness (Sintel): 1.93


== Bezique ==
== 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. The card game of bezique is played with two packs of 32 cards, hence the name.
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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 212: Line 281:
[[Comma list]]: 65625/65536, 847288609443/843308032000
[[Comma list]]: 65625/65536, 847288609443/843308032000


[[Mapping]]: [{{val| 32 0 125 -113 }}, {{val| 0 1 -1 4 }}]
{{Mapping|legend=1| 32 0 125 -113 | 0 1 -1 4 }}
: mapping generators: ~100352/98415, ~3
 
[[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 }}


Mapping generators: ~100352/98415, ~3
{{Optimal ET sequence|legend=1| 96d, 224, 544, 768, 1312, 2080bc }}


[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\32, ~3/2 = 701.610
[[Badness]] (Sintel): 6.82


=== 11-limit ===
=== 11-limit ===
Line 223: Line 299:
Comma list: 9801/9800, 46656/46585, 65625/65536
Comma list: 9801/9800, 46656/46585, 65625/65536


Mapping: [{{val| 32 0 125 -113 60 }}, {{val| 0 1 -1 4 1 }}]
Mapping: {{mapping| 32 0 125 -113 60 | 0 1 -1 4 1 }}


Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.601
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 ET sequence|legend=0| 96d, 224, 544, 768 }}
 
Badness (Sintel): 2.25


=== 13-limit ===
=== 13-limit ===
Line 232: Line 314:
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655


Mapping: [{{val| 32 0 125 -113 60 17 }}, {{val| 0 1 -1 4 1 2 }}]
Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
 
Optimal tunings:
* WE: ~45/44 = 37.5021{{c}}, ~3/2 = 701.5769{{c}}
* CWE: ~45/44 = 37.5000{{c}}, ~3/2 = 701.5490{{c}}
 
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}


Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.593
Badness (Sintel): 1.23


Vals: 224, 320, 544, 768, 992, 1216, 1312, ...
== References ==


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 16:03, 22 July 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.

This is a collection of rank-2 horwell temperaments, which temper out the horwell comma (monzo: [-16 1 5 1⟩, ratio: 65625/65536).

Temperaments discussed elsewhere are

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 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: [⟨1 -12 10 -22], ⟨0 23 -13 42]]

mapping generators: ~2, ~5488/3645

Optimal tunings:

  • WE: ~2 = 1200.0934 ¢, ~5488/3645 = 708.8291 ¢
error map: ⟨+0.093 -0.007 -0.158 -0.059]
  • CWE: ~2 = 1200.0000 ¢, ~5488/3645 = 708.7752 ¢
error map: ⟨0.000 -0.126 -0.391 -0.268]

Optimal ET sequence: 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: [⟨1 -12 10 -22 -3], ⟨0 23 -13 42 12]]

Optimal tunings:

  • WE: ~2 = 1200.0516 ¢, ~128/85 = 708.8057 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/85 = 708.7758 ¢

Optimal ET sequence: 22, 105d, 127d, 149, 171, 1219, 1390

Badness (Sintel): 0.421

Mutt

Main article: Mutt
For the 5-limit version, see Father–3 equivalence continuum #Mutt (5-limit).

Mutt tempers out the landscape comma in addition to the horwell comma, and may be described as the 84 & 87 temperament.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 250047/250000

Mapping: [⟨3 -2 6 20], ⟨0 7 1 -12]]

mapping generators: ~63/50, ~5/4

Optimal tunings:

  • WE: ~63/50 = 400.0351 ¢, ~5/4 = 385.9974 ¢ (~126/125 = 14.0377 ¢)
error map: ⟨+0.105 -0.043 -0.105 -0.092]
  • CWE: ~63/50 = 400.0000 ¢, ~5/4 = 385.9638 ¢ (~126/125 = 14.0362 ¢)
error map: ⟨0.000 -0.208 -0.350 -0.392]

Optimal ET sequence: 84, 87, 171

Badness (Sintel): 0.719

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨3 -2 6 20 21], ⟨0 7 1 -12 -11]]

Optimal tunings:

  • WE: ~44/35 = 399.9783 ¢, ~5/4 = 385.9993 ¢ (~126/125 = 13.9790 ¢)
  • CWE: ~44/35 = 400.0000 ¢, ~5/4 = 386.0208 ¢ (~126/125 = 13.9792 ¢)

Optimal ET sequence: 84, 87, 171, 258

Badness (Sintel): 1.93

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨3 -2 6 20 21 14], ⟨0 7 1 -12 -11 -3]]

Optimal tunings:

  • WE: ~44/35 = 399.9610 ¢, ~5/4 = 385.9842 ¢ (~126/125 = 13.9768 ¢)
  • CWE: ~44/35 = 400.0000 ¢, ~5/4 = 386.0231 ¢ (~126/125 = 13.9769 ¢)

Optimal ET sequence: 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.

The name oquatonic was given by Petr Pařízek in 2011 as an abbreviation of the Italian ottantaquatro ("eighty-four")[1].

Subgroup: 2.3.5.7

Comma list: 65625/65536, 390625/388962

Mapping: [⟨28 0 65 123], ⟨0 1 0 -1]]

mapping generators: ~128/125, ~3

Optimal tunings:

  • WE: ~128/125 = 42.8570 ¢, ~3/2 = 702.1112 ¢
error map: ⟨-0.004 +0.152 -0.609 +0.477]
  • CWE: ~128/125 = 42.8571 ¢, ~3/2 = 702.1132 ¢
error map: ⟨0.000 +0.158 -0.599 +0.489]

Optimal ET sequence: 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: [⟨28 0 65 123 230], ⟨0 1 0 -1 -3]]

Optimal tunings:

  • WE: ~128/125 = 42.8577 ¢, ~3/2 = 702.0275 ¢
  • CWE: ~128/125 = 42.8571 ¢, ~3/2 = 702.0174 ¢

Optimal ET sequence: 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: [⟨28 0 65 123 230 148], ⟨0 1 0 -1 -3 -1]]

Optimal tunings:

  • WE: ~40/39 = 42.8571 ¢, ~3/2 = 702.0289 ¢
  • CWE: ~40/39 = 42.8571 ¢, ~3/2 = 702.0288 ¢

Optimal ET sequence: 84, 140, 224, 364, 588

Badness (Sintel): 0.908

Emkay

For the 5-limit version, see Miscellaneous 5-limit temperaments #Emka.
Scale tree graph for emkay.

Emkay may be described as the 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

Mapping: [⟨1 -13 -2 39], ⟨0 27 8 -67]]

mapping generators: ~2, ~4536/3125

Optimal tunings:

  • WE: ~2 = 1200.0279 ¢, ~4536/3125 = 648.2405 ¢
error map: ⟨+0.028 +0.177 -0.445 +0.146]
  • CWE: ~2 = 1200.0000 ¢, ~4536/3125 = 648.2254 ¢
error map: ⟨0.000 +0.133 -0.510 +0.069]

Optimal ET sequence: 87, 137, 224, 311, 535, 1381c, 1916c

Badness (Sintel): 3.43

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -13 -2 39 4], ⟨0 27 8 -67 -1]]

Optimal tunings:

  • WE: ~2 = 1199.9958 ¢, ~16/11 = 648.2231 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 648.2254 ¢

Optimal ET sequence: 87, 137, 224, 311, 535

Badness (Sintel): 1.18

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -13 -2 39 4 1], ⟨0 27 8 -67 -1 5]]

Optimal tunings:

  • WE: ~2 = 1199.9694 ¢, ~16/11 = 648.2085 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 648.2251 ¢

Optimal ET sequence: 87, 137, 224, 311, 535

Badness (Sintel): 0.738

Kastro

For the 5-limit version, see Very high accuracy temperaments #Astro.

Kastro may be described as the 109 & 118 temperament, named by Petr Pařízek in 2011 as a variation of astro[1].

Subgroup: 2.3.5.7

Comma list: 65625/65536, 117649/116640

Mapping: [⟨1 -26 13 -23], ⟨0 31 -12 29]]

mapping generators: ~2, ~6272/3375

Optimal tunings:

  • WE: ~2 = 1200.1529 ¢, ~6272/3375 = 1067.9515 ¢
error map: ⟨+0.153 +0.567 +0.256 -1.749]
  • CWE: ~2 = 1200.0000 ¢, ~6272/3375 = 1067.8174 ¢
error map: ⟨0.000 +0.384 -0.122 -2.122]

Optimal ET sequence: 109, 118, 345d, 463d, 581dd

Badness (Sintel): 4.64

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -26 13 -23 -9], ⟨0 31 -12 29 14]]

Optimal tunings:

  • WE: ~2 = 1200.2427 ¢, ~224/121 = 1068.0296 ¢
  • CWE: ~2 = 1200.0000 ¢, ~224/121 = 1067.8166 ¢

Optimal ET sequence: 109, 118, 345de, 463de, 581dde

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -26 13 -23 -9 -23], ⟨0 31 -12 29 14 30]]

Optimal tunings:

  • WE: ~2 = 1200.4303 ¢, ~13/7 = 1068.2040 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/7 = 1067.8267 ¢

Optimal ET sequence: 109, 118f, 227f

Badness (Sintel): 1.93

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.

Subgroup: 2.3.5.7

Comma list: 65625/65536, 847288609443/843308032000

Mapping: [⟨32 0 125 -113], ⟨0 1 -1 4]]

mapping generators: ~100352/98415, ~3

Optimal tunings:

  • WE: ~100352/98415 = 37.5038 ¢, ~3/2 = 701.6058 ¢
error map: ⟨+0.120 -0.229 -0.071 +0.154]
  • CWE: ~100352/98415 = 37.5000 ¢, ~3/2 = 701.5544 ¢
error map: ⟨0.000 -0.401 -0.368 -0.108]

Optimal ET sequence: 96d, 224, 544, 768, 1312, 2080bc

Badness (Sintel): 6.82

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨32 0 125 -113 60], ⟨0 1 -1 4 1]]

Optimal tunings:

  • WE: ~45/44 = 37.5025 ¢, ~3/2 = 701.5912 ¢
  • CWE: ~45/44 = 37.5000 ¢, ~3/2 = 701.5566 ¢

Optimal ET sequence: 96d, 224, 544, 768

Badness (Sintel): 2.25

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨32 0 125 -113 60 17], ⟨0 1 -1 4 1 2]]

Optimal tunings:

  • WE: ~45/44 = 37.5021 ¢, ~3/2 = 701.5769 ¢
  • CWE: ~45/44 = 37.5000 ¢, ~3/2 = 701.5490 ¢

Optimal ET sequence: 96d, 224, 544, 768, 1312

Badness (Sintel): 1.23

References

  1. ↑ 1.0 1.1 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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