Horwell temperaments: Difference between revisions

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


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]]
* ''[[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) → [[Vishnuzmic family #Narayana|Vishnuzmic 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]]


== 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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: {{monzo| -44 -3 21 }}
[[Comma list]]: 65625/65536, 420175/419904


{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
: mapping generators: ~2, ~5488/3645


: mapping generators: ~98304/78125, ~393216/390625
[[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 tuning]] ([[POTE]]): ~98304/78125 = 1\3, ~5/4 = 385.980 (~393216/390625 = 14.020)
{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, …, 3613ccd }}


{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
[[Badness]] (Sintel): 0.654
 
== 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|legend=1| 3 5 7 8 | 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]] ([[POTE]]): ~63/50 = 1\3, ~5/4 = 385.964 (~126/125 = 14.036)
[[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 }}


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


[[Badness]]: 0.028406
[[Badness]] (Sintel): 0.719


=== 11-limit ===
=== 11-limit ===
Line 57: Line 72:
Comma list: 441/440, 4375/4356, 16384/16335
Comma list: 441/440, 4375/4356, 16384/16335


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


Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.020 (~126/125 = 13.980)
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 ET sequence|legend=1| 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 70: Line 87:
Comma list: 364/363, 441/440, 625/624, 2200/2197
Comma list: 364/363, 441/440, 625/624, 2200/2197


Mapping: {{mapping| 3 5 7 8 10 11 | 0 -7 -1 12 11 3 }}
Mapping: {{mapping| 3 -2 6 20 21 14 | 0 7 1 -12 -11 -3 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.022 (~126/125 = 13.978)
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=1| 84, 87, 171, 258, 429ef }}
{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}


Badness: 0.029089
Badness (Sintel): 1.20


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


[[Subgroup]]: 2.3.5.7
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.  


[[Comma list]]: 65625/65536, 420175/419904
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"/>.  
 
{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
 
{{Multival|legend=1| 23 -13 42 -74 2 134 }}
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5488/3645 = 708.774
 
{{Optimal ET sequence|legend=1| 22, 149, 171, 1903c, 2074c, 2245cd, 2416cd, 2587cd, 2758cd, 2929cd, 3100cd, 3271ccd, 3442ccd, 3613ccd }}
 
[[Badness]]: 0.025840
 
== 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.


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


[[Comma list]]: 65625/65536, 244140625/243045684
[[Comma list]]: 65625/65536, 390625/388962


{{Mapping|legend=1| 1 14 6 -28 | 0 -27 -8 67 }}
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}
: mapping generators: ~128/125, ~3


{{Multival|legend=1| 27 8 -67 -50 -182 -178 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~3125/2268 = 551.7745
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
 
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}


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


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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: {{mapping| 1 14 6 -28 3 6 | 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}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7749
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


{{Optimal ET sequence|legend=1| 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]].''


=== See also ===
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
* [[:File:Scale Tree Graph For Emkay.png]]


== Kastro ==
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.
{{See also| Very high accuracy temperaments #Astro }}


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


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


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


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 }}


{{Optimal ET sequence|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: {{mapping| 1 5 1 6 5 | 0 -31 12 -29 -14 }}
Mapping: {{mapping| 1 -13 -2 39 4 | 0 27 8 -67 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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: {{mapping| 1 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}
Mapping: {{mapping| 1 -13 -2 39 4 1 | 0 27 8 -67 -1 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 109, 118f, 227f }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness: 0.046695
Badness (Sintel): 0.738


== Oquatonic ==
== Kastro ==
: ''For the 5-limit version of this temperament, see [[28th-octave temperaments #Oquatonic (5-limit)]].''
: ''For the 5-limit version, see [[Very high accuracy temperaments #Astro]].''
 
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.


The name ''oquatonic'' was given by [[Petr Pařízek]] in 2011 as an abbreviation of the Italian [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 65625/65536, 390625/388962
[[Comma list]]: 65625/65536, 117649/116640


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


: mapping generators: ~128/125, ~3
[[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 }}


{{Multival|legend=1| 28 0 -28 -65 -123 -65 }}
{{Optimal ET sequence|legend=1| 109, 118, 345d, 463d, 581dd }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1137
[[Badness]] (Sintel): 4.64
 
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
 
[[Badness]]: 0.088286


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


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 239: Line 265:


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


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


{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
{{Optimal ET sequence|legend=1| 96d, 224, 544, 768, 1312, 2080bc }}


[[Badness]]: 0.270
[[Badness]] (Sintel): 6.82


=== 11-limit ===
=== 11-limit ===
Line 255: Line 284:
Mapping: {{mapping| 32 0 125 -113 60 | 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=1| 224, 544, 768 }}
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768 }}


Badness: 0.0680
Badness (Sintel): 2.25


=== 13-limit ===
=== 13-limit ===
Line 268: Line 299:
Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}
Mapping: {{mapping| 32 0 125 -113 60 17 | 0 1 -1 4 1 2 }}


Optimal tuning (CTE): ~45/44 = 1\32, ~3/2 = 701.593
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=1| 224, 544, 768, 1312 }}
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}


Badness: 0.0298
Badness (Sintel): 1.23


== Notes ==
== References ==


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

Latest revision as of 15:19, 22 June 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. The name fifthplus means using a sharp fifth interval (such as a superpyth fifth) as a generator. It is a restriction of prime archagall.

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, 149, 171, 1903c, 2074c, …, 3613ccd

Badness (Sintel): 0.654

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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