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Horwell temperaments: Difference between revisions

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{{Technical data page}}
Horwell temperaments temper out the horwell comma, {{monzo|-16 1 5 1}} = 65625/65536.
Horwell temperaments temper out the horwell comma, {{monzo|-16 1 5 1}} = 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  
* ''[[Semabila]]'' (+49/48) → [[Mabila family #Septimal mabila|Mabila family]]
* ''[[Worschmidt]]'' (+126/125) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Escaped]]'' (+245/243) → [[Escapade family #Escaped|Escapade family]]
* ''[[Maquiloid]]'' (+686/675) → [[Maquila family #Maquiloid|Maquila family]]
* ''[[Keen]]'' (+875/864) → [[Diaschismic family #Keen|Diaschismic family]]
* [[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]]
* ''[[Eris]]'' (+16875/16807) → [[Mirkwai clan #Eris|Mirkwai clan]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity family]]
* ''[[Kaboom]]'' (+4802000/4782969) → [[Vavoom family #Kaboom|Vavoom family]]
* ''[[Soviet ferris wheel]]'' (+{{monzo| -5 -9 -5 11 }}) → [[20th-octave temperaments #Soviet ferris wheel|20th-octave temperaments]]


== Mutt ==
== Mutt ==
{{Main|Mutt temperament}}
{{Main| Mutt temperament }}


Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val|3 5 7}}, {{val|0 -7 -1}}]
{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}


[[POTE generator]]: ~5/4 = 385.980
: mapping generators: ~98304/78125, ~393216/390625


{{Val list|legend=1| 3, 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 1\3, ~5/4 = 385.980 (~393216/390625 = 14.020)
 
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}


[[Badness]]: 0.162467
[[Badness]]: 0.162467


=== 7-limit ===
=== 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 5 7 8 | 0 -7 -1 12 }}
 
{{Multival|legend=1|21 3 -36 -44 -116 -92}}


[[POTE generator]]: ~5/4 = 385.964
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~5/4 = 385.964 (~126/125 = 14.036)


{{Val list|legend=1| 3, 84, 87, 171 }}
{{Optimal ET sequence|legend=1| 84, 87, 171 }}


[[Badness]]: 0.028406
[[Badness]]: 0.028406
Line 38: Line 55:
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 5 7 8 10 | 0 -7 -1 12 11 }}


POTE generator: ~5/4 = 386.020
Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.020 (~126/125 = 13.980)


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429e }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429e }}


Badness: 0.058344
Badness: 0.058344
Line 51: Line 68:
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 5 7 8 10 11 | 0 -7 -1 12 11 3 }}


POTE generator: ~5/4 = 386.022
Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.022 (~126/125 = 13.978)


Optimal GPV sequence: {{Val list| 3, 84, 87, 171, 258, 429ef }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429ef }}


Badness: 0.029089
Badness: 0.029089


== Fifthplus ==
== Fifthplus ==
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.
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
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val|1 11 -3 20}}, {{val|0 -23 13 -42}}]
{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
 
{{Multival|legend=1|23 -13 42 -74 2 134}}


[[POTE generator]]: ~5488/3645 = 708.774
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5488/3645 = 708.774


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


[[Badness]]: 0.025840
[[Badness]]: 0.025840


== Emkay ==
== Emkay ==
[[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.
[[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
[[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 14 6 -28 | 0 -27 -8 67 }}


{{Multival|legend=1|27 8 -67 -50 -182 -178}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3125/2268 = 551.7745


[[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]]: 0.135696
[[Badness]]: 0.135696
Line 98: Line 111:
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 14 6 -28 3 | 0 -27 -8 67 1 }}


POTE generator: ~11/8 = 551.7746
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7746


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381ce, 1916ce }}


Badness: 0.035586
Badness: 0.035586
Line 111: Line 124:
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 14 6 -28 3 6 | 0 -27 -8 67 1 -5 }}


POTE generator: ~11/8 = 551.7749
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7749


Optimal GPV sequence: {{Val list| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}


Badness: 0.017853
Badness: 0.017853
=== See also ===
* [[:File:Scale Tree Graph For Emkay.png]]


== Kastro ==
== Kastro ==
{{see also| Very high accuracy temperaments #Astro }}
{{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, 117649/116640


[[Mapping]]: [{{val| 1 5 1 6 }}, {{val| 0 -31 12 -29 }}]
{{Mapping|legend=1| 1 5 1 6 | 0 -31 12 -29 }}


[[POTE generator]]: ~3375/3136 = 132.1845
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3375/3136 = 132.1845


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


[[Badness]]: 0.183435
[[Badness]]: 0.183435
Line 139: Line 155:
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 5 1 6 5 | 0 -31 12 -29 -14 }}


POTE generator: ~121/112 = 132.1864
Optimal tuning (POTE): ~2 = 1\1, ~121/112 = 132.1864


Optimal GPV sequence: {{Val list| 109, 118, 345de, 463de, 581dde }}
{{Optimal ET sequence|legend=1| 109, 118, 345de, 463de, 581dde }}


Badness: 0.052693
Badness: 0.052693
Line 152: Line 168:
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 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}


POTE generator: ~13/12 = 132.1789
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 132.1789


Optimal GPV sequence: {{Val list| 109, 118f, 227f }}
{{Optimal ET sequence|legend=1| 109, 118f, 227f }}


Badness: 0.046695
Badness: 0.046695


== Oquatonic ==
== Oquatonic ==
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 of this temperament, see [[28th-octave temperaments #Oquatonic (5-limit)]].''


Subgroup: 2.3.5.7
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>.
 
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 28 0 65 123 }}, {{val| 0 1 0 -1 }}]
{{Mapping|legend=1| 28 0 65 123 | 0 1 0 -1 }}


Mapping generators: ~128/125, ~3
: mapping generators: ~128/125, ~3


{{Multival|legend=1| 28 0 -28 -65 -123 -65 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.1137


[[POTE generator]]: ~3/2 = 702.1137
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
 
{{Val list|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}


[[Badness]]: 0.088286
[[Badness]]: 0.088286
Line 184: Line 202:
Comma list: 1375/1372, 6250/6237, 65625/65536
Comma list: 1375/1372, 6250/6237, 65625/65536


Mapping: [{{val| 28 0 65 123 230 }}, {{val| 0 1 0 -1 -3 }}]
Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}


POTE generator: ~3/2 = 702.0186
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0186


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}
{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}


Badness: 0.047853
Badness: 0.047853
Line 197: Line 215:
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197
Comma list: 625/624, 1375/1372, 2080/2079, 2200/2197


Mapping: [{{val| 28 0 65 123 230 148 }}, {{val| 0 1 0 -1 -3 -1 }}]
Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}


POTE generator: ~3/2 = 702.0288
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.0288


Optimal GPV sequence: {{Val list| 84, 140, 224, 364, 588 }}
{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588 }}


Badness: 0.021968
Badness: 0.021968
Line 208: Line 226:
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. The card game of bezique is played with two packs of 32 cards, hence the name.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 65625/65536, 847288609443/843308032000
 
{{Mapping|legend=1| 32 0 125 -113 | 0 1 -1 4 }}


Comma list: 65625/65536, 847288609443/843308032000
: mapping generators: ~100352/98415, ~3


Mapping: 32 32 93 15 92, 0 1 -1 4 1
[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 1\32, ~3/2 = 701.610


Mapping generators: ~100352/98415 = 1\32, ~3/2 = 701.610
{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}


Optimal tuning (CTE): ~3/2 = 701.953
[[Badness]]: 0.270


=== 11-limit ===
=== 11-limit ===
Line 223: Line 245:
Comma list: 9801/9800, 46656/46585, 65625/65536
Comma list: 9801/9800, 46656/46585, 65625/65536


Mapping: 32 32 93 15 92, 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


Mapping generators: ~45/44 = 1\32, ~3/2 = 701.601
{{Optimal ET sequence|legend=1| 224, 544, 768 }}


Optimal tuning (CTE): ~3/2 = 701.953
Badness: 0.0680


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


Comma list: 729/728, 1575/1573, 67392/67375, 78975/78848
Comma list: 729/728, 1575/1573, 4225/4224, 6656/6655
 
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


Mapping: 32 32 93 15 92 81, 0 1 -1 4 1 2
{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}


Mapping generators: ~45/44 = 1\32, ~3/2 = 701.593
Badness: 0.0298


Optimal tuning (CTE): ~3/2 = 701.953
== Notes ==


Vals: 224, 320, 544, 768, 992, 1216, 1312, ...
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Pages with mostly numerical content]]
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell temperaments| ]] <!-- main article -->
[[Category:Horwell]]
[[Category:Horwell| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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