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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  
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* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
* ''[[Bisupermajor]]'' (+10976/10935) → [[Hemimage temperaments #Bisupermajor|Hemimage temperaments]]
* ''[[Eris]]'' (+16875/16807) → [[Mirkwai clan #Eris|Mirkwai clan]]
* ''[[Eris]]'' (+16875/16807) → [[Mirkwai clan #Eris|Mirkwai clan]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnuzmic family #Narayana|Vishnuzmic family]]
* ''[[Narayana]]'' (+321489/320000) → [[Vishnu family #Narayana|Vishnu family]]
* ''[[Paramity]]'' (+1600000/1594323) → [[Amity family #Paramity|Amity 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]]
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{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}
{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}
: mapping generators: ~98304/78125, ~393216/390625
: mapping generators: ~98304/78125, ~393216/390625


[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 1\3, ~5/4 = 385.980 (~393216/390625 = 14.020)
[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 400.000{{c}}, ~5/4 = 385.980{{c}} (~393216/390625 = 14.020{{c}})


{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}
{{Optimal ET sequence|legend=1| 84, 87, 171, 771, 942, 1113, 1284, 1455 }}


[[Badness]]: 0.162467
[[Badness]] (Smith): 0.162467


=== 7-limit ===
=== 7-limit ===
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{{Mapping|legend=1| 3 5 7 8 | 0 -7 -1 12 }}
{{Mapping|legend=1| 3 5 7 8 | 0 -7 -1 12 }}
: mapping generators: ~63/50, ~126/125


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~5/4 = 385.964 (~126/125 = 14.036)
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~5/4 = 385.964{{c}} (~126/125 = 14.036{{c}})


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


[[Badness]]: 0.028406
[[Badness]] (Smith): 0.028406


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 3 5 7 8 10 | 0 -7 -1 12 11 }}
Mapping: {{mapping| 3 5 7 8 10 | 0 -7 -1 12 11 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.020 (~126/125 = 13.980)
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~5/4 = 386.020{{c}} (~126/125 = 13.980{{c}})


{{Optimal ET sequence|legend=1| 84, 87, 171, 258, 429e }}
{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429e }}


Badness: 0.058344
Badness (Smith): 0.058344


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 3 5 7 8 10 11 | 0 -7 -1 12 11 3 }}
Mapping: {{mapping| 3 5 7 8 10 11 | 0 -7 -1 12 11 3 }}


Optimal tuning (POTE): ~44/35 = 1\3, ~5/4 = 386.022 (~126/125 = 13.978)
Optimal tuning (POTE): ~44/35 = 400.000{{c}}, ~5/4 = 386.022{{c}} (~126/125 = 13.978{{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 (Smith): 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. It is a restriction of [[24576/24565#2.3.5.7.17 subgroup (prime archagall)|prime archagall]].
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. It is a restriction of [[24576/24565 #2.3.5.7.17 subgroup (prime archagall)|prime archagall]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}


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


{{Optimal ET sequence|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]] (Smith): 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.
[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for 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.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 14 6 -28 | 0 -27 -8 67 }}
{{Mapping|legend=1| 1 14 6 -28 | 0 -27 -8 67 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3125/2268 = 551.7745
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3125/2268 = 551.7745{{c}}


{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}


[[Badness]]: 0.135696
[[Badness]] (Smith): 0.135696


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 14 6 -28 3 | 0 -27 -8 67 1 }}
Mapping: {{mapping| 1 14 6 -28 3 | 0 -27 -8 67 1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7746
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~11/8 = 551.7746{{c}}


{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535, 1381ce, 1916ce }}


Badness: 0.035586
Badness (Smith): 0.035586


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 14 6 -28 3 6 | 0 -27 -8 67 1 -5 }}
Mapping: {{mapping| 1 14 6 -28 3 6 | 0 -27 -8 67 1 -5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.7749
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~11/8 = 551.7749{{c}}
 
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}


Badness: 0.017853
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}


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


== 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
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 5 1 6 | 0 -31 12 -29 }}
{{Mapping|legend=1| 1 5 1 6 | 0 -31 12 -29 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3375/3136 = 132.1845
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3375/3136 = 132.1845{{c}}


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


[[Badness]]: 0.183435
[[Badness]] (Smith): 0.183435


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 5 1 6 5 | 0 -31 12 -29 -14 }}
Mapping: {{mapping| 1 5 1 6 5 | 0 -31 12 -29 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~121/112 = 132.1864
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~121/112 = 132.1864{{c}}


{{Optimal ET sequence|legend=1| 109, 118, 345de, 463de, 581dde }}
{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}


Badness: 0.052693
Badness (Smith): 0.052693


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}
Mapping: {{mapping| 1 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 132.1789
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~13/12 = 132.1789{{c}}


{{Optimal ET sequence|legend=1| 109, 118f, 227f }}
{{Optimal ET sequence|legend=0| 109, 118f, 227f }}


Badness: 0.046695
Badness (Smith): 0.046695


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


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 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.  
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{{Mapping|legend=1| 28 0 65 123 | 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


[[Optimal tuning]] ([[POTE]]): ~128/125 = 42.8571, ~3/2 = 702.1137
[[Optimal tuning]] ([[POTE]]): ~128/125 = 42.8571{{c}}, ~3/2 = 702.1137{{c}}


{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}


[[Badness]]: 0.088286
[[Badness]] (Smith): 0.088286


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}
Mapping: {{mapping| 28 0 65 123 230 | 0 1 0 -1 -3 }}


Optimal tuning (POTE): ~128/125 = 42.8571, ~3/2 = 702.0186
Optimal tuning (POTE): ~128/125 = 42.8571{{c}}, ~3/2 = 702.0186{{c}}


{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588, 1400cd, 1988cd, 2576ccdd }}


Badness: 0.047853
Badness (Smith): 0.047853


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}
Mapping: {{mapping| 28 0 65 123 230 148 | 0 1 0 -1 -3 -1 }}


Optimal tuning (POTE): ~40/39 = 42.8571, ~3/2 = 702.0288
Optimal tuning (POTE): ~40/39 = 42.8571{{c}}, ~3/2 = 702.0288{{c}}


{{Optimal ET sequence|legend=1| 84, 140, 224, 364, 588 }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


Badness: 0.021968
Badness (Smith): 0.021968


== Bezique ==
== Bezique ==
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{{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]] ([[CTE]]): ~100352/98415 = 37.500{{c}}, ~3/2 = 701.610{{c}}


{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}


[[Badness]]: 0.270
[[Badness]] (Smith): 0.270


=== 11-limit ===
=== 11-limit ===
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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 tuning (CTE): ~45/44 = 37.500{{c}}, ~3/2 = 701.601{{c}}


{{Optimal ET sequence|legend=1| 224, 544, 768 }}
{{Optimal ET sequence|legend=0| 224, 544, 768 }}


Badness: 0.0680
Badness (Smith): 0.0680


=== 13-limit ===
=== 13-limit ===
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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 tuning (CTE): ~45/44 = 37.500{{c}}, ~3/2 = 701.593{{c}}


{{Optimal ET sequence|legend=1| 224, 544, 768, 1312 }}
{{Optimal ET sequence|legend=0| 224, 544, 768, 1312 }}


Badness: 0.0298
Badness (Smith): 0.0298


== 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]]
Retrieved from "https://en.xen.wiki/w/Horwell_temperaments"