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Horwell temperaments: Difference between revisions

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[[File:Scale Tree Graph For Emkay.png|thumb|Scale tree graph for emkay.]]
[[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.
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
[[Subgroup]]: 2.3.5.7
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: mapping generators: ~2, ~4536/3125
: mapping generators: ~2, ~4536/3125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3125/2268 = 551.7745{{c}}
[[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| 87, 137, 224, 311, 535, 1381c, 1916c }}
{{Optimal ET sequence|legend=1| 87, 137, 224, 311, 535, 1381c, 1916c }}
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Comma list: 3025/3024, 4000/3993, 65625/65536
Comma list: 3025/3024, 4000/3993, 65625/65536


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


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~11/8 = 551.7746{{c}}
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=0| 87, 137, 224, 311, 535, 1381ce, 1916ce }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness (Smith): 0.035586
Badness (Sintel): 1.18


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


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


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~11/8 = 551.7749{{c}}
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=0| 87, 137, 224, 311, 535, 1916cef, 2451cceff, 2986cceeff }}
{{Optimal ET sequence|legend=0| 87, 137, 224, 311, 535 }}


Badness (Smith): 0.017853
Badness (Sintel): 0.738


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


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


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


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


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


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


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


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~121/112 = 132.1864{{c}}
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=0| 109, 118, 345de, 463de, 581dde }}
{{Optimal ET sequence|legend=0| 109, 118, 345de, 463de, 581dde }}


Badness (Smith): 0.052693
Badness (Sintel): 1.74


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


Mapping: {{mapping| 1 5 1 6 5 7 | 0 -31 12 -29 -14 -30 }}
Mapping: {{mapping| 1 -26 13 -23 -9 -23 | 0 31 -12 29 14 30 }}


Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~13/12 = 132.1789{{c}}
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=0| 109, 118f, 227f }}
{{Optimal ET sequence|legend=0| 109, 118f, 227f }}


Badness (Smith): 0.046695
Badness (Sintel): 1.93


== Oquatonic ==
== Oquatonic ==
: ''For the 5-limit version, 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.  
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 [[wiktionary: ottantaquatro|''ottantaquatro'' ("eighty-four")]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.  
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>.  
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: mapping generators: ~128/125, ~3
: mapping generators: ~128/125, ~3


[[Optimal tuning]] ([[POTE]]): ~128/125 = 42.8571{{c}}, ~3/2 = 702.1137{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 28, 56, 84, 140, 224, 364, 588, 952 }}


[[Badness]] (Smith): 0.088286
[[Badness]] (Sintel): 2.23


=== 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{{c}}, ~3/2 = 702.0186{{c}}
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, 1400cd, 1988cd, 2576ccdd }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


Badness (Smith): 0.047853
Badness (Sintel): 1.58


=== 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{{c}}, ~3/2 = 702.0288{{c}}
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 ET sequence|legend=0| 84, 140, 224, 364, 588 }}
{{Optimal ET sequence|legend=0| 84, 140, 224, 364, 588 }}


Badness (Smith): 0.021968
Badness (Sintel): 0.908


== Bezique ==
== Bezique ==
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: mapping generators: ~100352/98415, ~3
: mapping generators: ~100352/98415, ~3


[[Optimal tuning]] ([[CTE]]): ~100352/98415 = 37.500{{c}}, ~3/2 = 701.610{{c}}
[[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]] (Smith): 0.270
[[Badness]] (Sintel): 6.82


=== 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 = 37.500{{c}}, ~3/2 = 701.601{{c}}
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| 224, 544, 768 }}
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768 }}


Badness (Smith): 0.0680
Badness (Sintel): 2.25


=== 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 = 37.500{{c}}, ~3/2 = 701.593{{c}}
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| 224, 544, 768, 1312 }}
{{Optimal ET sequence|legend=0| 96d, 224, 544, 768, 1312 }}


Badness (Smith): 0.0298
Badness (Sintel): 1.23


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