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Horwell temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
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[[Comma list]]: {{monzo| -44 -3 21 }}
[[Comma list]]: {{monzo| -44 -3 21 }}


{{Mapping|legend=1| 3 5 7 | 0 -7 -1 }}
{{Mapping|legend=1| 3 -2 6 | 0 7 1 }}
: mapping generators: ~98304/78125, ~393216/390625
: mapping generators: ~98304/78125, ~5/4


[[Optimal tuning]] ([[POTE]]): ~98304/78125 = 400.000{{c}}, ~5/4 = 385.980{{c}} (~393216/390625 = 14.020{{c}})
[[Optimal tuning]]s:
* [[WE]]: ~98304/78125 = 400.0227{{c}}, ~5/4 = 386.0017{{c}} (~393216/390625 = 14.0210{{c}})
: [[error map]]: {{val| +0.068 +0.012 -0.176 }}
* [[CWE]]: ~98304/78125 = 400.0000{{c}}, ~5/4 = 385.9858{{c}} (~393216/390625 = 14.0142{{c}})
: error map: {{val| 0.000 -0.055 -0.328 }}


{{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, 4194cc, 5649cc }}


[[Badness]] (Smith): 0.162467
[[Badness]] (Sintel): 3.81


=== 7-limit ===
=== 7-limit ===
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[[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, ~126/125
: mapping generators: ~63/50, ~5/4


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~5/4 = 385.964{{c}} (~126/125 = 14.036{{c}})
[[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]] (Smith): 0.028406
[[Badness]] (Sintel): 0.719


=== 11-limit ===
=== 11-limit ===
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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 = 400.000{{c}}, ~5/4 = 386.020{{c}} (~126/125 = 13.980{{c}})
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=0| 84, 87, 171, 258, 429e }}
{{Optimal ET sequence|legend=0| 84, 87, 171, 258 }}


Badness (Smith): 0.058344
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
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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 = 400.000{{c}}, ~5/4 = 386.022{{c}} (~126/125 = 13.978{{c}})
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=0| 84, 87, 171, 258, 429ef }}
{{Optimal ET sequence|legend=0| 84, 87, 171, 258, 429ef }}


Badness (Smith): 0.029089
Badness (Sintel): 1.20


== 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]].
: ''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]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Comma list]]: 65625/65536, 420175/419904
[[Comma list]]: 65625/65536, 420175/419904


{{Mapping|legend=1| 1 11 -3 20 | 0 -23 13 -42 }}
{{Mapping|legend=1| 1 -12 10 -22 | 0 23 -13 42 }}
: mapping generators: ~2, ~5488/3645


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5488/3645 = 708.774{{c}}
[[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 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, …, 3613ccd }}


[[Badness]] (Smith): 0.025840
[[Badness]] (Sintel): 0.654


== Emkay ==
== Emkay ==
[[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]] (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]] 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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[[Comma list]]: 65625/65536, 244140625/243045684
[[Comma list]]: 65625/65536, 244140625/243045684


{{Mapping|legend=1| 1 14 6 -28 | 0 -27 -8 67 }}
{{Mapping|legend=1| 1 -13 -2 39 | 0 27 8 -67 }}
: mapping generators: ~2, ~4536/3125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3125/2268 = 551.7745{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3125/2268 = 551.7745{{c}}
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{{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]] (Smith): 0.135696
[[Badness]] (Sintel): 3.43


=== 11-limit ===
=== 11-limit ===
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