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Keemic temperaments: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
This is a collection of [[regular temperament|temperaments]] that [[tempering out|temper out]] the keema ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: [[875/864]]), with [[S-expression]] S5/S6. Its fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]]–6/5–5/4–[[9/7]] are set equidistant from each other. As the [[Keemic family #Undecimal supermagic|canonical extension]] of rank-3 keemic to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])<sup>2</sup> is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.
This is a collection of [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864), with [[S-expression]] S5/S6. Its fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]]–6/5–5/4–[[9/7]] are set equidistant from each other. As the [[Keemic family #Undecimal supermagic|canonical extension]] of rank-3 keemic to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])<sup>2</sup> is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.


Full [[7-limit]] keemic temperaments discussed elsewhere are:
Full [[7-limit]] keemic temperaments discussed elsewhere are:
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: Mapping generators: ~2, ~25/21
: Mapping generators: ~2, ~25/21


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/21 = 292.710
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/21 = 292.710{{c}}


{{Optimal ET sequence|legend=1| 4, 37, 41 }}
{{Optimal ET sequence|legend=1| 4, 37, 41 }}


[[Badness]]: 0.060269
[[Badness]] (Smith): 0.060269


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}
Mapping: {{mapping| 1 5 5 5 2 | 0 -14 -11 -9 6 }}


Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.547
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 292.547{{c}}


{{Optimal ET sequence|legend=1| 4, 37, 41, 119 }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}


Badness: 0.043209
Badness (Smith): 0.043209


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}
Mapping: {{mapping| 1 5 5 5 2 2 | 0 -14 -11 -9 6 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.457
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 292.457{{c}}


{{Optimal ET sequence|legend=1| 4, 37, 41, 78, 119f }}
{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}


Badness: 0.032913
Badness (Smith): 0.032913


=== Quato ===
=== Quato ===
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Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}
Mapping: {{mapping| 1 5 5 5 12 | 0 -14 -11 -9 -35 }}


Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 292.851
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~25/21 = 292.851{{c}}


{{Optimal ET sequence|legend=1| 41, 127cd, 168cd }}
{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}


Badness: 0.041170
Badness (Smith): 0.041170


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}
Mapping: {{mapping| 1 5 5 5 12 12 | 0 -14 -11 -9 -35 -34 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 292.928
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/11 = 292.928{{c}}


{{Optimal ET sequence|legend=1| 41, 86ce, 127cd }}
{{Optimal ET sequence|legend=0| 41, 86ce, 127cd }}


Badness: 0.030081
Badness (Smith): 0.030081


== Chromo ==
== Chromo ==
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: Mapping generators: ~2, ~25/24
: Mapping generators: ~2, ~25/24


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 53.816
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~25/24 = 53.816{{c}}


{{Optimal ET sequence|legend=1| 22, 45, 67c }}
{{Optimal ET sequence|legend=1| 22, 45, 67c }}


[[Badness]]: 0.090769
[[Badness]] (Smith): 0.090769


== Barbad ==
== Barbad ==
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: Mapping generators: ~2, ~98/75
: Mapping generators: ~2, ~98/75


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98/75 = 468.331
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~98/75 = 468.331{{c}}


{{Optimal ET sequence|legend=1| 18, 23d, 41 }}
{{Optimal ET sequence|legend=0| 18, 23d, 41 }}


[[Badness]]: 0.110448
[[Badness]] (Smith): 0.110448


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}
Mapping: {{mapping| 1 9 7 11 14 | 0 -19 -12 -21 -27 }}


Optimal tuning (POTE): ~2 = 1\1, ~98/75 = 468.367
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~98/75 = 468.367{{c}}


{{Optimal ET sequence|legend=1| 18e, 23de, 41, 228ccdd }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41, 228ccdd }}


Badness: 0.050105
Badness (Smith): 0.050105


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}
Mapping: {{mapping| 1 9 7 11 14 8 | 0 -19 -12 -21 -27 -11 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 468.270
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~13/10 = 468.270{{c}}


{{Optimal ET sequence|legend=1| 18e, 23de, 41 }}
{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}


Badness: 0.039183
Badness (Smith): 0.039183


== Hyperkleismic ==
== Hyperkleismic ==
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: Mapping generators: ~2, ~6/5
: Mapping generators: ~2, ~6/5


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 323.780
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 323.780{{c}}


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=1| 26, 37, 63 }}


[[Badness]]: 0.157830
[[Badness]] (Smith): 0.157830


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}
Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.796
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 323.796{{c}}


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness: 0.065356
Badness (Smith): 0.065356


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}
Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}


Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 323.790
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 323.790{{c}}


{{Optimal ET sequence|legend=1| 26, 37, 63 }}
{{Optimal ET sequence|legend=0| 26, 37, 63 }}


Badness: 0.035724
Badness (Smith): 0.035724


== Sevond ==
== Sevond ==
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: Mapping generators: ~10/9, ~3
: Mapping generators: ~10/9, ~3


[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~3/2 = 705.613
[[Optimal tuning]] ([[POTE]]): ~10/9 = 171.429{{c}}, ~3/2 = 705.613{{c}}


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}


[[Badness]]: 0.206592
[[Badness]] (Smith): 0.206592


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}
Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}


Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.518
Optimal tuning (POTE): ~10/9 = 171.429{{c}}, ~3/2 = 705.518{{c}}


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness: 0.070437
Badness (Smith): 0.070437


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}
Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}


Optimal tuning (POTE): ~10/9 = 1\7, ~3/2 = 705.344
Optimal tuning (POTE): ~10/9 = 171.429{{c}}, ~3/2 = 705.344{{c}}


{{Optimal ET sequence|legend=1| 7, 56, 63, 119 }}
{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}


Badness: 0.041238
Badness (Smith): 0.041238


[[Category:Temperament collections]]
[[Category:Temperament collections]]
Retrieved from "https://en.xen.wiki/w/Keemic_temperaments"