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{{Main| Decic }}
{{Main| Decic }}


The ''decic'' temperament (10 & 50) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. This temperament is supported by [[10edo|10-]], [[50edo|50-]], and [[60edo]], and can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.
Decic tempers out the [[cloudy comma]], 16807/16384 and the [[marvel comma]], 225/224 in the 7-limit, as well as the [[linus comma]], {{monzo| 11 -10 -10 10 }}. It maybe described as the {{nowrap| 50 & 60 }} temperament, [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]]. It can be extended naturally to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
: mapping generators: ~15/14, ~3
: mapping generators: ~15/14, ~3


[[Optimal tuning]] ([[POTE]]): ~15/14 = 1\10, ~3/2 = 698.696 (~49/48 = 21.304)
[[Optimal tuning]] ([[POTE]]): ~15/14 = 120.000{{c}}, ~3/2 = 698.696{{c}} (~49/48 = 21.304{{c}})


{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}


[[Badness]]: 0.089135
[[Badness]] (Smith): 0.089135


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}


: mapping generators: ~15/14, ~3
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 696.791{{c}} (~56/55 = 23.209{{c}})
 
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 696.791 (~56/55 = 23.209)


{{Optimal ET sequence|legend=1| 10, 30be, 40, 50 }}
{{Optimal ET sequence|legend=0| 10, 30be, 40, 50 }}


Badness: 0.063900
Badness (Smith): 0.063900


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}


: mapping generators: ~14/13, ~3
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 696.993{{c}} (~78/77 = 23.007{{c}})
 
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 696.993 (~78/77 = 23.007)


{{Optimal ET sequence|legend=1| 10, 30be, 40, 50 }}
{{Optimal ET sequence|legend=0| 10, 30be, 40, 50 }}


Badness: 0.036880
Badness (Smith): 0.036880


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}


: mapping generators: ~14/13, ~3
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 697.085{{c}} (~78/77 = 22.915{{c}})
 
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 697.085 (~78/77 = 22.915)


{{Optimal ET sequence|legend=1| 10, 30beg, 40, 50 }}
{{Optimal ET sequence|legend=0| 10, 30beg, 40, 50 }}


Badness: 0.025064
Badness (Smith): 0.025064


=== Splendecic ===
=== Splendecic ===
''Splendecic'' (10e & 50) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[Marvel family #Fantastic|fantastic rank-3 temperament]], its name is a portmanteau of "[[wiktionary:splendid|splendid]]" (synonym of "[[wiktionary:fantastic|fantastic]]") and "decic".
Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.518 (~99/98 = 21.482)
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 698.518{{c}} (~99/98 = 21.482{{c}})


{{Optimal ET sequence|legend=1| 10e, 30bee, 40e, 50, 60e, 110de }}
{{Optimal ET sequence|legend=0| 10e, 30bee, 40e, 50, 60e, 110de }}


Badness: 0.059802
Badness (Smith): 0.059802


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.365 (~91/90 = 21.635)
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 698.365{{c}} (~91/90 = 21.635{{c}})


{{Optimal ET sequence|legend=1| 10e, 30bee, 40e, 50, 60e, 110de }}
{{Optimal ET sequence|legend=0| 10e, 30bee, 40e, 50, 60e, 110de }}


Badness: 0.037989
Badness (Smith): 0.037989


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
Mapping: {{mapping| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 698.375 (~91/90 = 21.625)
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 698.375{{c}} (~91/90 = 21.625{{c}})


{{Optimal ET sequence|legend=1| 10e, 30beeg, 40e, 50, 60e, 110deg }}
{{Optimal ET sequence|legend=0| 10e, 30beeg, 40e, 50, 60e, 110deg }}


Badness: 0.026050
Badness (Smith): 0.026050


=== Prodecic ===
=== Prodecic ===
''Prodecic'' (10 & 50e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[Marvel family #Prodigy|prodigy rank-3 temperament]], its name is a portmanteau of "[[wiktionary:prodigy|prodigy]]" and "decic".
Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.210 (~55/54 = 19.790)
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 700.210{{c}} (~55/54 = 19.790{{c}})


{{Optimal ET sequence|legend=1| 10, 40ee, 50e, 60e }}
{{Optimal ET sequence|legend=0| 10, 40ee, 50e, 60e }}


Badness: 0.066600
Badness (Smith): 0.066600


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}


Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.503 (~91/90 = 19.497)
Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~3/2 = 700.503{{c}} (~91/90 = 19.497{{c}})


{{Optimal ET sequence|legend=1| 10, 40ee, 50e, 60e }}
{{Optimal ET sequence|legend=0| 10, 40ee, 50e, 60e }}


Badness: 0.041788
Badness (Smith): 0.041788


==== 17-limit ====
==== 17-limit ====
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Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.474 (~91/90 = 19.526)
Optimal tuning (POTE): ~15/14 = 1\10, ~3/2 = 700.474 (~91/90 = 19.526)


{{Optimal ET sequence|legend=1| 10, 40ee, 50e, 60e }}
{{Optimal ET sequence|legend=0| 10, 40ee, 50e, 60e }}


Badness: 0.027590
Badness (Smith): 0.027590


== Pentadecal ==
== Pentadecal ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Pentadecal]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Pentadecal]].''


The ''pentadecal'' temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), {{monzo|-11 -15 15}}. This temperament can be described as 15&60 temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[875/864|keema]], 875/864 in the 7-limit.
Pentadecal tempers out the 15-5/3-comma, {{monzo| -11 -15 15 }} in the 5-limit. This temperament can be described as {{nowrap| 15 & 60 }} temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[875/864|keema]], 875/864 in the 7-limit.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 15 0 11 42 | 0 1 1 0 }}
{{Mapping|legend=1| 15 0 11 42 | 0 1 1 0 }}
: mapping generators: ~21/20, ~3
: mapping generators: ~21/20, ~3


[[Optimal tuning]] ([[POTE]]): ~21/20 = 1\15, ~3/2 = 700.223 (~126/125 = 19.777)
[[Optimal tuning]] ([[POTE]]): ~21/20 = 80.000{{c}}, ~3/2 = 700.223{{c}} (~126/125 = 19.777{{c}})


{{Optimal ET sequence|legend=1| 15, 45, 60 }}
{{Optimal ET sequence|legend=1| 15, 45, 60 }}


[[Badness]]: 0.114833
[[Badness]] (Smith): 0.114833


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}
Mapping: {{mapping| 15 0 11 42 52 | 0 1 1 0 0 }}


: mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 80.000{{c}}, ~3/2 = 702.733{{c}} (~56/55 = 17.267{{c}})
 
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 702.733 (~56/55 = 17.267)


{{Optimal ET sequence|legend=1| 15, 45, 60, 75de, 135de }}
{{Optimal ET sequence|legend=0| 15, 45, 60, 75de, 135de }}


Badness: 0.077420
Badness (Smith): 0.077420


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}
Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}


: mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 80.000{{c}}, ~3/2 = 701.715{{c}} (~91/90 = 18.285{{c}})
 
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.715 (~91/90 = 18.285)


{{Optimal ET sequence|legend=1| 15, 45f, 60, 135de, 195cddee }}
{{Optimal ET sequence|legend=0| 15, 45f, 60, 135de, 195cddee }}


Badness: 0.051740
Badness (Smith): 0.051740


=== Quindecal ===
=== Quindecal ===
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Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}
Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}


: mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 80.000{{c}}, ~3/2 = 700.318{{c}} (~126/125 = 19.682{{c}})
 
Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 700.318 (~126/125 = 19.682)


{{Optimal ET sequence|legend=1| 15, 45e, 60e }}
{{Optimal ET sequence|legend=0| 15, 45e, 60e }}


Badness: 0.044405
Badness (Smith): 0.044405


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}
Mapping: {{mapping| 15 0 11 42 28 103 | 0 1 1 0 1 -2 }}


: mapping generators: ~21/20, ~3
Optimal tuning (POTE): ~21/20 = 80.000{{c}}, ~3/2 = 701.647{{c}} (~65/64 = 18.353{{c}})


Optimal tuning (POTE): ~21/20 = 1\15, ~3/2 = 701.647 (~65/64 = 18.353)
{{Optimal ET sequence|legend=0| 15f, 45e, 60e }}


{{Optimal ET sequence|legend=1| 15f, 45e, 60e }}
Badness (Smith): 0.055361
 
Badness: 0.055361


== Quinkee ==
== Quinkee ==
The ''quinkee'' temperament (15 & 40) has a period of 1/5 octave and tempers out the keega (1029/1000), and from this it derives its name.
Quinkee has a period of 1/5 octave and tempers out the keega (1029/1000), from which it derives its name. It may be described as the {{nowrap| 15 & 40 }} temperament.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 243: Line 227:
{{Mapping|legend=1| 5 9 12 14 | 0 -3 -1 0 }}
{{Mapping|legend=1| 5 9 12 14 | 0 -3 -1 0 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~5/4 = 393.439 (~25/24 = 86.561)
[[Optimal tuning]] ([[POTE]]): ~8/7 = 240.000{{c}}, ~5/4 = 393.439{{c}} (~25/24 = 86.561{{c}})


{{Optimal ET sequence|legend=1| 15, 40, 55, 125cd, 180ccdd }}
{{Optimal ET sequence|legend=1| 15, 40, 55, 125cd, 180ccdd }}


[[Badness]]: 0.184222
[[Badness]] (Smith): 0.184222


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 5 9 12 14 18 | 0 -3 -1 0 -2 }}
Mapping: {{mapping| 5 9 12 14 18 | 0 -3 -1 0 -2 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 393.438 (~22/21 = 86.562)
Optimal tuning (POTE): ~8/7 = 240.000{{c}}, ~5/4 = 393.438{{c}} (~22/21 = 86.562{{c}})


{{Optimal ET sequence|legend=1| 15, 40, 55, 125cd, 180ccdde }}
{{Optimal ET sequence|legend=0| 15, 40, 55, 125cd, 180ccdde }}


Badness: 0.065192
Badness (Smith): 0.065192


=== 13-limit ===
=== 13-limit ===
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Mapping: {{mapping| 5 9 12 14 18 20 | 0 -3 -1 0 -2 -4 }}
Mapping: {{mapping| 5 9 12 14 18 20 | 0 -3 -1 0 -2 -4 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~5/4 = 392.798 (~22/21 = 87.202)
Optimal tuning (POTE): ~8/7 = 240.000{{c}}, ~5/4 = 392.798{{c}} (~22/21 = 87.202{{c}})
 
{{Optimal ET sequence|legend=1| 15, 40, 55 }}


Badness: 0.048440
{{Optimal ET sequence|legend=0| 15, 40, 55 }}


== References ==
Badness (Smith): 0.048440


[[Category:Temperament clans]]
[[Category:Temperament clans]]
Retrieved from "https://en.xen.wiki/w/Cloudy_clan"