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The '''keemic family''' of rank-3 temperaments are [[planar temperament]]s tempering out the keema, [[875/864]].
{{Technical data page}}
The '''keemic family''' is a family of [[rank-3 temperament]]s which [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864). For [[rank-2 temperament]]s, see [[keemic temperaments]].
 
== Keemic ==
Keemic, also known as ''supermagic'' in earlier materials, has the same lattice structure as [[5-limit]] JI, and [[7/4]] is found by a stack of three [[~]][[6/5]]'s.
 
[[Subgroup]]: 2.3.5.7


= Supermagic =
[[Comma list]]: 875/864
[[Comma list]]: 875/864


[[Mapping]]: [{{val| 1 0 0 5 }}, {{val| 0 1 0 3 }}, {{val| 0 0 1 -3 }}]
{{Mapping|legend=1| 1 0 0 5 | 0 1 0 3 | 0 0 1 -3 }}
 
: mapping generators: ~2, ~3, ~5
Mapping generators: ~2, ~3, ~5


Map to lattice: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]
[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]


Lattice basis:  
Lattice basis:  
: 6/5 length = 0.8879, 3/2 length = 1.3391
: 6/5 length = 0.8879, 3/2 length = 1.3391
: Angle (6/5, 3/2) = 77.834
: Angle (6/5, 3/2) = 77.834
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0490{{c}}, ~3/2 = 702.4868{{c}}, ~5/4 = 380.8010{{c}}
: [[error map]]: {{val| +1.049 +1.581 -3.415 -1.670 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8060{{c}}, ~5/4 = 381.0536{{c}}
: error map: {{val| 0.000 +0.851 -5.260 -3.569 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and 9-odd-limit minimax
* [[7-odd-limit|7-]] and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 5/4 3/4 1/4 -1/4 }}, {{monzo| 5/4 3/4 -3/4 3/4 }}]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/4 3/4 1/4 -1/4 | 5/4 3/4 -3/4 3/4 }}
: Eigenmonzos: 2, 4/3, 7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


{{Val list|legend=1| 7, 15, 19, 41 }}
{{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}
 
[[Badness]] (Sintel): 0.937


[[Projection pair]]: 7 864/125
[[Projection pair]]: 7 864/125
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Scales: [[supermagic15]]
Scales: [[supermagic15]]


== 11-limit ==
== Undecimal keemic ==
Comma list: 100/99, 385/384
{{See also| Ptolemismic clan #Keemic }}
 
Keemic is naturally an [[11-limit]] temperament due to the identity 875/864 = ([[100/99]])⋅([[385/384]]). This identifies a stack of two ~6/5's as ~[[16/11]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 100/99, 385/384
 
{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9868{{c}}, ~3/2 = 704.1908{{c}}, ~5/4 = 381.5352{{c}}
: [[error map]]: {{val| -0.013 +2.223 -4.805 -0.885 +3.318 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.1922{{c}}, ~5/4 = 381.5339{{c}}
: error map: {{val| 0.000 +2.237 -4.780 -0.851 +3.365 }}
 
{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}
 
[[Badness]] (Sintel): 0.770
 
Scales: [[supermagic15]]
 
== Supernatural ==
Supernatural uses the same mapping as [[7-limit]] [[magic]] with an independent generator for prime 11.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 225/224, 245/243
 
{{Mapping|legend=1| 1 0 2 -1 0 | 0 5 1 12 0 | 0 0 0 0 1 }}
: mapping generators: ~2, ~5, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0786{{c}}, ~5/4 = 380.6939{{c}}, ~11/8 = 548.0690{{c}}
: [[error map]]: {{val| +1.079 +1.514 -3.463 -1.578 -0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.4576{{c}}, ~11/8 = 548.8962{{c}}
: error map: {{val| 0.000 +0.333 -5.856 -3.335 -2.422 }}
 
{{Optimal ET sequence|legend=1| 19, 22, 38d, 41, 60e, 101cd, 164c, 224ccde }} *
 
<nowiki>*</nowiki> [[optimal patent val]]: [[104edo|104]]


Mapping: [{{val| 1 0 0 5 2 }}, {{val| 0 1 0 3 -2 }}, {{val| 0 0 1 -3 2 }}]
[[Badness]] (Sintel): 1.07


{{Val list|legend=1| 7, 15, 19, 22, 37, 41, 104, 145c, 201ce, 264bce, 305bcce }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.641 × 10<sup>-3</sup>
Comma list: 105/104, 196/195, 245/243


= Supernatural =
{{Mapping|legend=0| 1 0 2 -1 0 -2 | 0 5 1 12 0 18 | 0 0 0 0 1 0 }}
Comma list: 225/224, 245/243


Mapping: [{{val| 1 0 2 -1 0 }}, {{val| 0 5 1 12 0 }}, {{val| 0 0 0 0 1 }}]
Optimal tunings:  
* WE: ~2 = 1201.3389{{c}}, ~5/4 = 380.4641{{c}}, ~11/8 = 547.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 380.1532{{c}}, ~11/8 = 548.2678{{c}}


{{Val list|legend=1| 19, 22, 41, 82e, 101cd, (104), 224ccde, 265ccde }}
{{Optimal ET sequence|legend=0| 19, 22f, 38df, 41, 60e, 79d, 101cd }}


Badness: 0.888 × 10<sup>-3</sup>
Badness (Sintel): 1.07


[[Category:Theory]]
[[Category:Keemic family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Keemic]]
[[Category:Catalogs of rank-3 temperaments]]
[[Category:Supermagic]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Keemic_family"