Keemic family: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
-The '''keemic family''' of 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]].
-= Supermagic =
+== Keemic ==
-Comma: 875/864
+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.
-and 9 limit minimax
+[[Subgroup]]: 2.3.5.7
-[|1 0 0 0&gt;, |0 1 0 0&gt;, |5/4 3/4 1/4 -1/4&gt;,
+[[Comma list]]: 875/864
-|5/4 3/4 -3/4 3/4&gt;]
-Eigenmonzos: 2, 4/3, 7/5
+{{Mapping|legend=1| 1 0 0 5 | 0 1 0 3 | 0 0 1 -3 }}
+: mapping generators: ~2, ~3, ~5
-Lattice basis: 6/5 length=0.8879 3/2 length=1.3391
+[[Mapping to lattice]]: [{{val| 0 0 -1 3 }}, {{val| 0 1 1 0 }}]
-Angle(6/5, 3/2) = 77.834
+Lattice basis:
+: 6/5 length = 0.8879, 3/2 length = 1.3391
+: Angle (6/5, 3/2) = 77.834
-Map to lattice: [&lt;0 0 -1 3|, &lt;0 1 1 0|]
+[[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 }}
-Map: [&lt;1 0 0 5|, &lt;0 1 0 3|, &lt;0 0 1 -3|]
+[[Minimax tuning]]:
+* [[7-odd-limit|7-]] and [[9-odd-limit]]
+: {{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 }}
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
-Generators: 2, 3, 5
+{{Optimal ET sequence|legend=1| 7, 12d, 15, 19, 41, 142cd, 183cd, 224ccd }}
-{{EDOs|legend=1| 7, 15, 19, 41 }}
+[[Badness]] (Sintel): 0.937
 [[Projection pair]]: 7 864/125
@@ Line 27: / Line 37: @@
 Scales: [[supermagic15]]
-== 11-limit ==
+== Undecimal keemic ==
-Commas: 100/99, 385/384
+{{See also| Ptolemismic clan #Keemic }}
-Map: [&lt;1 0 0 5 2|, &lt;0 1 0 3 -2|, &lt;0 0 1 -3 2|]
+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]].
-{{EDOs|legend=1| 7, 15, 19, 22, 37, 41, 104, 145c, 201ce, 264bce, 305bcce }}
+[[Subgroup]]: 2.3.5.7.11
-Badness: 0.000641
+[[Comma list]]: 100/99, 385/384
-= Supernatural =
+{{Mapping|legend=1| 1 0 0 5 2 | 0 1 0 3 -2 | 0 0 1 -3 2 }}
-Commas: 225/224, 245/243
-Map:[&lt;1 0 2 -1 0|, &lt;0 5 1 12 0|, &lt;0 0 0 0 1|]
+[[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 }}
-{{EDOs|legend=1| 19, 22, 41, 82e, 101cd, (104), 224ccde, 265ccde }}
+{{Optimal ET sequence|legend=1| 7, 15, 19, 22, 37, 41, 104 }}
-Badness: 0.000888
+[[Badness]] (Sintel): 0.770
-[[Category:Theory]]
+Scales: [[supermagic15]]
-[[Category:Temperament family]]
-[[Category:Keemic]]
+== Supernatural ==
-[[Category:Supermagic]]
+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]]
+[[Badness]] (Sintel): 1.07
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 105/104, 196/195, 245/243
+Mapping: {{mapping| 1 0 2 -1 0 -2 | 0 5 1 12 0 18 | 0 0 0 0 1 0 }}
+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}}
+{{Optimal ET sequence|legend=0| 19, 22f, 38df, 41, 60e, 79d, 101cd }}
+Badness (Sintel): 1.07
+[[Category:Temperament families]]
+[[Category:Keemic family| ]] <!-- main article -->
 [[Category:Rank 3]]