Gamelismic clan: Difference between revisions

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 * [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
 * ''[[Echidnic]]'' (+686/675} → [[Diaschismic family #Echidnic|Diaschismic family]]
-* [[Valentine]] (+126/125) → [[Starling temperaments #Valentine|Starling temperaments]]
 * [[Superkleismic]] (+875/864) → [[Shibboleth family #Superkleismic|Shibboleth family]]
 * ''[[Blacksmith]]'' (+28/27) → [[Limmic temperaments #Blacksmith|Limmic temperaments]]
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 Badness (Smith): 0.015701
+== Valentine ==
+{{Main| Valentine }}
+Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of ~21/20, three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[The Seven Limit Symmetrical Lattices|lattice of 7-limit tetrads]]. Valentine can also be described as the {{nowrap|31 & 46}} temperament, and [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)<sup>1/9</sup> as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit as {{multival| 9 5 -3 7 … }}, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)<sup>1/10</sup>.
+[[Subgroup]]: 2.3.5
+[[Comma list]]: 1990656/1953125
+{{Mapping|legend=1| 1 1 2 | 0 9 5 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~25/24 = 78.039
+{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 123 }}
+[[Badness]]: 0.122765
+=== 7-limit ===
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 126/125, 1029/1024
+{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
+: mapping generators: ~2, ~21/20
+Wedgie: {{multival| 9 5 -3 -13 -30 -21 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 77.864
+[[Minimax tuning]]:
+* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
+: [{{monzo| 1 0 0 0 }}, {{monzo| 5/2 3/4 0 -3/4 }}, {{monzo| 17/6 5/12 0 -5/12 }}, {{monzo| 5/2 -1/4 0 1/4 }}]
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
+* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
+: [{{monzo| 1 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 }}, {{monzo| 47/21 10/21 0 -5/21 }}, {{monzo| 20/7 -2/7 0 1/7 }}]
+: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+[[Algebraic generator]]: smaller root of ''x''<sup>2</sup> - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.
+{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185, 262cd }}
+[[Badness]]: 0.031056
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 126/125, 176/175
+Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}
+: mapping generators: ~2, ~21/20
+Wedgie: {{multival| 9 5 -3 7 -13 -30 -20 -21 -1 30 }}
+Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.881
+Minimax tuning:
+* [[11-odd-limit]]: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
+: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
+: eigenmonzo (unchanged-interval) basis: 2.11/7
+Algebraic generator: positive root of 4''x''<sup>3</sup> + 15''x''<sup>2</sup> - 21, or else Gontrand2, the smallest positive root of 4''x''<sup>7</sup> - 8''x''<sup>6</sup> + 5.
+{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 262cdee, 339cdeee }}
+Badness: 0.016687
+==== Dwynwen ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 91/90, 121/120, 126/125, 176/175
+Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}
+Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.219
+{{Optimal ET sequence|legend=1| 15, 31f, 46 }}
+Badness: 0.023461
+==== Lupercalia ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 66/65, 105/104, 121/120, 126/125
+Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}
+Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.709
+{{Optimal ET sequence|legend=1| 15, 31, 77ff, 108eff, 139efff }}
+Badness: 0.021328
+==== Valentino ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 176/175, 196/195
+Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}
+Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.958
+{{Optimal ET sequence|legend=1| 15f, 31, 46, 77 }}
+Badness: 0.020665
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 121/120, 126/125, 154/153, 176/175, 196/195
+Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}
+Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 78.003
+{{Optimal ET sequence|legend=1| 15f, 31, 46, 77, 123e, 200ceg }}
+Badness: 0.016768
+==== Semivalentine ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 169/168, 176/175
+Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
+Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 77.839
+{{Optimal ET sequence|legend=1| 16, 30, 46, 62, 108ef }}
+Badness: 0.032749
+==== Hemivalentine ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 121/120, 126/125, 176/175, 343/338
+Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
+Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 39.044
+{{Optimal ET sequence|legend=1| 30, 31, 61, 92f, 123f }}
+Badness: 0.047059
+=== Hemivalentino ===
+Subgroup: 2.3.5.7.11
+Comma list: 126/125, 243/242, 1029/1024
+Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}
+Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.921
+{{Optimal ET sequence|legend=1| 31, 92e, 123, 154, 185 }}
+Badness: 0.061275
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 196/195, 243/242, 1029/1024
+Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}
+Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 38.948
+{{Optimal ET sequence|legend=1| 31, 92e, 123, 154 }}
+Badness: 0.057919
+==== Hemivalentoid ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 126/125, 144/143, 243/242, 343/338
+Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}
+Optimal tuning (POTE): ~2 = 1\1, ~40/39 = 38.993
+{{Optimal ET sequence|legend=1| 31, 92ef, 123f }}
+Badness: 0.057931
 == Unidec ==