@@ Line 1: / Line 1: @@
-The '''sensamagic family''' of rank-3 temperaments tempers out [[245/243]].
+{{Technical data page}}
+The '''sensamagic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the sensamagic comma, [[245/243]].
-= Sensamagic =
+For a list of rank-2 temperaments, see [[Sensamagic clan]].
-{{main| Sensamagic }}
-Subgroup: 2.3.5.7
+== Sensamagic ==
+{{Main| Sensamagic }}
+Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].
+Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 245/243
-[[Mapping]]: [{{val| 1 0 0 0 }}, {{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
+{{Mapping|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}
+: mapping generators: ~2, ~3, ~9/7
-Mapping generators: ~2, ~3, ~9/7
+[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
-Map to lattice: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]
 Lattice basis:
@@ Line 18: / Line 24: @@
 : Angle (3/2, 9/7) = 86.5288°
-[[POTE generator]]s: ~3/2 = 703.7424, ~9/7 = 440.9020
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9983{{c}}, ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}}
+: [[error map]]: {{val| -0.002 +1.785 -0.771 -2.248 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}
+: error map: {{val| 0.000 +1.786 -0.769 -2.245 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 0 0 1/5 2/5 }}, {{monzo| 0 0 1 0 }}, {{monzo| 0 0 0 1 }}]
+: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
-: [[Eigenmonzo]]s: 2, 8/7, 5/4
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
 * [[9-odd-limit]]
-: [{{monzo| 1 0 0 0 }}, {{monzo| 0 1 0 0 }}, {{monzo| 0 5/3 2/3 -2/3 }}, {{monzo| 0 5/3 -1/3 1/3 }}]
+: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
-: [[Eigenmonzo]]s: 2, 4/3, 7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5
-{{Val list|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
+{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}
-[[Badness]]: 0.129 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.570
 [[Projection pair]]: 5 243/49 to 2.3.7
-<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
+{{Databox|[[Minkowski block]]s|
-<div style="line-height:1.6;">[[Minkowski blocks]]</div>
+.3.7 subgroup
-<div class="mw-collapsible-content">
-{2, 3, 7} subgroup
 * 12: 729/686, 64/63
 * 17: 64/63, 19683/19208
@@ Line 44: / Line 51: @@
 * 22: 64/63, 537824/531441
 * 24: 64/63, 15059072/14348907
+}}
+=== Overview to extensions ===
+The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.
+Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.
-</div></div>
+== Undecimal sensamagic ==
+{{Main| Sensamagic }}
-= Undecimal sensamagic =
+Undecimal sensamagic tempers out not only [[385/384]], but [[896/891]], making itself a [[strong extension]] of [[parapyth]].
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 245/243, 385/384
-[[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 1 2 -2 }}, {{val| 0 0 2 -1 -1 }}]
+{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}
-[[POTE generator]]s: ~3/2 = 703.8004, ~9/7 = 440.9178
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9667{{c}}, ~3/2 = 703.7809{{c}}, ~9/7 = 440.9056{{c}}
+: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}
+: error map: {{val| 0.000 +1.840 -0.683 -2.154 +0.175 }}
 [[Minimax tuning]]:
 * [[11-odd-limit]]
 : [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
-: [[Eigenmonzo]]s: 2, 11/9, 7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9
-{{Val list|legend=1| 17, 19, 22, 41, 68, 87, 196, 283, 607bd, 694bd }}
+{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283 }}
-[[Badness]]: 0.722 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.868
 [[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7
-== 13-limit ==
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
-Subgroup: 2.3.5.7.11
-Comma list: 245/243, 352/351, 385/384
+Comma list: 245/243, 352/351, 364/363
-Mapping: [{{val| 1 0 0 0 7 12 }}, {{val| 0 1 1 2 -2 -5 }}, {{val| 0 0 2 -1 -1 -1 }}]
+Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }}
-Vals: {{Val list| 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd }}
+Optimal tunings:
+* WE: ~2 = 1199.9905{{c}}, ~3/2 = 703.7325{{c}}, ~9/7 = 440.9149{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7381{{c}}, ~9/7 = 440.9184{{c}}
-Badness: 1.20 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=0| 17, 19f, 22, 41, 46, 63, 87, 237, 283 }}
-= Sensawer =
+Badness (Sintel): 1.12
-Subgroup: 2.3.5.7.11
+== Sensawer ==
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 245/243, 441/440
-[[Mapping]]: [{{val| 1 0 0 0 -3 }}, {{val| 0 1 1 2 5 }}, {{val| 0 0 2 -1 -4 }}]
+{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}
-[[POTE generator]]s: ~3/2 = 703.1900, ~9/7 = 441.1359
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1654{{c}}, ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}}
+: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2917{{c}}, ~9/7 = 441.1849{{c}}
+: error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}
-{{Val list|legend=1| 14c, 19e, 27e, 41, 60e, 87, 302d, 389d, 476bd }}
+{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87 }}
-[[Badness]]: 0.796 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 0.957
-== 13-limit ==
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
 Comma list: 196/195, 245/243, 352/351
-Mapping: [{{val| 1 0 0 0 -3 2 }}, {{val| 0 1 1 2 5 2 }}, {{val| 0 0 2 -1 -4 -4 }}]
+Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }}
-Vals: {{Val list| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
+Optimal tunings:
+* WE: ~2 = 1199.9800{{c}}, ~3/2 = 703.4468{{c}}, ~9/7 = 441.3705{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4494{{c}}, ~9/7 = 441.3758{{c}}
-Badness: 0.928 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=0| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}
-= Octarod =
+Badness (Sintel): 0.868
-Subgroup: 2.3.5.7.11
+== Octarod ==
+Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 100/99, 245/243
-[[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 1 2 0 }}, {{val| 0 0 2 -1 4 }}]
+{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }}
-[[POTE generator]]s: ~3/2 = 705.0464, ~9/7 = 439.5050
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.2854{{c}}, ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}}
+: [[error map]]: {{val| -0.715 +1.957 -3.915 -0.245 +4.226 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5246{{c}}, ~9/7 = 439.2798{{c}}
+: error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}
-{{Val list|legend=1| 8d, 14c, 19, 22, 27e, 41, 104, 131e }}
+{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*
-[[Badness]]: 0.581 × 10<sup>-3</sup>
+<nowiki/>*[[Optimal patent val]]: [[104edo|104]]
+[[Badness]] (Sintel): 0.698
 Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]
-= Shrusus =
+== Shrusus ==
+[[Subgroup]]: 2.3.5.7.11
-Subgroup: 2.3.5.7.11
 [[Comma list]]: 176/175, 245/243
-[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 1 2 4 }}, {{val| 0 0 2 -1 3 }}]
+{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}
-[[POTE generator]]s: ~3/2 = 706.3702, ~9/7 = 442.1147
-{{Val list|legend=1| 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1198.9114{{c}}, ~3/2 = 705.7294{{c}}, ~9/7 = 441.7137{{c}}
+: [[error map]]: {{val| -1.089 +2.686 +1.754 -1.258 -3.259 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.8402{{c}}, ~9/7 = 442.1064{{c}}
+: error map: {{val| 0.000 +3.885 +3.739 +0.748 -1.638 }}
-[[Badness]]: 0.877 × 10<sup>-3</sup>
+{{Optimal ET sequence|legend=1| 19e, 22, 27e, 46, 68, 95, 141bc, 163bc }}
-== Shrusic ==
+[[Badness]] (Sintel): 1.05
+=== Shrusic ===
 Subgroup: 2.3.5.7.11.13
 Comma list: 91/90, 176/175, 245/243
-Mapping: [{{val| 1 0 0 0 -4 1 }}, {{val| 0 1 1 2 4 1 }}, {{val| 0 0 2 -1 3 3 }}]
+Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}
-Vals: {{Val list| 22, 46, 211bcf, 233bcf, 257bcf, 279bcf }}
-Badness: 1.125 × 10<sup>-3</sup>
-= Sensigh =
-Subgroup: 2.3.5.7.11.13
-Comma list: 91/90, 126/125, 169/168
-Mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}, {{val| 0 0 0 0 1 0 }}]
-Vals: {{Val list| 19, 27, 46, 111df, 157df }}
-Badness: 0.939 × 10<sup>-3</sup>
-== 17-limit ==
+Optimal tunings:
+* WE: ~2 = 1199.7256{{c}}, ~3/2 = 704.9071{{c}}, ~9/7 = 443.1303{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9572{{c}}, ~9/7 = 443.2018{{c}}
-Subgroup: 2.3.5.7.11.13.17
+{{Optimal ET sequence|legend=0| 19e, 22, 27e, 46 }}
-Comma list: 91/90, 126/125, 154/153, 169/168
+Badness (Sintel): 1.05
-Mapping: [{{val| 1 6 8 11 0 10 0 }}, {{val| 0 -7 -9 -13 0 -10 1 }}, {{val| 0 0 0 0 1 0 1 }}]
+== Bisector ==
+[[Subgroup]]: 2.3.5.7.11
-Vals: {{Val list| 19, 27, 46 }}
-= Bisector =
-Subgroup: 2.3.5.7.11
 [[Comma list]]: 121/120, 245/243
-[[Mapping]]: [{{val| 2 0 0 0 3 }}, {{val| 0 1 1 2 1 }}, {{val| 0 0 2 -1 1 }}]
+{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}
+: mapping generators: ~77/54, ~3, ~9/7
-Mapping generators: ~77/54, ~3, ~9/7
-[[POTE generator]]s: ~3/2 = 703.0884, ~9/7 = 441.1060
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 600.3096{{c}}, ~3/2 = 703.4512{{c}}, ~9/7 = 441.3336{{c}}
+: [[error map]]: {{val| +0.619 +2.115 +0.424 -2.019 -4.985 }}
+* [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 703.5671{{c}}, ~9/7 = 441.2436{{c}}
+: error map: {{val| 0.000 +1.612 -0.259 -2.935 -6.507 }}
-{{Val list|legend=1| 22, 46, 68, 82e, 106de, 114, 128e }}
+{{Optimal ET sequence|legend=1| 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e }}
-[[Badness]]: 1.089 × 10<sup>-3</sup>
+[[Badness]] (Sintel): 1.31
-[[Category:Regular temperament theory]]
+[[Category:Temperament families]]
-[[Category:Temperament family]]
+[[Category:Sensamagic family| ]] <!-- main article -->
-[[Category:Sensamagic]]
 [[Category:Rank 3]]

Sensamagic family: Difference between revisions