Canou family: Difference between revisions

Line 20:

: Angle (3/2, 81/70) = 73.88 deg

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220

: [[error map]]: {{val| 0.0000 +0.3625 -0.1667 -0.3249 }}

* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237

: error map: {{val| 0.0000 +0.3904 -0.1175 -0.2640 }}

[[Minimax tuning]]:

Line 45:

Line 49:

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215

: [[error map]]: {{val| 0.0000 +0.2565 -0.3768 -0.5383 +0.2980 }}

* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186

: error map: {{val| 0.0000 +0.2279 -0.4221 -0.6043 +0.1069 }}

{{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}

Line 60:

Line 68:

Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 0 0 -4 3 -3 4 }}

Optimal ~~tuning (~~CTE): ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183

Optimal tunings:

* CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183

* CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222

{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}

Line 73:

Line 83:

Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}

Optimal ~~tuning (~~CTE): ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012

Optimal tunings:

* CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012

* CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066

{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}

Line 84:

Line 96:

Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215

Mapping: {{mapping|| 1 0 0 -1 -7 -13 -5 -6 | 0 1 2 2 7 10 6 7 | 0 0 -4 3 -3 4 -2 -4 }}

Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 -6 | 0 1 2 2 7 10 6 7 | 0 0 -4 3 -3 4 -2 -4 }}

Optimal ~~tuning (~~CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930

Optimal tunings:

* CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930

* CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983

{{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}

Badness (Smith): 1.00 × 10-3

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155

Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 -6 4 | 0 1 2 2 7 10 6 7 1 | 0 0 -4 3 -3 4 -2 -4 -5 }}

Optimal tunings:

* CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222

* CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223

Badness (Smith): 0.948 × 10-3

== Canta ==

Line 101:

Line 130:

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378

: [[error map]]: {{val| 0.0000 +0.8543 +1.9537 -0.1940 +6.0769 }}

* [[CWE]]: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492

: error map: {{val| 0.0000 +1.5699 +2.5393 +1.8714 +5.2799 }}

{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}

[[Badness]]: 4.~~523~~ × 10-3

[[Badness]] (Smith): 4.52 × 10-3

=== 13-limit ===

Line 114:

Line 147:

Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}

Optimal ~~tuning (~~CTE): ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447

Optimal tunings:

* CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447

* CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887

Badness: 4.~~781~~ × 10-3

Badness (Smith): 4.78 × 10-3

== Semicanou ==

Line 133:

Line 168:

: mapping generators: ~99/70, ~3, ~81/70

[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191

[[Optimal tuning]]s:

* [[CTE]]: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191

: [[error map]]: {{val| 0.0000 +0.4712 +0.0625 -0.1163 -1.0846 }}

* [[CWE]]: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179

: error map: {{val| 0.0000 +0.4498 +0.0245 -0.1627 -1.1262 }}

{{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}

[[Badness]]: 2.~~197~~ × 10-3

[[Badness]] (Smith): 2.20 × 10-3

=== 13-limit ===

Line 146:

Line 185:

Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}

Optimal ~~tuning (~~CTE): ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526

Optimal tunings:

* CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526

* CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511

Badness: 2.~~974~~ × 10-3

Badness (Smith): 2.97 × 10-3

<!-- debatable canonicity

Line 160:

Line 201:

Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}

Optimal ~~tuning (~~CTE): ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663

Optimal tunings:

* CTE: ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663

{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}

{{Optimal ET sequence|legend=0| 94, 118f, 198g, 212g, 292, 410 }}

Badness: 2.~~421~~ × 10-3

Badness (Smith): 2.42 × 10-3

==== 19-limit ====

Line 173:

Line 215:

Mapping: {{mapping| 2 0 0 -2 1 -11 -10 -12 | 0 1 2 2 2 5 6 7 | 0 0 -4 3 -1 6 -2 -4 }}

Optimal ~~tuning (~~CTE): ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870

Optimal tunings:

* CTE: ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870

{{Optimal ET sequence|legend=1| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}

{{Optimal ET sequence|legend=0| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}

Badness: 2.~~177~~ × 10-3

Badness (Smith): 2.18 × 10-3

-->

=== Semicanoumint ===

Line 188:

Line 231:

Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}

Optimal ~~tuning (~~CTE): ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819

Optimal tunings:

* CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819

* CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704

Badness: 2.~~701~~ × 10-3

Badness (Smith): 2.70 × 10-3

=== Semicanouwolf ===

Line 205:

Line 250:

Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}

Optimal ~~tuning (~~CTE): ~99/70 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382

Optimal tunings:

* CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382

* CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396

Badness: 3.~~511~~ × 10-3

Badness (Smith): 3.51 × 10-3

[[Category:Temperament families]]

@@ Line 20: / Line 20: @@
 : Angle (3/2, 81/70) = 73.88 deg
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.3175, ~81/70 = 254.6220
+: [[error map]]: {{val| 0.0000 +0.3625 -0.1667 -0.3249 }}
+* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.3455, ~81/70 = 254.6237
+: error map: {{val| 0.0000 +0.3904 -0.1175 -0.2640 }}
 [[Minimax tuning]]:
@@ Line 45: / Line 49: @@
 {{Mapping|legend=1| 1 0 0 -1 -7 | 0 1 2 2 7 | 0 0 -4 3 -3 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.2115, ~81/70 = 254.6215
+: [[error map]]: {{val| 0.0000 +0.2565 -0.3768 -0.5383 +0.2980 }}
+* [[CWE]]: ~2 = 1200.0000, ~3/2 = 702.1829, ~81/70 = 254.6186
+: error map: {{val| 0.0000 +0.2279 -0.4221 -0.6043 +0.1069 }}
 {{Optimal ET sequence|legend=1| 94, 99e, 118, 193, 212, 311, 740, 1051d }}
@@ Line 60: / Line 68: @@
 Mapping: {{mapping| 1 0 0 -1 -7 -13 | 0 1 2 2 7 10 | 0 0 -4 3 -3 4 }}
-Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
+Optimal tunings:
+* CTE: ~2 = 1200.0000, ~3/2 = 702.2075, ~81/70 = 254.6183
+* CWE: ~2 = 1200.0000, ~3/2 = 702.1889, ~81/70 = 254.6222
 {{Optimal ET sequence|legend=0| 94, 118f, 193f, 212, 217, 311, 740, 1051d }}
@@ Line 73: / Line 83: @@
 Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 | 0 1 2 2 7 10 6 | 0 0 -4 3 -3 4 -2 }}
-Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
+Optimal tunings:
+* CTE: ~2 = 1200.0000, ~3/2 = 702.2296, ~51/44 = 254.6012
+* CWE: ~2 = 1200.0000, ~3/2 = 702.2055, ~51/44 = 254.6066
 {{Optimal ET sequence|legend=0| 94, 118f, 193f, 212g, 217, 311, 740g, 1051dg }}
@@ Line 84: / Line 96: @@
 Comma list: 595/594, 833/832, 969/968, 1156/1155, 1216/1215
-Mapping: {{mapping|| 1 0 0 -1 -7 -13 -5 -6 | 0 1 2 2 7 10 6 7 | 0 0 -4 3 -3 4 -2 -4 }}
+Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 -6 | 0 1 2 2 7 10 6 7 | 0 0 -4 3 -3 4 -2 -4 }}
-Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
+Optimal tunings:
+* CTE: ~2 = 1200.0000, ~3/2 = 702.2355, ~22/19 = 254.5930
+* CWE: ~2 = 1200.0000, ~3/2 = 702.2117, ~22/19 = 254.5983
 {{Optimal ET sequence|legend=0| 94, 118f, 193f, 212gh, 217, 311, 740g, 1051dgh }}
 Badness (Smith): 1.00 × 10<sup>-3</sup>
+=== 23-limit ===
+Subgroup: 2.3.5.7.11.13.17.19.23
+Comma list: 595/594, 760/759, 833/832, 875/874, 969/968, 1156/1155
+Mapping: {{mapping| 1 0 0 -1 -7 -13 -5 -6 4 | 0 1 2 2 7 10 6 7 1 | 0 0 -4 3 -3 4 -2 -4 -5 }}
+Optimal tunings:
+* CTE: ~2 = 1200.0000, ~3/2 = 702.2361, ~22/19 = 254.6222
+* CWE: ~2 = 1200.0000, ~3/2 = 702.2359, ~22/19 = 254.6223
+{{Optimal ET sequence|legend=0| 94, 193f, 212gh, 217, 311 }}
+Badness (Smith): 0.948 × 10<sup>-3</sup>
 == Canta ==
@@ Line 101: / Line 130: @@
 {{Mapping|legend=1| 1 0 0 -1 6 | 0 1 2 2 -2 | 0 0 4 -3 -3 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.0000, ~3/2 = 702.8093, ~64/55 = 254.3378
+: [[error map]]: {{val| 0.0000 +0.8543 +1.9537 -0.1940 +6.0769 }}
+* [[CWE]]: ~2 = 1200.0000, ~3/2 = 703.5249, ~64/55 = 254.5492
+: error map: {{val| 0.0000 +1.5699 +2.5393 +1.8714 +5.2799 }}
-{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e }}
+{{Optimal ET sequence|legend=1| 75e, 80, 99e, 179e, 457bcddeeee }}
-[[Badness]]: 4.523 × 10<sup>-3</sup>
+[[Badness]] (Smith): 4.52 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 114: / Line 147: @@
 Mapping: {{mapping| 1 0 0 -1 6 11 | 0 1 2 2 -2 -5 | 0 0 4 -3 -3 -3 }}
-Optimal tuning (CTE): ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
+Optimal tunings:
+* CTE: ~2 = 1200.0000, ~3/2 = 703.6228, ~64/55 = 254.3447
+* CWE: ~2 = 1200.0000, ~3/2 = 703.8323, ~64/55 = 254.5887
-{{Optimal ET sequence|legend=1| 75e, 80, 99ef, 179ef }}
+{{Optimal ET sequence|legend=0| 75e, 80, 99ef, 179ef }}
-Badness: 4.781 × 10<sup>-3</sup>
+Badness (Smith): 4.78 × 10<sup>-3</sup>
 == Semicanou ==
@@ Line 133: / Line 168: @@
 : mapping generators: ~99/70, ~3, ~81/70
-[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
+[[Optimal tuning]]s:
+* [[CTE]]: ~99/70 = 600.0000, ~3/2 = 702.4262, ~81/70 = 254.6191
+: [[error map]]: {{val| 0.0000 +0.4712 +0.0625 -0.1163 -1.0846 }}
+* [[CWE]]: ~99/70 = 600.0000, ~3/2 = 702.4048, ~81/70 = 254.6179
+: error map: {{val| 0.0000 +0.4498 +0.0245 -0.1627 -1.1262 }}
 {{Optimal ET sequence|legend=1| 80, 94, 118, 198, 212, 292, 330e, 410 }}
-[[Badness]]: 2.197 × 10<sup>-3</sup>
+[[Badness]] (Smith): 2.20 × 10<sup>-3</sup>
 === 13-limit ===
@@ Line 146: / Line 185: @@
 Mapping: {{mapping| 2 0 0 -2 1 -11 | 0 1 2 2 2 5 | 0 0 -4 3 -1 6 }}
-Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
+Optimal tunings:
+* CTE: ~99/70 = 600.0000, ~3/2 = 702.4802, ~81/70 = 254.6526
+* CWE: ~99/70 = 600.0000, ~3/2 = 702.4945, ~81/70 = 254.6511
-{{Optimal ET sequence|legend=1| 80f, 94, 118f, 198, 410 }}
+{{Optimal ET sequence|legend=0| 80f, 94, 118f, 198, 410 }}
-Badness: 2.974 × 10<sup>-3</sup>
+Badness (Smith): 2.97 × 10<sup>-3</sup>
 <!-- debatable canonicity
@@ Line 160: / Line 201: @@
 Mapping: {{mapping| 2 0 0 -2 1 -11 -10 | 0 1 2 2 2 5 6 | 0 0 -4 3 -1 6 -2 }}
-Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663
+Optimal tunings:
+* CTE: ~99/70 = 600.0000, ~3/2 = 702.4415, ~81/70 = 254.6663
-{{Optimal ET sequence|legend=1| 94, 118f, 198g, 212g, 292, 410 }}
+{{Optimal ET sequence|legend=0| 94, 118f, 198g, 212g, 292, 410 }}
-Badness: 2.421 × 10<sup>-3</sup>
+Badness (Smith): 2.42 × 10<sup>-3</sup>
 ==== 19-limit ====
@@ Line 173: / Line 215: @@
 Mapping: {{mapping| 2 0 0 -2 1 -11 -10 -12 | 0 1 2 2 2 5 6 7 | 0 0 -4 3 -1 6 -2 -4 }}
-Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870
+Optimal tunings:
+* CTE: ~99/70 = 600.0000, ~3/2 = 702.4030, ~81/70 = 254.6870
-{{Optimal ET sequence|legend=1| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
+{{Optimal ET sequence|legend=0| 94, 118f, 198gh, 212gh, 292h, 410, 622ef }}
-Badness: 2.177 × 10<sup>-3</sup>
+Badness (Smith): 2.18 × 10<sup>-3</sup>
 -->
 === Semicanoumint ===
@@ Line 188: / Line 231: @@
 Mapping: {{mapping| 2 0 0 -2 1 11 | 0 1 2 2 2 -1 | 0 0 -4 3 -1 -1 }}
-Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
+Optimal tunings:
+* CTE: ~99/70 = 600.0000, ~3/2 = 702.5374, ~81/70 = 254.6819
+* CTE: ~99/70 = 600.0000, ~3/2 = 702.7916, ~81/70 = 254.6704
-{{Optimal ET sequence|legend=1| 80, 94, 118, 174d, 198, 490f }}
+{{Optimal ET sequence|legend=0| 80, 94, 118, 174d, 198, 490f }}
-Badness: 2.701 × 10<sup>-3</sup>
+Badness (Smith): 2.70 × 10<sup>-3</sup>
 === Semicanouwolf ===
@@ Line 205: / Line 250: @@
 Mapping: {{mapping| 2 0 0 -2 1 0 | 0 1 2 2 2 3 | 0 0 -4 3 -1 -5 }}
-Optimal tuning (CTE): ~99/70 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
+Optimal tunings:
+* CTE: ~55/39 = 600.0000, ~3/2 = 702.7417, ~15/13 = 254.3382
+* CWE: ~55/39 = 600.0000, ~3/2 = 702.8092, ~15/13 = 254.3396
-{{Optimal ET sequence|legend=1| 80, 104c, 118f, 198f, 420cff }}
+{{Optimal ET sequence|legend=0| 80, 104c, 118f, 198f, 420cff }}
-Badness: 3.511 × 10<sup>-3</sup>
+Badness (Smith): 3.51 × 10<sup>-3</sup>
 [[Category:Temperament families]]