Catalog of rank-4 temperaments: Difference between revisions

Latest revision as of 05:05, 5 June 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

A rank-4 temperament has a period and three additional independent generators, for a total of four dimensions. Typical examples include 7-limit JI, full 11-limit temperament with a one-dimensional comma basis, and full 13-limit temperament with a two-dimensional comma basis.

Cake (45/44)

Subgroup: 2.3.5.7.11

Comma list: 45/44

Mapping: [⟨1 0 0 0 -2], ⟨0 1 0 0 2], ⟨0 0 1 0 1], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~5, ~7

@@ Line 308: / Line 308: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.620{{c}}, ~5/4 = 386.673{{c}}, ~7/4 = 967.775{{c}}
+* [[WE]]: ~2 = 1200.1734{{c}}, ~3/2 = 701.4907{{c}}, ~5/4 = 386.2784{{c}}, ~7/4 = 967.5667{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.496{{c}}, ~5/4 = 386.432{{c}}, ~7/4 = 967.588{{c}}
+: [[error map]]: {{val| +0.173 -0.291 +0.311 -0.912 +0.692 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4963{{c}}, ~5/4 = 386.4317{{c}}, ~7/4 = 967.5883{{c}}
+: error map: {{val| 0.000 -0.459 +0.118 -1.238 +0.420 }}
 [[Minimax tuning]]:
@@ Line 316: / Line 318: @@
 {{Optimal ET sequence|legend=1| 10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d }}
+[[Badness]] (Sintel): 0.189
 === Commas 364/363, 441/440 ===
@@ Line 329: / Line 333: @@
 : 3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.880{{c}}, ~5/4 = 386.523{{c}}, ~7/4 = 967.841{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0145{{c}}, ~3/2 = 701.8745{{c}}, ~5/4 = 386.4853{{c}}, ~7/4 = 967.8244{{c}}
+: [[error map]]: {{val| +0.014 -0.066 +0.201 -0.972 +1.609 -0.667 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8716{{c}}, ~5/4 = 386.4975{{c}}, ~7/4 = 967.8241{{c}}
+: error map: {{val| 0.000 -0.083 +0.184 -1.002 +1.576 -0.692 }}
 [[Minimax tuning]]:
 * [[13-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 0 }}, ~3 = {{monzo| 5/3 0 1/3 -1/3 -1/3 1/3 }}, ~5 = {{monzo| 1/6 0 5/6 2/3 -5/6 1/3 }}, ~7 = {{monzo| 0 0 0 1 0 0 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/10, 8/7, 16/13
+: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7.11/5.13
 * [[15-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 0 }}, ~3 = {{monzo| 5/4 1/4 1/4 -1/4 -1/4 1/4 }}, ~5 = {{monzo| 5/4 -3/4 5/4 -1/4 -1/4 1/4 }}, ~7 = {{monzo| 17/8 -11/8 5/8 -1/8 3/8 1/8 }}
-: [[Eigenmonzo]]s (unchanged-intervals): 2, 14/13, 6/5, 11/9
+: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5/3.11/9.13/7
 {{Optimal ET sequence|legend=1| 12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289 }}
+[[Badness]] (Sintel): 0.452
 === Commas 351/350, 441/440 ===
@@ Line 346: / Line 356: @@
 {{Mapping|legend=1| 1 0 0 0 -3 1 | 0 1 0 0 2 -3 | 0 0 1 0 -1 2 | 0 0 0 1 2 1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.175{{c}}, ~5/4 = 387.330{{c}}, ~7/4 = 968.480{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.2006{{c}}, ~3/2 = 701.0354{{c}}, ~5/4 = 386.8579{{c}}, ~7/4 = 968.2226{{c}}
+: [[error map]]: {{val| +0.201 -0.719 +0.945 -0.202 +0.541 -0.893 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0371{{c}}, ~5/4 = 387.0417{{c}}, ~7/4 = 968.2546{{c}}
+: error map: {{val| 0.000 -0.918 +0.728 -0.571 +0.224 -1.301 }}
 {{Optimal ET sequence|legend=1| 12f, 14cf, 19e, 26, 27e, 31, 45ef, 46, 58, 72, 103, 130, 233, 279, 409, 512bf, 642bf }}
+[[Badness]] (Sintel): 0.524
 === Commas 196/195, 352/351 ===
@@ Line 357: / Line 373: @@
 {{Mapping|legend=1| 1 0 0 0 -3 2 | 0 1 0 0 2 -1 | 0 0 1 0 -1 -1 | 0 0 0 1 2 2 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.601{{c}}, ~5/4 = 387.137{{c}}, ~7/4 = 966.418{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.6830{{c}}, ~3/2 = 702.6689{{c}}, ~5/4 = 387.7765{{c}}, ~7/4 = 967.0260{{c}}
+: [[error map]]: {{val| -0.317 +0.397 +0.829 -2.434 -0.022 +2.128 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7536{{c}}, ~5/4 = 387.5834{{c}}, ~7/4 = 966.9373{{c}}
+: error map: {{val| 0.000 +0.799 +1.270 -1.889 +0.480 +3.010 }}
 {{Optimal ET sequence|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}
+[[Badness]] (Sintel): 0.548
 === Tannic ===
@@ Line 367: / Line 389: @@
 {{Mapping|legend=1| 1 0 0 0 -3 11 | 0 1 0 0 2 -4 | 0 0 1 0 -1 2 | 0 0 0 1 2 -2 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7398{{c}}, ~3/2 = 700.6733{{c}}, ~5/4 = 385.4125{{c}}, ~7/4 = 966.9648{{c}}
+: [[error map]]: {{val| +0.740 -0.542 +0.578 -0.381 -0.714 -1.147 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.3274{{c}}, ~5/4 = 385.8974{{c}}, ~7/4 = 966.8107{{c}}
+: error map: {{val| 0.000 -1.628 -0.416 -2.015 -2.939 -3.664 }}
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.555{{c}}, ~5/4 = 387.816{{c}}, ~7/4 = 967.984{{c}}
 {{Optimal ET sequence|legend=1| 17c, 26, 29, 31, 43, 46, 60e, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 545bddeefff }}
+[[Badness]] (Sintel): 1.23
 ==== 17-limit ====
-[[Subgroup]]: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 273/272, 441/440, 561/560
+Comma list: 273/272, 441/440, 561/560
-{{Mapping|legend=1| 1 0 0 0 -3 11 7 | 0 1 0 0 2 -4 -3 | 0 0 1 0 -1 2 2 | 0 0 0 1 2 -2 -1 }}
+Mapping: {{mapping| 1 0 0 0 -3 11 7 | 0 1 0 0 2 -4 -3 | 0 0 1 0 -1 2 2 | 0 0 0 1 2 -2 -1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.606{{c}}, ~5/4 = 387.655{{c}}, ~7/4 = 967.876{{c}}
+Optimal tunings:
+* WE: ~2 = 1200.7320{{c}}, ~3/2 = 700.7374{{c}}, ~5/4 = 385.2303{{c}}, ~7/4 = 966.8369{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.3909{{c}}, ~5/4 = 385.7215{{c}}, ~7/4 = 966.6922{{c}}
-{{Optimal ET sequence|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
+{{Optimal ET sequence|legend=0| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
+Badness (Sintel): 0.701
 === Commas 441/440, 847/845 ===
@@ Line 391: / Line 425: @@
 : Mapping generators: ~2, ~3, ~5, ~13/11
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.611{{c}}, ~5/4 = 386.882{{c}}, ~13/11 = 290.287{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.1778{{c}}, ~3/2 = 701.4785{{c}}, ~5/4 = 385.4755{{c}}, ~13/11 = 290.3844{{c}}
+: [[error map]]: {{val| +0.178 -0.299 +0.517 -1.226 -0.170 +1.005 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4843{{c}}, ~5/4 = 386.6332{{c}}, ~13/11 = 290.3162{{c}}
+: error map: {{val| 0.000 -0.471 +0.319 -1.560 -0.451 +0.655 }}
 {{Optimal ET sequence|legend=1| 12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d }}
+[[Badness]] (Sintel): 1.11
 == Swetismic (540/539) ==
@@ Line 404: / Line 444: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.6167{{c}}, ~5/4 = 386.0717{{c}}, ~7/4 = 969.5334{{c}}
+* [[WE]]: ~2 = 1199.9149{{c}}, ~3/2 = 701.7196{{c}}, ~5/4 = 386.2545{{c}}, ~7/4 = 969.6659{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.6950{{c}}, ~5/4 = 386.1796{{c}}, ~7/4 = 969.6366{{c}}
+: [[error map]]: {{val| -0.085 -0.321 -0.229 +0.670 +0.508 }}
+* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.6950{{c}}, ~5/4 = 386.1795{{c}}, ~7/4 = 969.6366{{c}}
+: error map: {{val| 0.000 -0.260 -0.134 +0.811 +0.673 }}
 {{Optimal ET sequence|legend=1| 8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde }}
-[[Badness]] (Smith): 0.0105 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.191
 === Commas 540/539, 729/728 ===
@@ Line 420: / Line 462: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.6687{{c}}, ~5/4 = 386.0441{{c}}, ~7/4 = 969.5668{{c}}
+* [[WE]]: ~2 = 1199.9170{{c}}, ~3/2 = 701.7128{{c}}, ~5/4 = 386.2521{{c}}, ~7/4 = 969.6598{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.7230{{c}}, ~5/4 = 386.1818{{c}}, ~7/4 = 969.6607{{c}}
+: [[error map]]: {{val| -0.083 -0.325 -0.228 +0.668 +0.504 +0.006 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7231{{c}}, ~5/4 = 386.1819{{c}}, ~7/4 = 969.6608{{c}}
+: error map: {{val| 0.000 -0.232 -0.132 +0.835 +0.712 +0.150 }}
 {{Optimal ET sequence|legend=1| 12e, 14cf, 17c, 19, 22f, 31f, 39df, 41, 53, 58, 72, 111, 130, 183, 224, 354, 578, 985d, 1267ddef }}
-[[Badness]] (Smith): 1.73 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.454
-=== Commas 540/539, 847/845 ===
-[[Subgroup]]: 2.3.5.7.11.13
-[[Comma list]]: 540/539, 847/845
-{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}
-: Mapping generators: ~2, ~3, ~5, ~13/11
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.284{{c}}, ~5/4 = 386.211{{c}}, ~13/11 = 291.733{{c}}
-{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
-[[Badness]] (Smith): 3.97 × 10<sup>-6</sup>
 === Commas 540/539, 625/624 ===
@@ Line 448: / Line 478: @@
 {{Mapping|legend=1| 1 0 0 0 2 -4 | 0 1 0 0 3 -1 | 0 0 1 0 1 4 | 0 0 0 1 -2 0 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.702{{c}}, ~5/4 = 385.631{{c}}, ~7/4 = 969.467{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0035{{c}}, ~3/2 = 701.6973{{c}}, ~5/4 = 385.6264{{c}}, ~7/4 = 969.4616{{c}}
+: [[error map]]: {{val| +0.004 -0.254 -0.680 +0.643 +0.488 +0.291 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6983{{c}}, ~5/4 = 385.6278{{c}}, ~7/4 = 969.4622{{c}}
+: error map: {{val| 0.000 -0.257 -0.686 +0.636 +0.480 +0.285 }}
 {{Optimal ET sequence|legend=1| 19, 22, 31, 49f, 50, 53, 72, 103, 121, 152f, 193, 224 }}
-[[Badness]] (Smith): 3.59 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.943
 === Commas 540/539, 676/675 ===
@@ Line 462: / Line 496: @@
 : Mapping generators: ~2, ~26/15, ~5, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~26/15 = 950.906{{c}}, ~5/4 = 386.474{{c}}, ~7/4 = 969.892{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7990{{c}}, ~26/15 = 950.9130{{c}}, ~5/4 = 386.8478{{c}}, ~7/4 = 970.1529{{c}}
+: [[error map]]: {{val| -0.201 -0.129 +0.132 +0.925 +0.702 -1.142 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.9953{{c}}, ~5/4 = 386.7094{{c}}, ~7/4 = 970.1192{{c}}
+: error map: {{val| 0.000 +0.036 +0.396 +1.293 +1.125 -0.832 }}
 {{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
-[[Badness]] (Smith): 3.06 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.802
+=== Commas 540/539, 847/845 ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 540/539, 847/845
+{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 0 0 3 3 | 0 0 1 1 -1 -1 | 0 0 0 2 -4 -3 }}
+: Mapping generators: ~2, ~3, ~5, ~13/11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8339{{c}}, ~3/2 = 701.4961{{c}}, ~5/4 = 386.5624{{c}}, ~13/11 = 291.6840{{c}}
+: [[error map]]: {{val| -0.166 -0.625 -0.084 +0.772 -0.627 +1.848 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4439{{c}}, ~5/4 = 386.4201{{c}}, ~13/11 = 291.7287{{c}}
+: error map: {{val| 0.000 -0.511 +0.106 +1.052 -0.321 +2.198 }}
+{{Optimal ET sequence|legend=1| 8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f }}
+[[Badness]] (Sintel): 1.04
 == Pentacircle (896/891) ==
@@ Line 477: / Line 533: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.576{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 968.126{{c}}
+* [[WE]]: ~2 = 1199.3774{{c}}, ~3/2 = 703.4693{{c}}, ~5/4 = 387.5573{{c}}, ~7/4 = 969.3690{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 703.743{{c}}, ~5/4 = 387.245{{c}}, ~7/4 = 969.048{{c}}
+: [[error map]]: {{val| -0.623 +0.892 -0.002 -0.702 +1.061 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7426{{c}}, ~5/4 = 387.2445{{c}}, ~7/4 = 969.0476{{c}}
+: error map: {{val| 0.000 +1.788 +0.931 +0.222 +2.759 }}
 {{Optimal ET sequence|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
-[[Badness]] (Smith): 0.0658 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 1.20
-=== Tridecimal pentacircle a.k.a. gentle ===
+=== Tridecimal pentacircle (gentle) ===
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 491: / Line 549: @@
 {{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 -4 -7 | 0 0 1 0 0 0 | 0 0 0 1 1 1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 703.786{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 967.665{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.3706{{c}}, ~3/2 = 703.4872{{c}}, ~5/4 = 387.5708{{c}}, ~7/4 = 969.3987{{c}}
+: [[error map]]: {{val| -0.629 +0.903 -0.002 -0.686 +0.985 +0.055 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.8328{{c}}, ~5/4 = 387.2881{{c}}, ~7/4 = 969.1612{{c}}
+: error map: {{val| 0.000 +1.878 +0.974 +0.335 +2.512 +1.804 }}
 {{Optimal ET sequence|legend=1| 12f, 17c, 22, 29, 34d, 41, 46, 58, 80, 87, 121, 167, 179ef, 208, 266ef, 433bceef, 641bbceeeff, 699bbcdeeeff }}
-[[Badness]] (Smith): 3.375 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.885
 == Topsy (847/845, 1001/1000) ==
@@ Line 502: / Line 564: @@
 [[Comma list]]: 847/845, 1001/1000
-{{Mapping|legend=1| 1 0 0 2 0 1 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 1 }}
+{{Mapping|legend=1| 1 0 0 -2 3 2 | 0 1 0 0 0 0 | 0 0 1 1 1 1 | 0 0 0 4 -3 -1 }}
-: Mapping generators: ~2, ~3, ~5, ~13/10
+: Mapping generators: ~2, ~3, ~5, ~20/13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0343{{c}}, ~3/2 = 701.9205{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
+: [[error map]]: {{val| +0.034 -0.000 +0.365 -0.353 -0.934 +0.753 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9261{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}
+: error map: {{val| 0.000 -0.029 +0.332 -0.408 -1.001 +0.675 }}
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.955{{c}}, ~5/4 = 386.699{{c}}, ~13/10 = 454.558{{c}}
 {{Optimal ET sequence|legend=1| 16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e }}
+[[Badness]] (Sintel): 1.04
 == Lehmerismic (3025/3024) ==
@@ Line 518: / Line 588: @@
 [[Optimal tuning]]:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.998{{c}}, ~5/4 = 386.252{{c}}, ~55/36 = 733.436{{c}}
+* [[WE]]: ~2 = 1200.0197{{c}}, ~3/2 = 701.9725{{c}}, ~5/4 = 386.2210{{c}}, ~55/36 = 733.4363{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.978{{c}}, ~5/4 = 386.232{{c}}, ~55/36 = 733.430{{c}}
+: [[error map]]: {{val| +0.020 +0.037 -0.053 +0.039 -0.118 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9782{{c}}, ~5/4 = 386.2319{{c}}, ~55/36 = 733.4299{{c}}
+: error map: {{val| 0.000 +0.023 -0.082 +0.012 -0.164 }}
+{{Optimal ET sequence|legend=1| 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
-{{Optimal ET sequence|legend=1| 7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee }}
+[[Badness]] (Sintel): 0.0802
 === Commas 2080/2079, 3025/3024 ===
@@ Line 531: / Line 605: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.076{{c}}, ~5/4 = 386.208{{c}}, ~55/36 = 733.438{{c}}
+* [[WE]]: ~2 = 1199.9717{{c}}, ~3/2 = 702.0958{{c}}, ~5/4 = 386.2618{{c}}, ~55/36 = 733.4380{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.099{{c}}, ~5/4 = 386.248{{c}}, ~55/36 = 733.449{{c}}
+: [[error map]]: {{val| -0.028 +0.113 -0.108 +0.118 -0.007 -0.134 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0990{{c}}, ~5/4 = 386.2477{{c}}, ~55/36 = 733.4490{{c}}
+: error map: {{val| 0.000 +0.144 -0.066 +0.171 +0.081 -0.082 }}
-{{Optimal ET sequence|legend=1| 15, 26, 41, 46, 72, 87, 111, 183, 224, 270, 311, 494 }}
+{{Optimal ET sequence|legend=1| 15, 24d, 26, 31f, 39df, 41, 46, 65d, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1075, 1569, 1839, 2333, 3408d }}
+[[Badness]] (Sintel): 0.272
 == Trimitone (8019/8000) ==
@@ Line 545: / Line 623: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5449{{c}}, ~5/4 = 386.7538{{c}}, ~7/4 = 968.8259{{c}}
+* [[WE]]: ~2 = 1200.1317{{c}}, ~3/2 = 701.4920{{c}}, ~5/4 = 386.4054{{c}}, ~7/4 = 968.5622{{c}}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4729{{c}}, ~5/4 = 386.5374{{c}}, ~7/4 = 968.6210{{c}}
+: [[error map]]: {{val| +0.132 -0.331 +0.355 -0.000 -0.263 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4730{{c}}, ~5/4 = 386.5375{{c}}, ~7/4 = 968.6211{{c}}
+: error map: {{val| 0.000 -0.482 +0.224 -0.205 -0.543 }}
 {{Optimal ET sequence|legend=1| 12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be }}
-[[Badness]] (Smith): 0.0820 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 1.50
 === Commas 729/728, 1001/1000 ===
@@ Line 558: / Line 638: @@
 {{Mapping|legend=1| 1 0 0 0 6 -3 | 0 1 0 0 -6 6 | 0 0 1 0 3 0 | 0 0 0 1 0 -1 }}
-: Mapping generators: ~2, ~3, ~5, ~7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5537{{c}}, ~5/4 = 386.7680{{c}}, ~7/4 = 968.8144{{c}}
+* [[WE]]: ~2 = 1200.1318{{c}}, ~3/2 = 701.4926{{c}}, ~5/4 = 386.4063{{c}}, ~7/4 = 968.5613{{c}}
+: [[error map]]: {{val| +0.132 -0.331 +0.356 -0.001 -0.264 -0.002 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4770{{c}}, ~5/4 = 386.5437{{c}}, ~7/4 = 968.6150{{c}}
+: error map: {{val| 0.000 -0.478 +0.230 -0.211 -0.549 -0.280 }}
 {{Optimal ET sequence|legend=1| 53, 58, 72, 111, 130, 183, 190, 243e, 248, 301, 373, 804, 1177be }}
-[[Badness]] (Smith): 3.33 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.875
+=== Commas 351/350, 2080/2079 ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 351/350, 2080/2079
+{{Mapping|legend=1| 1 0 0 0 6 1 | 0 1 0 0 -6 -3 | 0 0 1 0 3 2 | 0 0 0 1 0 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9538{{c}}, ~3/2 = 701.5852{{c}}, ~5/4 = 387.1356{{c}}, ~7/4 = 969.7495{{c}}
+: [[error map]]: {{val| -0.046 -0.416 +0.730 +0.831 +0.301 -1.447 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.5936{{c}}, ~5/4 = 387.0975{{c}}, ~7/4 = 969.7451{{c}}
+: error map: {{val| 0.000 -0.361 +0.784 +0.919 +0.413 -1.369 }}
+{{Optimal ET sequence|legend=1| 19, 26, 39df, 46, 53, 58, 72, 111, 130, 183, 313 }}
+[[Badness]] (Sintel): 0.677
 == Kalismic (9801/9800) ==
@@ Line 577: / Line 675: @@
 [[Optimal tuning]]:
-* [[CTE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.942{{c}}, ~5/4 = 386.327{{c}}, ~7/4 = 968.846{{c}}
+* [[WE]]: ~99/70 = 600.0018{{c}}, ~3/2 = 701.9397{{c}}, ~5/4 = 386.3193{{c}}, ~7/4 = 968.8374{{c}}
-* [[CWE]]: ~99/70 = 600.000{{c}}, ~3/2 = 701.940{{c}}, ~5/4 = 386.323{{c}}, ~7/4 = 968.840{{c}}
+: [[error map]]: {{val| +0.004 -0.012 +0.013 +0.019 -0.028 }}
+* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9399{{c}}, ~5/4 = 386.3226{{c}}, ~7/4 = 968.8399{{c}}
+: error map: {{val| 0.000 -0.015 +0.009 +0.014 -0.035 }}
-{{Optimal ET sequence|legend=1| 8d, 10, 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
+{{Optimal ET sequence|legend=1| 12, 22, 34d, 46, 58, 72, 118, 130, 152, 224, 270, 342, 612, 836, 1084, 1106, 1236, 1506, 1578, 1848, 2684, 4038, 4190, 4532, 11254, 15786e, 21896e }}
+[[Badness]] (Sintel): 0.0415
 === Commas 1716/1715, 2080/2079 ===
@@ Line 592: / Line 694: @@
 : 3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299
-[[Optimal tuning]] ([[CTE]]): ~99/70 = 600.000{{c}}, ~3/2 = 702.024{{c}}, ~5/4 = 386.294{{c}}, ~7/4 = 969.114{{c}}
+[[Optimal tuning]]:
+* [[WE]]: ~99/70 = 599.9732{{c}}, ~3/2 = 702.0465{{c}}, ~5/4 = 386.4214{{c}}, ~7/4 = 969.1757{{c}}
+: [[error map]]: {{val| -0.054 +0.038 +0.001 +0.243 -0.001 -0.210 }}
+* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 702.0554{{c}}, ~5/4 = 386.3788{{c}}, ~7/4 = 969.1756{{c}}
+: error map: {{val| 0.000 +0.100 +0.065 +0.350 +0.126 -0.121 }}
 [[Minimax tuning]]:
 * [[13-odd-limit|13-]] and [[15-odd-limit]]: ~99/70 = {{monzo| 1/2 0 0 0 0 0 }}, ~3 = {{monzo| 7/10 4/5 0 -2/5 0 1/5 }}, ~5 = {{monzo| 7/10 -1/5 1 -2/5 0 1/5 }}, ~7 = {{monzo| 7/5 -2/5 0 1/5 0 2/5 }}
-: [[Eigenmonzo basis|Eigenmonzo (unchanged-intervals) basis]]: 2, 6/5, 16/13, 9/7
+: [[eigenmonzo basis|eigenmonzo (unchanged-intervals) basis]]: 2.5/3.9/7.13
-{{Optimal ET sequence|legend=1| 10e, 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}
+{{Optimal ET sequence|legend=1| 12f, 14cf, 22f, 26, 36ce, 46, 58, 72, 130, 198, 224, 270, 494, 764, 1258, 1810d, 2304d, 2574d }}
+[[Badness]] (Sintel): 0.256
 == Unisquary (12005/11979) ==
@@ Line 605: / Line 713: @@
 [[Comma list]]: [[12005/11979]]
-{{Mapping|legend=1| 1 0 0 0 0 | 0 1 0 2 2 | 0 0 1 -1 -1 | 0 0 0 3 4 }}
+{{Mapping|legend=1| 1 0 0 -6 -8 | 0 1 0 2 2 | 0 0 1 2 3 | 0 0 0 3 4 }}
-: Mapping generators: ~2, ~3, ~5, ~11/7
+: Mapping generators: ~2, ~3, ~5, ~44/35
 [[Optimal tuning]]s:
-* [[CTE]] and [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.031{{c}}, ~5/4 = 386.233{{c}}, ~11/7 = 783.508{{c}}
+* [[WE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2749{{c}}
+: [[error map]]: {{val| -0.000 +0.076 -0.081 +0.243 -0.475 +0.541 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0306{{c}}, ~5/4 = 386.2325{{c}}, ~44/35 = 397.2750{{c}}
+: error map: {{val| 0.000 +0.076 -0.081 -0.475 +0.541 }}
 {{Optimal ET sequence|legend=1| 12, 21, 23de, 26, 37, 46, 58, 72, 118, 130, 190, 239, 311, 441, 559, 752e, 870 }}
+[[Badness]] (Sintel): 2.43
 === Hensquary ===
@@ Line 618: / Line 731: @@
 [[Comma list]]: 364/363, 1716/1715
-{{Mapping|legend=1| 1 0 0 0 0 -2 | 0 1 0 2 2 3 | 0 0 1 -1 -1 -1 | 0 0 0 3 4 5 }}
+{{Mapping|legend=1| 1 0 0 -6 -8 -12 | 0 1 0 2 2 3 | 0 0 1 2 3 4 | 0 0 0 3 4 5 }}
-: Mapping generators: ~2, ~3, ~5, ~11/7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.072{{c}}, ~5/4 = 386.528{{c}}, ~11/7 = 783.823{{c}}
+* [[WE]]: ~2 = 1199.7826{{c}}, ~3/2 = 702.2831{{c}}, ~5/4 = 386.9208{{c}}, ~44/35 = 396.8581{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.253{{c}}, ~5/4 = 386.776{{c}}, ~11/7 = 783.875{{c}}
+: [[error map]]: {{val| -0.217 +0.111 +0.172 +0.156 +1.443 -1.487 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2535{{c}}, ~5/4 = 386.7762{{c}}, ~44/35 = 397.0993{{c}}
+: error map: {{val| 0.000 +0.298 +0.463 +0.531 +1.915 -1.166 }}
 {{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}
+[[Badness]] (Sintel): 0.988
 === Ekasquary ===
@@ Line 632: / Line 748: @@
 [[Comma list]]: 1575/1573, 4459/4455
-{{Mapping|legend=1| 1 0 0 0 0 0 | 0 1 0 2 2 0 | 0 0 1 -1 -1 3 | 0 0 0 3 4 -5 }}
+{{Mapping|legend=1| 1 0 0 -6 -8 10 | 0 1 0 2 2 0 | 0 0 1 2 3 -2 | 0 0 0 3 4 -5 }}
-: Mapping generators: ~2, ~3, ~5, ~11/7
 [[Optimal tuning]]s:
-* [[CTE]] and [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.952{{c}}, ~5/4 = 386.139{{c}}, ~11/7 = 783.548{{c}}
+* [[WE]]: ~2 = 1199.9999{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4088{{c}}
+: [[error map]]: {{val| -0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9524{{c}}, ~5/4 = 386.1388{{c}}, ~44/35 = 397.4089{{c}}
+: error map: {{val| 0.000 -0.003 -0.175 -0.417 +0.639 +0.150 }}
 {{Optimal ET sequence|legend=1| 46f, 49f, 58, 72, 118, 121, 130, 190, 193, 248, 311, 441, 752e, 1072 }}
+[[Badness]] (Sintel): 1.12
 == Semicanousmic (14641/14580) ==
@@ Line 645: / Line 765: @@
 [[Comma list]]: [[14641/14580]]
-{{Mapping|legend=1| 1 0 2 0 1 | 0 1 2 0 2 | 0 0 -4 0 -1 | 0 0 0 1 0 }}
+{{Mapping|legend=1| 1 0 -2 0 0 | 0 1 2 0 2 | 0 0 4 0 1 | 0 0 0 1 0 }}
-: Mapping generators: ~2, ~3, ~18/11, ~7
+: Mapping generators: ~2, ~3, ~11/9, ~7
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.334{{c}}, ~18/11 = 854.555{{c}}, ~7/4 = 968.826{{c}}
+* [[WE]]: ~2 = 1200.0495{{c}}, ~3/2 = 702.2793{{c}}, ~11/9 = 345.4721{{c}}, ~7/4 = 968.7265{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.292{{c}}, ~18/11 = 854.548{{c}}, ~7/4 = 968.756{{c}}
+: [[error map]]: {{val| +0.049 +0.374 +0.133 -0.000 -1.188 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2918{{c}}, ~11/9 = 345.4516{{c}}, ~7/4 = 968.7556{{c}}
+: error map: {{val| 0.000 +0.337 +0.077 -0.070 -1.283 }}
 {{Optimal ET sequence|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
-[[Badness]] (Smith): 0.351 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 6.40
-=== Tridecimal semicanousmic ===
+== Semiporwellismic (16384/16335) ==
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11
-Comma list: 2080/2079, 14641/14580
+[[Comma list]]: 16384/16335
-Mapping: {{mapping| 1 0 2 0 1 -6 | 0 1 2 0 2 3 | 0 0 -4 0 -1 3 | 0 0 0 1 0 1 }}
+{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}
+: Mapping generators: ~2, ~3, ~128/99, ~7
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.469{{c}}, ~18/11 = 854.644{{c}}, ~7/4 = 968.959{{c}}
+* [[WE]]: ~2 = 1199.7329{{c}}, ~3/2 = 702.3657{{c}}, ~128/99 = 442.1589{{c}}, ~7/4 = 969.3599{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.483{{c}}, ~18/11 = 854.642{{c}}, ~7/4 = 968.989{{c}}
+: [[error map]]: {{val| -0.267 +0.144 +0.103 -0.000 +0.456 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5107{{c}}, ~128/99 = 442.2243{{c}}, ~7/4 = 969.4120{{c}}
+: error map: {{val| 0.000 +0.556 +0.646 +0.586 +1.436 }}
-{{Optimal ET sequence|legend=0| 87, 111, 181, 198, 323, 410 }}
+{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
-Badness (Smith): 17.1 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 4.00
-==== 17-limit ====
+=== Tridecimal semiporwellismic ===
-Subgroup: 2.3.5.7.11.13.17
+[[Subgroup]]: 2.3.5.7.11.13
-Comma list: 715/714, 1089/1088, 14641/14580
+[[Comma list]]: 2080/2079, 3584/3575
-Mapping: {{mapping| 1 0 2 0 1 -6 -4 | 0 1 2 0 2 3 6 | 0 0 -4 0 -1 3 -2 | 0 0 0 1 0 1 0 }}
+{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 1 0 -2 0 | 0 0 2 0 -1 -3 | 0 0 0 1 0 1 }}
-Optimal tunings:
+[[Optimal tuning]]s:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.413{{c}}, ~18/11 = 854.633{{c}}, ~7/4 = 969.032{{c}}
+* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.411{{c}}, ~18/11 = 854.634{{c}}, ~7/4 = 969.026{{c}}
+: [[error map]]: {{val| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3927{{c}}, ~84/65 = 442.4436{{c}}, ~7/4 = 969.1894{{c}}
-{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
+: error map: {{val| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}
-Badness (Smith): 34.0 × 10<sup>-6</sup>
-==== 19-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444
-Mapping: {{mapping| 1 0 2 0 1 -6 -4 -4 | 0 1 2 0 2 3 6 7 | 0 0 -4 0 -1 3 -2 -4 | 0 0 0 1 0 1 0 0 }}
-Optimal tunings:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.368{{c}}, ~18/11 = 854.639{{c}}, ~7/4 = 969.075{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.351{{c}}, ~18/11 = 854.644{{c}}, ~7/4 = 969.010{{c}}
-{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
+{{Optimal ET sequence|legend=1| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}
-Badness (Smith): 41.9 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 1.13
-== Semiporwellismic (16384/16335) ==
-[[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: 16384/16335
-{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 0 -2 | 0 0 2 0 -1 | 0 0 0 1 0 }}
-: Mapping generators: ~2, ~3, ~128/99, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.470{{c}}, ~128/99 = 442.106{{c}}, ~7/4 = 968.826{{c}}
-{{Optimal ET sequence|legend=1| 19, 22, 41, 65d, 68, 84, 87, 111, 130, 152, 239, 282, 328, 369, 521e, 1370bcdeee, 1411bcdeee, 1609bccdeee }}
-[[Badness]] (Smith): 0.219 × 10<sup>-6</sup>
 == Symbiotic (19712/19683) ==
@@ Line 724: / Line 822: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 968.736{{c}}
+* [[WE]]: ~2 = 1199.9290{{c}}, ~3/2 = 702.2265{{c}}, ~5/4 = 386.4556{{c}}, ~7/4 = 968.8979{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.244{{c}}, ~5/4 = 386.406{{c}}, ~7/4 = 968.859{{c}}
+: [[error map]]: {{val| -0.071 +0.201 -0.000 -0.070 -0.106 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2440{{c}}, ~5/4 = 386.4064{{c}}, ~7/4 = 968.8593{{c}}
+: error map: {{val| 0.000 +0.289 +0.093 +0.033 +0.019 }}
 {{Optimal ET sequence|legend=1| 17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b }}
-[[Badness]] (Smith): 0.120 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 2.19
 === Tridecimal symbiotic ===
@@ Line 739: / Line 839: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.228{{c}}, ~5/4 = 386.284{{c}}, ~7/4 = 968.789{{c}}
+* [[WE]]: ~2 = 1199.9268{{c}}, ~3/2 = 702.2293{{c}}, ~5/4 = 386.4554{{c}}, ~7/4 = 968.9114{{c}}
-* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.254{{c}}, ~5/4 = 386.401{{c}}, ~7/4 = 968.898{{c}}
+: [[error map]]: {{val| -0.073 +0.201 -0.005 -0.061 -0.093 -0.012 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2544{{c}}, ~5/4 = 386.4007{{c}}, ~7/4 = 968.8979{{c}}
+: error map: {{val| 0.000 +0.299 +0.087 +0.072 +0.074 +0.125 }}
 {{Optimal ET sequence|legend=1| 17c, 34dff, 36ce, 41, 53, 58, 94, 111, 152f, 212, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bf }}
-[[Badness]] (Smith): 3.31 × 10<sup>-6</sup>
+[[Badness]] (Sintel): 0.868
 === 2.3.5.7.11.13.19 subgroup ===
@@ Line 754: / Line 856: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000{{c}}, ~3/2 = 702.230{{c}}, ~5/4 = 386.304{{c}}, ~7/4 = 968.797{{c}}
+* WE: ~2 = 1199.9273{{c}}, ~3/2 = 702.2289{{c}}, ~5/4 = 386.4508{{c}}, ~7/4 = 968.9092{{c}}
-* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.255{{c}}, ~5/4 = 386.408{{c}}, ~7/4 = 968.902{{c}}
+* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.2554{{c}}, ~5/4 = 386.4080{{c}}, ~7/4 = 968.9015{{c}}
 {{Optimal ET sequence|legend=0| 17ch, 24f, 34dffh, 36ceh, 41, 53, 58h, 94, 111, 152f, 212h, 217, 270, 581, 851, 1003, 1273, 1854, 2124b, 3127bfh }}
-Badness (Smith): 4.37 × 10<sup>-6</sup>
+Badness (Sintel): 0.375
+== Argyric (41503/41472) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[41503/41472]]
+{{Mapping|legend=1| 1 0 0 1 3 | 0 1 0 0 2 | 0 0 1 0 0 | 0 0 0 2 -3 }}
+: Mapping generators: ~2, ~3, ~5, ~144/77
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0485{{c}}, ~3/2 = 701.9606{{c}}, ~5/4 = 386.2167{{c}}, ~144/77 = 1084.3250{{c}}
+: [[error map]]: {{val| +0.049 +0.054 -0.000 -0.127 -0.129 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9739{{c}}, ~5/4 = 386.2460{{c}}, ~144/77 = 1084.2953{{c}}
+: error map: {{val| 0.000 +0.019 -0.068 -0.235 -0.256 }}
+{{Optimal ET sequence|legend=1| 19, 22, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
+[[Badness]] (Sintel): 0.777
+== Chrysic (43923/43904) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[43923/43904]]
+{{Mapping|legend=1| 1 0 0 3 4 | 0 1 0 3 2 | 0 0 1 0 0 | 0 0 0 -4 -3 }}
+: Mapping generators: ~2, ~3, ~5, ~33/14
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0167{{c}}, ~3/2 = 701.9323{{c}}, ~5/4 = 386.2803{{c}}, ~33/28 = 284.2370{{c}}
+: [[error map]]: {{val| +0.017 -0.006 -0.000 +0.056 -0.114 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9341{{c}}, ~5/4 = 386.2927{{c}}, ~33/28 = 284.2352{{c}}
+: error map: {{val| 0.000 -0.021 -0.021 +0.036 -0.155 }}
+{{Optimal ET sequence|legend=1| 17c, 26, 29, 34d, 43, 46, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 2715, 2985, 3327e, 3479, 3821, 4091e, 4433e, 5539e }}
+[[Badness]] (Sintel): 0.770
+== Elysic (117649/117612) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: [[117649/117612]]
+{{Mapping|legend=1| 1 0 0 0 -1 | 0 2 0 0 -5 | 0 0 1 0 0 | 0 0 0 1 3 }}
+: Mapping generators: ~2, ~343/198, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0027{{c}}, ~343/198 = 950.9861{{c}}, ~5/4 = 386.3082{{c}}, ~7/4 = 968.7558{{c}}
+: [[error map]]: {{val| +0.003 +0.017 -0.000 -0.065 +0.033 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9850{{c}}, ~5/4 = 386.3105{{c}}, ~7/4 = 968.7571{{c}}
+: error map: {{val| 0.000 +0.015 -0.003 -0.069 +0.028 }}
+{{Optimal ET sequence|legend=1| 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d }}
+[[Badness]] (Sintel): 0.901
 == Olympic (131072/130977) ==
+Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])<sup>2</sup>, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 769: / Line 927: @@
 : Mapping generators: ~2, ~3, ~5, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.103{{c}}, ~5/4 = 386.314{{c}}, ~7/4 = 969.012{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9460{{c}}, ~3/2 = 702.0489{{c}}, ~5/4 = 386.4217{{c}}, ~7/4 = 968.9839{{c}}
+: [[error map]]: {{val| -0.054 +0.040 -0.000 +0.050 +0.038 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0853{{c}}, ~5/4 = 386.4125{{c}}, ~7/4 = 969.0242{{c}}
+: error map: {{val| 0.000 +0.130 +0.099 +0.198 +0.207 }}
 {{Optimal ET sequence|legend=1| 41, 53, 84, 87, 130, 183, 224, 270, 494, 764, 1164, 1205, 1475, 1969, 2239, 3133de, 4608cde, 5102bcde, 10474bbccdddeee }}
+[[Badness]] (Sintel): 2.05
 === Tridecimal olympic ===
@@ Line 780: / Line 944: @@
 {{Mapping|legend=1| 1 0 0 0 17 12 | 0 1 0 0 -5 -2 | 0 0 1 0 0 -1 | 0 0 0 1 -2 -1 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.103{{c}}, ~5/4 = 386.298{{c}}, ~7/4 = 969.007{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9638{{c}}, ~3/2 = 702.0645{{c}}, ~5/4 = 386.3208{{c}}, ~7/4 = 968.9712{{c}}
+: [[error map]]: {{val| -0.036 +0.073 -0.065 +0.073 +0.128 -0.166 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0884{{c}}, ~5/4 = 386.3269{{c}}, ~7/4 = 969.0015{{c}}
+: error map: {{val| 0.000 +0.133 +0.013 +0.176 +0.237 -0.033 }}
 {{Optimal ET sequence|legend=1| 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e }} *
 <nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]
+[[Badness]] (Sintel): 0.292
+== Alphaxenic (131769/131072) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 131769/131072
+{{Mapping|legend=1| 2 1 0 0 8 | 0 2 0 0 -1 | 0 0 1 0 0 | 0 0 0 1 0 }}
+: mapping generators: ~363/256, ~16/11, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}, ~5/4 = 385.6773{{c}}, ~7/4 = 968.1894{{c}}
+: [[error map]]: {{val| +0.318 -0.094 -0.000 -0.000 -0.897 }}
+* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}, ~5/4 = 385.7900{{c}}, ~7/4 = 968.1927{{c}}
+: error map: {{val| 0.000 -0.491 -0.524 -0.633 -2.050 }}
+{{Optimal ET sequence|legend=1| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}
+[[Badness]] (Sintel): 23.0
 == Seascape (160083/160000) ==
 [[Subgroup]]: 2.3.5.7.11
-[[Comma list]]: [[Seascape comma|160083/160000]]
+[[Comma list]]: [[160083/160000]]
 {{Mapping|legend=1| 1 0 0 0 4 | 0 2 0 0 -3 | 0 0 1 0 2 | 0 0 0 1 -1 }}
 : Mapping generators: ~2, ~400/231, ~5, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~400/231 = 950.960{{c}}, ~5/4 = 386.417{{c}}, ~7/4 = 968.751{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0285{{c}}, ~400/231 = 950.9641{{c}}, ~5/4 = 386.3335{{c}}, ~7/4 = 968.7129{{c}}
+: [[error map]]: {{val| +0.028 -0.027 +0.077 -0.056 -0.085 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~400/231 = 950.9492{{c}}, ~5/4 = 386.3638{{c}}, ~7/4 = 969.7159{{c}}
+: error map: {{val| 0.000 -0.057 +0.050 -0.110 -0.154 }}
+{{Optimal ET sequence|legend=1| 14c, 15, 19, 29, 39d, 43, 53, 58, 72, 130, 183, 198, 212, 255, 270, 342, 612, 954, 1084, 1354, 1696, 2833e, 3575, 3917e, 4187e, 4529de, 5883de }}
-{{Optimal ET sequence|legend=1| 15, 19, 29, 43, 53, 58, 68, 72 }}
+[[Badness]] (Sintel): 0.596
 == Fifthchromismic (1610510/1594323) ==
@@ Line 803: / Line 997: @@
 [[Comma list]]: [[1610510/1594323]]
-{{Mapping|legend=1| 1 0 4 0 -1 | 0 1 3 0 2 | 0 0 -5 0 1 | 0 0 0 1 0 }}
+{{Mapping|legend=1| 1 0 -6 0 1 | 0 1 3 0 2 | 0 0 5 0 -1 | 0 0 0 1 0 }}
-: Mapping generators: ~2, ~3, ~22/9, ~7
+: Mapping generators: ~2, ~3, ~18/11, ~7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.738{{c}}, ~11/9 = 344.406{{c}}, ~7/4 = 968.826{{c}}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9757{{c}}, ~3/2 = 702.7610{{c}}, ~18/11 = 855.5656{{c}}, ~7/4 = 968.8735{{c}}
+: [[error map]]: {{val| -0.024 +0.782 -0.130 -0.001 -1.434 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7565{{c}}, ~18/11 = 855.5885{{c}}, ~7/4 = 968.8591{{c}}
+: error map: {{val| 0.000 +0.801 -0.102 +0.033 -1.393 }}
-{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}
+{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 251, 258e, 338, 432, 519 }}
+[[Badness]] (Sintel): 97.2
 === Fifthchroma ===
-{{see also| Fifth-chroma temperaments }}
+{{See also| Fifth-chroma temperaments }}
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 817: / Line 1,017: @@
 [[Comma list]]: 352/351, 951665/944784
-{{Mapping|legend=1| 1 0 4 0 -1 4 | 0 1 3 0 2 -1 | 0 0 -5 0 1 1 | 0 0 0 1 0 0 }}
+{{Mapping|legend=1| 1 0 -6 0 1 6 | 0 1 3 0 2 -1 | 0 0 5 0 -1 -1 | 0 0 0 1 0 0 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8431{{c}}, ~3/2 = 702.9005{{c}}, ~18/11 = 855.4090{{c}}, ~7/4 = 969.1386{{c}}
+: [[error map]]: {{val| -0.157 +0.789 -0.097 -0.001 -1.396 +0.379 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.9326{{c}}, ~18/11 = 855.5400{{c}}, ~7/4 = 969.1437{{c}}
+: error map: {{val| 0.000 +0.978 +0.184 +0.318 -0.993 +1.000 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.776{{c}}, ~11/9 = 344.410{{c}}, ~7/4 = 968.826{{c}}
+{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 181, 244, 251, 261, 338, 599b }}
-{{Optimal ET sequence|legend=1| 77, 80, 87, 94, 167 }}
+[[Badness]] (Sintel): 7.63
 ==== 17-limit ====
@@ Line 828: / Line 1,034: @@
 Comma list: 352/351, 715/714, 155771/155520
-Mapping: {{mapping| 1 0 4 0 -1 4 6 | 0 1 3 0 2 -1 3 | 0 0 -5 0 1 1 -3 | 0 0 0 1 0 0 -1 }}
+Mapping: {{mapping| 1 0 -6 0 1 6 0 | 0 1 3 0 2 -1 3 | 0 0 5 0 -1 -1 3 | 0 0 0 1 0 0 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.8126{{c}}, ~3/2 = 702.8981{{c}}, ~11/9 = 855.3764{{c}}, ~7/4 = 969.3543{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9376{{c}}, ~11/9 = 855.5306{{c}}, ~7/4 = 969.4313{{c}}
-Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~3/2 = 702.713{{c}}, ~11/9 = 344.403{{c}}, ~7/4 = 969.193{{c}}
+{{Optimal ET sequence|legend=0| 70, 73g, 77, 80, 87, 94, 157, 167, 181, 251, 261, 338, 432, 599bg }}
-{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167 }}
+Badness (Sintel): 3.65
 ==== 19-limit ====
@@ Line 839: / Line 1,049: @@
 Comma list: 352/351, 715/714, 1463/1458, 2057/2052
-Mapping: {{mapping| 1 0 4 0 -1 4 6 2 | 0 1 3 0 2 -1 3 4 | 0 0 -5 0 1 1 -3 -1 | 0 0 0 1 0 0 -1 -1 }}
+Mapping: {{mapping| 1 0 -6 0 1 6 0 0 | 0 1 3 0 2 -1 3 4 | 0 0 5 0 -1 -1 3 1 | 0 0 0 1 0 0 -1 -1 }}
+Optimal tunings:
+* WE: ~2 = 1199.8128{{c}}, ~3/2 = 702.9226{{c}}, ~11/9 = 855.3621{{c}}, ~7/4 = 969.3187{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.9618{{c}}, ~11/9 = 855.5163{{c}}, ~7/4 = 969.3961{{c}}
+{{Optimal ET sequence|legend=0| 70, 73gh, 77, 80, 87, 94, 157, 167h, 181, 251, 261, 338, 599bg }}
+Badness (Sintel): 2.63
+== Quartismic (117440512/117406179) ==
+[[Subgroup]]: 2.3.5.7.11
+[[Comma list]]: 117440512/117406179
+{{Mapping|legend=1| 1 0 0 1 5 | 0 1 0 1 -1 | 0 0 1 0 0 | 0 0 0 5 1 }}
+: mapping generators: ~2, ~3, ~5, ~33/32
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~5/4 = 386.3387{{c}}, ~33/32 = 53.3743{{c}}
+: [[error map]]: {{val| -0.012 +0.008 -0.000 -0.004 +0.031 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~5/4 = 386.3343{{c}}, ~33/32 = 53.3730{{c}}
+: error map: {{val| 0.000 +0.025 +0.021 +0.019 +0.075 }}
+{{Optimal ET sequence|legend=1| 21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee }}
+[[Badness]] (Sintel): 5.00
+=== Tridecimal quartismic ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 6656/6655, 123201/123200
+{{Mapping|legend=1| 1 0 0 1 5 6 | 0 1 0 1 -1 -3 | 0 0 1 0 0 1 | 0 0 0 5 1 3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9884{{c}}, ~3/2 = 701.9674{{c}}, ~5/4 = 386.3432{{c}}, ~33/32 = 53.3763{{c}}
+: [[error map]]: {{val| -0.012 +0.001 +0.006 -0.000 +0.045 -0.016 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9728{{c}}, ~5/4 = 386.3385{{c}}, ~33/32 = 53.3748{{c}}
+: error map: {{val| 0.000 +0.018 +0.025 +0.021 +0.084 +0.017 }}
-Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~3/2 = 702.736{{c}}, ~11/9 = 344.417{{c}}, ~7/4 = 969.159{{c}}
+{{Optimal ET sequence|legend=1| 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee }}
-{{Optimal ET sequence|legend=0| 77, 80, 87, 94, 167h }}
+[[Badness]] (Sintel): 0.456
 [[Category:Temperaments]]

Catalog of rank-4 temperaments: Difference between revisions

Latest revision as of 05:05, 5 June 2026

Cake (45/44)

Mothwellsmic (99/98)

Ptolemismic (100/99)

Biyatismic (121/120)

Valinorsmic (176/175)

Tridecimal valinorsmic

Rastmic (243/242)

Frostmic (245/242)

Akua (352/351, 847/845)

Keenanismic (385/384)

Martwin

Ancient

Commas 351/350, 385/384

Zaxa

Commas 364/363, 385/384

Commas 385/384, 847/845

Werckismic (441/440)

Commas 364/363, 441/440

Commas 351/350, 441/440

Commas 196/195, 352/351

Tannic

17-limit

Commas 441/440, 847/845

Swetismic (540/539)

Commas 540/539, 729/728

Commas 540/539, 625/624

Commas 540/539, 676/675

Commas 540/539, 847/845

Pentacircle (896/891)

Tridecimal pentacircle (gentle)

Topsy (847/845, 1001/1000)

Lehmerismic (3025/3024)

Commas 2080/2079, 3025/3024

Trimitone (8019/8000)

Commas 729/728, 1001/1000

Commas 351/350, 2080/2079

Kalismic (9801/9800)

Commas 1716/1715, 2080/2079

Unisquary (12005/11979)

Hensquary

Ekasquary

Semicanousmic (14641/14580)

Semiporwellismic (16384/16335)

Tridecimal semiporwellismic

Symbiotic (19712/19683)

Tridecimal symbiotic

2.3.5.7.11.13.19 subgroup

Argyric (41503/41472)

Chrysic (43923/43904)

Elysic (117649/117612)

Olympic (131072/130977)

Tridecimal olympic

Alphaxenic (131769/131072)

Seascape (160083/160000)

Fifthchromismic (1610510/1594323)

Fifthchroma

17-limit

19-limit

Quartismic (117440512/117406179)

Tridecimal quartismic

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