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Catalog of rank-4 temperaments: Difference between revisions

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[[Optimal tuning]]s:  
[[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]]:  
[[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 }}
{{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 ===
=== Commas 364/363, 441/440 ===
Line 329: Line 333:
: 3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657
: 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]]:  
[[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 }}
* [[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 }}
* [[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 }}
{{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 ===
=== 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 }}
{{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 }}
{{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 ===
=== 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 }}
{{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 }}
{{Optimal ET sequence|legend=1| 10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232 }}
[[Badness]] (Sintel): 0.548


=== Tannic ===
=== 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 }}
{{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 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 }}
{{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 ====
==== 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: {{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 }}


{{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 }}
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 tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 700.606{{c}}, ~5/4 = 387.655{{c}}, ~7/4 = 967.876{{c}}
{{Optimal ET sequence|legend=0| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}


{{Optimal ET sequence|legend=1| 17cg, 26, 29g, 31, 43, 46, 60e, 72, 103, 149, 221ef }}
Badness (Sintel): 0.701


=== Commas 441/440, 847/845 ===
=== Commas 441/440, 847/845 ===
Line 391: Line 425:
: Mapping generators: ~2, ~3, ~5, ~13/11
: 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 }}
{{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) ==
== Swetismic (540/539) ==
Line 404: Line 444:


[[Optimal tuning]]s:  
[[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 }}
{{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 ===
=== Commas 540/539, 729/728 ===
Line 420: Line 462:


[[Optimal tuning]]s:  
[[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 }}
{{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 ===
=== Commas 540/539, 847/845 ===
Line 435: Line 479:
: Mapping generators: ~2, ~3, ~5, ~13/11
: 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 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 }}
{{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>
[[Badness]] (Sintel): 1.04


=== Commas 540/539, 625/624 ===
=== Commas 540/539, 625/624 ===
Line 448: Line 496:
{{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 }}
{{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 }}
{{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 ===
=== Commas 540/539, 676/675 ===
Line 462: Line 514:
: Mapping generators: ~2, ~26/15, ~5, ~7
: 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 }}
{{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


== Pentacircle (896/891) ==
== Pentacircle (896/891) ==
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