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Catalog of rank-4 temperaments: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (1/)
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.873{{c}}, ~5/4 = 386.213{{c}}, ~11/8 = 550.306{{c}}
* [[WE]]: ~2 = 1199.6918{{c}}, ~3/2 = 702.7270{{c}}, ~5/4 = 386.9729{{c}}, ~11/8 = 551.3122{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.896{{c}}, ~5/4 = 386.784{{c}}, ~11/8 = 551.068{{c}}
: [[error map]]: {{val| -0.308 +0.464 +0.043 -0.064 -0.930 +1.062 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8960{{c}}, ~5/4 = 386.7835{{c}}, ~11/8 = 551.0679{{c}}
: error map: {{val| 0.000 +0.941 +0.470 +0.581 -0.025 +1.852 }}


{{Optimal ET sequence|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
{{Optimal ET sequence|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}


[[Badness]] (Smith): 2.550 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.669


== Keenanismic (385/384) ==
== Keenanismic (385/384) ==
Line 174: Line 176:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.363{{c}}, ~5/4 = 385.439{{c}}, ~7/4 = 967.547{{c}}
* [[WE]]: ~2 = 1200.4104{{c}}, ~3/2 = 701.6916{{c}}, ~5/4 = 385.1761{{c}}, ~7/4 = 967.5421{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.693{{c}}, ~5/4 = 385.149{{c}}, ~7/4 = 967.300{{c}}
: [[error map]]: {{val| +0.410 +0.147 -0.317 -0.463 -0.703 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6933{{c}}, ~5/4 = 385.1491{{c}}, ~7/4 = 967.3004{{c}}
: error map: {{val| 0.000 -0.262 -1.165 -1.525 -2.074 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 0 1 0 0 0 }}, ~5 = {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, ~7 = {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}
* [[11-odd-limit]]: ~2 = {{monzo| 1 0 0 0 0 }}, ~3 = {{monzo| 0 1 0 0 0 }}, ~5 = {{monzo| 7/3 1/3 2/3 -1/3 -1/3 }}, ~7 = {{monzo| 7/3 1/3 -1/3 2/3 -1/3 }}
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.7/5.11/5


{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}


[[Badness]] (Smith): 15.2 × 10<sup>-9</sup>
[[Badness]] (Sintel): 0.277


=== Martwin ===
=== Martwin ===
Line 195: Line 199:
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.629{{c}}, ~5/4 = 385.219{{c}}, ~7/4 = 967.741{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3494{{c}}, ~3/2 = 702.2131{{c}}, ~5/4 = 384.8666{{c}}, ~7/4 = 967.8487{{c}}
: [[error map]]: {{val| +0.349 +0.608 -0.748 -0.278 -0.422 -0.710 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1659{{c}}, ~5/4 = 384.8721{{c}}, ~7/4 = 967.6109{{c}}
: error map: {{val| 0.000 +0.211 -1.442 -1.215 -1.635 -1.608 }}


[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Keenanismic family -->
[[Minimax tuning]]: {{confirm}} <!-- contradiction with the data in Keenanismic family -->
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~2 = {{val| 1 0 0 0 0 0 }}, ~3 = {{val| 0 1 0 0 0 0 }}, ~5 = {{val| 2/3 4/3 1/3 0 0 -1/3 }}, ~7 = {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}
* [[13-odd-limit|13-]] and [[15-odd-limit]]: ~2 = {{val| 1 0 0 0 0 0 }}, ~3 = {{val| 0 1 0 0 0 0 }}, ~5 = {{val| 2/3 4/3 1/3 0 0 -1/3 }}, ~7 = {{val| 19/6 -1/6 -1/6 1/2 -1/2 1/6 }}
: [[Eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.3.11/7.13/5


{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}
{{Optimal ET sequence|legend=1| 12e, 15, 19, 22f, 26, 31f, 41, 46, 53, 72, 87, 125f, 140, 159, 212, 299, 371df, 465cef, 677cdeeff, 764cdeeff }}


[[Badness]] (Smith): 2.21 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.579


=== Ancient ===
=== Ancient ===
Line 212: Line 220:
{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}
{{Mapping|legend=1| 1 0 0 0 7 -4 | 0 1 0 0 1 -1 | 0 0 1 0 -1 4 | 0 0 0 1 -1 0 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.358{{c}}, ~5/4 = 385.674{{c}}, ~7/4 = 967.452{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4130{{c}}, ~3/2 = 701.6863{{c}}, ~5/4 = 385.2319{{c}}, ~7/4 = 967.5188{{c}}
: [[error map]]: {{val| +0.413 +0.144 -0.256 -0.481 -0.730 -0.047 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6838{{c}}, ~5/4 = 385.2497{{c}}, ~7/4 = 967.2556{{c}}
: error map: {{val| 0.000 -0.271 -1.064 -1.570 -2.139 -1.213 }}


{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}
{{Optimal ET sequence|legend=1| 15, 19, 22, 31, 50, 53, 72, 87, 103, 140, 159, 190, 243e, 315ef, 330e, 402def }}


[[Badness]] (Smith): 2.57 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.675


=== Commas 351/350, 385/384 ===
=== Commas 351/350, 385/384 ===
Line 225: Line 237:
{{Mapping|legend=1| 1 0 0 0 7 1 | 0 1 0 0 1 -3 | 0 0 1 0 -1 2 | 0 0 0 1 -1 1 }}
{{Mapping|legend=1| 1 0 0 0 7 1 | 0 1 0 0 1 -3 | 0 0 1 0 -1 2 | 0 0 0 1 -1 1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 701.205{{c}}, ~5/4 = 386.715{{c}}, ~7/4 = 967.646{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6793{{c}}, ~3/2 = 700.7455{{c}}, ~5/4 = 385.5609{{c}}, ~7/4 = 967.5816{{c}}
: [[error map]]: {{val| +0.679 -0.530 +0.606 +0.114 -0.998 -1.343 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.5141{{c}}, ~5/4 = 385.6072{{c}}, ~7/4 = 967.1521{{c}}
: error map: {{val| 0.000 -1.441 -0.706 -1.674 -3.563 -3.703 }}


{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}
{{Optimal ET sequence|legend=1| 19, 22, 26, 31, 46, 53, 72, 103, 149, 221ef, 324bdef, 473bdeeff, 495bdeefff, 545bddeefff, 598bcdeeefff }}


[[Badness]] (Smith): 2.98 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.781


=== Zaxa ===
=== Zaxa ===
Line 239: Line 255:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.657{{c}}, ~5/4 = 385.632{{c}}, ~7/4 = 967.829{{c}}
* [[WE]]: ~2 = 1200.1117{{c}}, ~3/2 = 702.6588{{c}}, ~5/4 = 385.6821{{c}}, ~7/4 = 968.0061{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.603{{c}}, ~5/4 = 385.644{{c}}, ~7/4 = 967.903{{c}}
: [[error map]]: {{val| +0.112 +0.815 -0.408 -0.596 -1.901 +1.137 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 702.6029{{c}}, ~5/4 = 385.6441{{c}}, ~7/4 = 967.9030{{c}}
: error map: {{val| 0.000 +0.648 -0.670 -0.923 -2.262 +0.720 }}


{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}
{{Optimal ET sequence|legend=1| 22, 31, 41, 46, 53, 77, 87, 118, 140, 258e, 461e }}


[[Badness]] (Smith): 3.35 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.880


=== Commas 364/363, 385/384 ===
=== Commas 364/363, 385/384 ===
Line 253: Line 271:
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 3 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 -3 }}
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 3 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 -3 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.761{{c}}, ~5/4 = 385.159{{c}}, ~7/4 = 966.298{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4593{{c}}, ~3/2 = 701.9714{{c}}, ~5/4 = 384.8903{{c}}, ~7/4 = 966.4090{{c}}
: [[error map]]: {{val| +0.459 +0.476 -0.505 -1.498 +1.191 -1.325 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 701.9809{{c}}, ~5/4 = 384.8523{{c}}, ~7/4 = 966.1074{{c}}
: error map: {{val| 0.000 +0.026 -1.461 -2.719 -0.297 -2.612 }}


{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}
{{Optimal ET sequence|legend=1| 9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159 }}


[[Badness]] (Smith): 3.32 × 10<sup>-6</sup>
[[Badness]] (Sintel): 0.871


=== Commas 385/384, 847/845 ===
=== Commas 385/384, 847/845 ===
Line 267: Line 289:
: Mapping generators: ~2, ~3, ~5, ~13/11
: Mapping generators: ~2, ~3, ~5, ~13/11


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.368{{c}}, ~5/4 = 385.416{{c}}, ~13/11 = 291.066{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3530{{c}}, ~3/2 = 701.7283{{c}}, ~5/4 = 385.4633{{c}}, ~13/11 = 291.0352{{c}}
: [[error map]]: {{val| +0.353 +0.126 -0.144 -0.586 -1.175 +0.651 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7205{{c}}, ~5/4 = 385.3656{{c}}, ~13/11 = 290.9769{{c}}
: error map: {{val| 0.000 -0.235 -0.948 -1.507 -2.282 -0.515 }}


{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}
{{Optimal ET sequence|legend=1| 34, 37, 41, 46, 53, 87, 103, 140, 190, 243e, 330e, 520de, 573dee }}


[[Badness]] (Smith): 4.15 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.09


== Werckismic (441/440) ==
== Werckismic (441/440) ==
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