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Catalog of rank-4 temperaments: Difference between revisions

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== Keenanismic (385/384) ==
== Keenanismic (385/384) ==
{{Main| Keenanismic family }}
{{Main| Keenanismic family }}
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: [[385/384]]
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 0 0 1 | 0 0 1 0 -1 | 0 0 0 1 -1 }}
: Mapping generators: ~2, ~3, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4104{{c}}, ~3/2 = 701.6916{{c}}, ~5/4 = 385.1761{{c}}, ~7/4 = 967.5421{{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]]:
* [[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
{{Optimal ET sequence|legend=1| 9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de }}
[[Badness]] (Sintel): 0.277
=== Martwin ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 325/324, 385/384
{{Mapping|legend=1| 1 0 0 0 7 2 | 0 1 0 0 1 4 | 0 0 1 0 -1 -2 | 0 0 0 1 -1 0 }}
Lattice basis:
: 4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247
[[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]]:
* [[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
{{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]] (Sintel): 0.579
=== Ancient ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 385/384, 625/624
{{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]]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 }}
[[Badness]] (Sintel): 0.675


=== Commas 351/350, 385/384 ===
=== Commas 351/350, 385/384 ===
Line 246: Line 184:


[[Badness]] (Sintel): 0.781
[[Badness]] (Sintel): 0.781
=== Zaxa ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 352/351, 385/384
{{Mapping|legend=1| 1 0 0 0 7 12 | 0 1 0 0 1 -2 | 0 0 1 0 -1 -1 | 0 0 0 1 -1 -1 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1117{{c}}, ~3/2 = 702.6588{{c}}, ~5/4 = 385.6821{{c}}, ~7/4 = 968.0061{{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 }}
[[Badness]] (Sintel): 0.880


=== Commas 364/363, 385/384 ===
=== Commas 364/363, 385/384 ===
Line 407: Line 328:
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=0| 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:  
Optimal tunings:  
Line 633: Line 554:


== Vishdelismic (5632/5625) ==
== Vishdelismic (5632/5625) ==
Vishdelismic tempers out 5632/5625, the [[vishdelisma]], and adds to [[vishdelic]] an independent generator for prime 7. From 5632/5625 = ([[176/175]])/([[225/224]]), vishdelismic has the feature of making [[14/9]], [[25/16]], and [[11/7]] equidistant.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 911: Line 834:
Comma list: 1216/1215, 1540/1539, 2080/2079
Comma list: 1216/1215, 1540/1539, 2080/2079


Mapping: {{mapping| 1 0 0 0 -8 -13 -6 | 0 1 0 0 9 12 5 | 0 0 1 0 0 -1 1 | 0 0 0 1 -1 0 0 }}
{{Mapping|legend=0| 1 0 0 0 -8 -13 -6 | 0 1 0 0 9 12 5 | 0 0 1 0 0 -1 1 | 0 0 0 1 -1 0 0 }}


Optimal tunings:  
Optimal tunings:  
Line 1,022: Line 945:
Comma list: 2080/2079, 3510/3509, 4096/4095
Comma list: 2080/2079, 3510/3509, 4096/4095


Mapping: {{mapping| 1 0 0 0 17 12 -21 | 0 1 0 0 -5 -2 11 | 0 0 1 0 0 -1 0 | 0 0 0 1 -2 -1 3 }}
{{Mapping|legend=0| 1 0 0 0 17 12 -21 | 0 1 0 0 -5 -2 11 | 0 0 1 0 0 -1 0 | 0 0 0 1 -2 -1 3 }}


Optimal tunings:  
Optimal tunings:  
Line 1,041: Line 964:
Comma list: 2080/2079, 3250/3249, 4096/4095
Comma list: 2080/2079, 3250/3249, 4096/4095


Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}
{{Mapping|legend=0| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}


Optimal tunings:
Optimal tunings:
Line 1,058: Line 981:
Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095
Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095


Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}
{{Mapping|legend=0| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}


Optimal tunings:
Optimal tunings:
Line 1,077: Line 1,000:
Comma list: 2080/2079, 4096/4095, 35000/34969
Comma list: 2080/2079, 4096/4095, 35000/34969


Mapping: {{mapping| 1 0 0 1 15 11 -13 | 0 1 0 0 -5 -2 5 | 0 0 1 0 0 -1 2 | 0 0 0 2 -4 -2 5 }}
{{Mapping|legend=0| 1 0 0 1 15 11 -13 | 0 1 0 0 -5 -2 5 | 0 0 1 0 0 -1 2 | 0 0 0 2 -4 -2 5 }}
: mapping generators: ~2, ~3, ~5, ~187/100
: mapping generators: ~2, ~3, ~5, ~187/100


Line 1,095: Line 1,018:
Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199
Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199


Mapping: {{mapping| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}
{{Mapping|legend=0| 1 0 0 1 15 11 -13 6 | 0 1 0 0 -5 -2 5 -2 | 0 0 1 0 0 -1 2 1 | 0 0 0 2 -4 -2 5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 1,112: Line 1,035:
Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199
Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199


Mapping: {{mapping| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}
{{Mapping|legend=0| 1 0 0 1 15 11 -13 6 -18 | 0 1 0 0 -5 -2 5 -2 -11 | 0 0 1 0 0 -1 2 1 0 | 0 0 0 2 -4 -2 5 -1 6 }}


Optimal tunings:
Optimal tunings:
Line 1,202: Line 1,125:
Comma list: 352/351, 715/714, 155771/155520
Comma list: 352/351, 715/714, 155771/155520


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 }}
{{Mapping|legend=0| 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:  
Optimal tunings:  
Line 1,217: Line 1,140:
Comma list: 352/351, 715/714, 1463/1458, 2057/2052
Comma list: 352/351, 715/714, 1463/1458, 2057/2052


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 }}
{{Mapping|legend=0| 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:  
Optimal tunings:  
Line 1,236: Line 1,159:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~121/96 = 400.0154{{c}}, ~3/2 = 950.9883{{c}}, ~5/4 = 386.2216{{c}}, ~7/4 = 968.7338{{c}}
* [[WE]]: ~121/96 = 400.0154{{c}}, ~1331/768 = 950.9883{{c}}, ~5/4 = 386.2216{{c}}, ~7/4 = 968.7338{{c}}
: [[error map]]: {{val| +0.046 +0.022 -0.000 -0.000 -0.207 }}
: [[error map]]: {{val| +0.046 +0.022 -0.000 -0.000 -0.207 }}
* [[CWE]]: ~121/96 = 400.0000{{c}}, ~3/2 = 950.9685{{c}}, ~5/4 = 386.2485{{c}}, ~7/4 = 968.7470{{c}}
* [[CWE]]: ~121/96 = 400.0000{{c}}, ~1331/768 = 950.9685{{c}}, ~5/4 = 386.2485{{c}}, ~7/4 = 968.7470{{c}}
: error map: {{val| 0.000 -0.018 -0.065 -0.079 -0.349 }}
: error map: {{val| 0.000 -0.018 -0.065 -0.079 -0.349 }}


Line 1,250: Line 1,173:
Comma list: 676/675, 6656/6655
Comma list: 676/675, 6656/6655


Mapping: {{mapping| 3 0 0 0 8 -3 | 0 2 0 0 1 3 | 0 0 1 0 0 1 | 0 0 0 1 0 0 }}
{{Mapping|legend=0| 3 0 0 0 8 -3 | 0 2 0 0 1 3 | 0 0 1 0 0 1 | 0 0 0 1 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~121/96 = 400.0003{{c}}, ~3/2 = 951.0845{{c}}, ~5/4 = 386.5832{{c}}, ~7/4 = 968.8238{{c}}
* WE: ~121/96 = 400.0003{{c}}, ~1331/768 = 951.0845{{c}}, ~5/4 = 386.5832{{c}}, ~7/4 = 968.8238{{c}}
* CWE: ~121/96 = 400.0000{{c}}, ~3/2 = 951.0840{{c}}, ~5/4 = 386.5837{{c}}, ~7/4 = 968.8241{{c}}
* CWE: ~121/96 = 400.0000{{c}}, ~1331/768 = 951.0840{{c}}, ~5/4 = 386.5837{{c}}, ~7/4 = 968.8241{{c}}


{{Optimal ET sequence|legend=0| 9, 15, 24, 39c/df, 48c, 57f, 63, 72, 87, 111, 159, 183, 198, 270 }}
{{Optimal ET sequence|legend=0| 9, 15, 24, 39c/df, 48c, 57f, 63, 72, 87, 111, 159, 183, 198, 270 }}


Badness (Sintel): 1.69
Badness (Sintel): 1.69
== Microorganismic (67110351/67108864) ==
It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]]. It has three valid 13-limit extensions, each of them basically just as accurate, but only koch is a strong extension.
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: [[67110351/67108864]]
[[Mapping]]: {{Mapping|1 0 0 7 7|0 1 0 1 -2|0 0 1 0 0|0 0 0 3 -5}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 = 1200.001014{{C}}, ~3 = 1901.953889{{C}}, ~5 =  2786.311685{{C}}, ~539/512 = 88.956151{{C}}
: [[error map]]: ⟨+0.001014 −0.000098 0.000000 −0.001537 −0.001401]
* [[CWE]]: ~2 = 1200.000000{{C}}, ~3 =  1901.953627{{C}}, ~5 =  2786.311919{{C}}, ~539/512 = 88.955923{{C}}
: error map: ⟨0.000000 −0.001374 −0.001795 −0.004512 −0.004808]
[[Optimal ET sequence]]: {{Edos|11c, 12e, 15, 26, 38, 41, 53, 68, 94, 161, 176, 202, 217, 255, 270, 566, 742, 836, 1106, 1578, 1848, 2684, 3737, 4007, 4843, 5585, 6421, 6691, 12276, 14960…}}
[[Badness]] (Sintel): 0.387
Notable temperaments not appearing in the optimal ET sequence are {{Edos|8269, 8539, 16808}}.
=== Leeuwenhoek ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: [[67110351/67108864]], [[123201/123200]]
[[Mapping]]: {{Mapping|1 0 0 1 7 7|0 2 0 2 −4 −7|0 0 1 0 0 1|0 0 0 3 −5 −1}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 = 1200.001182{{C}}, ~207515/119808 = 950.977093{{C}}, ~5 = 2786.311871{{C}}, ~539/512 = 88.956523{{C}}
: [[error map]]: {{Val|+0.001182 −0.000815 +0.000522 −0.000971 −0.000653 −0.001325}}
* [[CWE]]: ~2 = 1200.000000{{C}}, ~207515/119808 =  950.976202{{C}}, ~5 =  2786.312215{{C}}, ~539/512 = 88.956363{{C}}
: error map: {{Val|0.000000 −0.002596 −0.001499 −0.004414 −0.004565 −0.005226}}
[[Optimal ET sequence]]: {{Edos|15f, 38, 53, 120c, 135f, 188cf, 188, 202f, 217, 270, 742, 1012, 1619, 1889, 2414, 2631, 2684, 4573, 4843, 5585, 8269, 8539, 13112, 16808…}}
[[Badness]] (Sintel): 0.349
=== Pasteur ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: [[67110351/67108864]], [[5767168/5767125]]
[[Mapping]]: {{Mapping|1 0 0 7 −3 3|0 1 0 4 −7 −4|0 0 1 0 0 −1|0 0 0 9 −15 −8}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 = 1200.000392{{C}}, ~3 = 1901.954644{{C}}, ~5 = 2786.314673{{C}}, ~512/455 = 204.332922{{C}}
: [[error map]]: {{Val|+0.000392 +0.000035 +0.001743 −0.002847 −0.004657 +0.004426}}
* [[CWE]]: ~2 = 1200.000000{{C}}, ~207515/119808 =  950.976202{{C}}, ~5 =  2786.312215{{C}}, ~539/512 = 88.956363{{C}}
: error map: {{Val|0.000000 −0.000511 +0.000884 −0.004026 −0.005906 +0.002962}}
[[Optimal ET sequence]]: {{Edos|12e, 37b, 41, 53, 94, 176, 217, 270, 581, 851, 1469, 1563, 1739, 1780, 1833, 2009, 2050, 2103, 2320, 2414, 2631, 2684, 5274, 5315, 5585, 8269, 8539, 14124, 16808…}}
[[Badness]] (Sintel): 0.924
=== Koch ===
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: [[67110351/67108864]], [[1990656/1990625]]
[[Mapping]]: {{Mapping|1 0 0 7 7 11|0 1 0 1 -2 3|0 0 1 0 0 -5|0 0 0 3 -5 -6}}
[[Optimal tuning|Optimal tunings]]:
* [[WE]]: ~2 = 1200.001014{{C}}, ~3 = 1901.953889{{C}}, ~5 =  2786.311685{{C}}, ~539/512 = 88.956151{{C}}
: [[error map]]: {{Val|+0.000612 −0.002463 +0.004690 −0.001614 −0.003969 +0.002382}}
* [[CWE]]: ~2 = 1200.000000{{C}}, ~3 =  1901.953627{{C}}, ~5 =  2786.311919{{C}}, ~539/512 = 88.955923{{C}}
: error map: {{Val|0.000000 −0.003186 +0.003468 −0.003438 −0.005995 +0.000566}}
[[Optimal ET sequence]]: {{Edos|11c, 15, 38f, 53, 121e, 149e, 202, 217, 270, 783f, 836, 1038, 1106, 1308, 1578, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808…}}
[[Badness]] (Sintel): 0.912


== Quartismic (117440512/117406179) ==
== Quartismic (117440512/117406179) ==
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