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Catalog of rank-4 temperaments: Difference between revisions

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[[Badness]] (Sintel): 0.454
[[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]]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


=== Commas 540/539, 625/624 ===
=== Commas 540/539, 625/624 ===
Line 523: Line 505:


[[Badness]] (Sintel): 0.802
[[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) ==
== Pentacircle (896/891) ==
Line 638: 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|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:  
[[Optimal tuning]]s:  
Line 649: Line 648:


[[Badness]] (Sintel): 0.875
[[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) ==
== Kalismic (9801/9800) ==
Line 697: Line 713:
[[Comma list]]: [[12005/11979]]
[[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:  
[[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 }}
{{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 ===
=== Hensquary ===
Line 710: Line 731:
[[Comma list]]: 364/363, 1716/1715
[[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:  
[[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 }}
{{Optimal ET sequence|legend=1| 26, 37, 46, 58, 72, 121, 130, 193, 239, 251e, 323e, 369 }}
[[Badness]] (Sintel): 0.988


=== Ekasquary ===
=== Ekasquary ===
Line 724: Line 748:
[[Comma list]]: 1575/1573, 4459/4455
[[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:  
[[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 }}
{{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) ==
== Semicanousmic (14641/14580) ==
Line 737: Line 765:
[[Comma list]]: [[14641/14580]]
[[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:  
[[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 }}
{{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}}
: error map: {{val| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}


{{Optimal ET sequence|legend=0| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
{{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): 34.0 × 10<sup>-6</sup>
[[Badness]] (Sintel): 1.13
 
==== 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 }}
 
Badness (Smith): 41.9 × 10<sup>-6</sup>
 
== 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) ==
== Symbiotic (19712/19683) ==
Line 816: Line 822:


[[Optimal tuning]]s:  
[[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 }}
{{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 ===
=== Tridecimal symbiotic ===
Line 831: Line 839:


[[Optimal tuning]]s:  
[[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 }}
{{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 ===
=== 2.3.5.7.11.13.19 subgroup ===
Line 846: Line 856:


Optimal tunings:  
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 }}
{{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 (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
[[Subgroup]]: 2.3.5.7.11


Line 861: Line 927:
: Mapping generators: ~2, ~3, ~5, ~7
: 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 }}
{{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 ===
=== Tridecimal olympic ===
Line 872: 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 }}
{{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 }} *
{{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]]
<nowiki>*</nowiki> [[optimal patent val]]: [[3044edo|3044]]
[[Badness]] (Sintel): 0.292
=== Metaolympic ===
Metaolympic is a [[weak extension]] of olympic, requiring the split of the prime 7 in half; or alternatively, the septimal comma or the garischisma. It is better without prime 17, but prime 17 can be added easily into the temperament by also tempering out 4200/4199, which also takes advantage of the split prime 7, at the cost of higher complexity.
Subgroup: 2.3.5.7.11.13.17
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 generators: ~2, ~3, ~5, ~187/100
Optimal tunings:
* WE: ~2 = 1199.9530{{c}}, ~3/2 = 702.0493{{c}}, ~5/4 = 386.2541{{c}}, ~187/100 = 1084.4393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0798{{c}}, ~5/4 = 386.2658{{c}}, ~187/100 = 1084.4811{{c}}
{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2815eg }}
Badness (Sintel): 1.59
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
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 }}
Optimal tunings:
* WE: ~2 = 1199.9539{{c}}, ~3/2 = 702.0464{{c}}, ~5/4 = 386.2644{{c}}, ~187/100 = 1084.4405{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0768{{c}}, ~5/4 = 386.2655{{c}}, ~187/100 = 1084.4813{{c}}
{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2451eg }}
Badness (Sintel): 0.764
== 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) ==
== Seascape (160083/160000) ==
[[Subgroup]]: 2.3.5.7.11
[[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|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
: 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) ==
== Fifthchromismic (1610510/1594323) ==
Line 895: Line 1,030:
[[Comma list]]: [[1610510/1594323]]
[[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 ===
=== Fifthchroma ===
{{see also| Fifth-chroma temperaments }}
{{See also| Fifth-chroma temperaments }}


[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13
Line 909: Line 1,050:
[[Comma list]]: 352/351, 951665/944784
[[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]] ([[CTE]]): ~2 = 1200.000{{c}}, ~3/2 = 702.776{{c}}, ~11/9 = 344.410{{c}}, ~7/4 = 968.826{{c}}
[[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 ET sequence|legend=1| 77, 80, 87, 94, 167 }}
{{Optimal ET sequence|legend=1| 70, 73, 77, 80, 87, 157, 167, 171e, 181, 244, 251, 261, 338, 599b }}
 
[[Badness]] (Sintel): 7.63


==== 17-limit ====
==== 17-limit ====
Line 920: Line 1,067:
Comma list: 352/351, 715/714, 155771/155520
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 tuning (CTE): ~2 = 1200.000{{c}}, ~3/2 = 702.713{{c}}, ~11/9 = 344.403{{c}}, ~7/4 = 969.193{{c}}
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 ET sequence|legend=0| 77, 80, 87, 94, 167 }}
{{Optimal ET sequence|legend=0| 70, 73g, 77, 80, 87, 94, 157, 167, 181, 251, 261, 338, 432, 599bg }}
 
Badness (Sintel): 3.65


==== 19-limit ====
==== 19-limit ====
Line 931: Line 1,082:
Comma list: 352/351, 715/714, 1463/1458, 2057/2052
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]]
[[Category:Temperaments]]
Retrieved from "https://en.xen.wiki/w/Catalog_of_rank-4_temperaments"