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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 790: Line 806:
* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
: [[error map]]: {{val| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
: [[error map]]: {{val| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.4546{{c}}, ~84/65 = 442.3626{{c}}, ~7/4 = 968.9590{{c}}
* [[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 +0.500 +0.866 +0.133 +1.410 +1.344 }}
: error map: {{val| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}


{{Optimal ET sequence|legend=1| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}
{{Optimal ET sequence|legend=1| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}
Line 846: Line 862:


Badness (Sintel): 0.375
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 883: Line 955:


[[Badness]] (Sintel): 0.292
[[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) ==
== Seascape (160083/160000) ==
Line 968: Line 1,058:


Badness (Sintel): 2.63
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 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 }}
[[Badness]] (Sintel): 0.456


[[Category:Temperaments]]
[[Category:Temperaments]]
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