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Catalog of rank-4 temperaments: Difference between revisions

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Move quartismic here
→Olympic (131072/130977): correct a ratio. Move part of the description of metaolympic to the no-17 extension
 
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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 668: Line 684:
[[Badness]] (Sintel): 0.0415
[[Badness]] (Sintel): 0.0415


=== Commas 1716/1715, 2080/2079 ===
=== Briggs ===
Briggs tempers out [[1716/1715]] and [[2080/2079]], and is the join of [[pessoal]] and [[greenland]], among many others.
 
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


Line 691: Line 709:


[[Badness]] (Sintel): 0.256
[[Badness]] (Sintel): 0.256
=== Kalimera ===
Kalimera is a [[weak extension]] of kalismic by adding the [[harmonisma]], and also an extremely high accuracy detemperament of [[donar]] achieved by observing the [[ragisma]].
[[Subgroup]]: 2.3.5.7.11.13
[[Comma list]]: 9801/9800, 10648/10647
{{Mapping|legend=1| 2 0 1 0 4 9 | 0 1 0 0 -2 -4 | 0 0 2 0 2 3 | 0 0 0 1 1 1 }}
: mapping generators: ~99/70, ~3, ~1053/560, ~7
[[Optimal tuning]]:
* [[WE]]: ~99/70 = 600.0006{{c}}, ~3/2 = 701.9470{{c}}, ~1053/560 = 1093.1617{{c}}, ~7/4 = 968.8431{{c}}
: [[error map]]: {{val| +0.001 -0.007 +0.010 +0.020 -0.043 +0.015 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9469{{c}}, ~1053/560 = 1093.1614{{c}}, ~7/4 = 968.8438{{c}}
: error map: {{val| 0.000 -0.008 +0.009 +0.018 -0.045 +0.013 }}
{{Optimal ET sequence|legend=1| 12f, 22, 34d, 46, 58, 80, 104c, 132de, 144, 166, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 5144, 6380, 7828, 9334, 12018 }}
[[Badness]] (Sintel): 0.164
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 9801/9800, 10648/10647, 194481/194480
Mapping: {{mapping| 2 0 1 0 4 9 -22 | 0 1 0 0 -2 -4 10 | 0 0 2 0 2 3 -7 | 0 0 0 1 1 1 2 }}
Optimal tuning:
* WE: ~99/70 = 600.0011{{c}}, ~3/2 = 701.9430{{c}}, ~1053/560 = 1093.1625{{c}}, ~7/4 = 968.8308{{c}}
: error map: {{val| +0.002 -0.010 +0.012 +0.009 -0.044 +0.024 +0.006 }}
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.9426{{c}}, ~1053/560 = 1093.1618{{c}}, ~7/4 = 968.8316{{c}}
: error map: {{val| 0.000 -0.012 +0.010 +0.006 -0.048 +0.019 +0.001 }}
{{Optimal ET sequence|legend=0| 46, 58, 104c, 166, 178, 212, 224, 270, 494g, 552g, 684g, 896ef, 908, 954, 1178, 1236, 1506, 2414, 2460, 3638, 3920, 4874, 5144, 6380, 10018e, 11524, 16398e }}
Badness (Sintel): 0.655


== Unisquary (12005/11979) ==
== Unisquary (12005/11979) ==
Line 744: Line 799:
[[Badness]] (Sintel): 1.12
[[Badness]] (Sintel): 1.12


== Semicanousmic (14641/14580) ==
== Semicathartismic (14641/14580) ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 882: Line 937:


[[Badness]] (Sintel): 0.770
[[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.
It also has a [[strong extension]] to include [[29/1|prime 29]] though with higher complexity.
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 902: Line 979:


=== Tridecimal olympic ===
=== Tridecimal olympic ===
==== 13-limit ====
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


[[Comma list]]: 2080/2079, 4096/4095
[[Comma list]]: [[2080/2079]], [[4096/4095]]


{{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 }}
Line 919: Line 997:


[[Badness]] (Sintel): 0.292
[[Badness]] (Sintel): 0.292
===== 2.3.5.7.11.13.29 subgroup =====
Subgroup: 2.3.5.7.11.13.29
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 }}
Optimal tunings:
* WE: ~2 = 1199.960{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.247{{c}}, ~7/4 = 968.895{{c}}
: error map: {{val| -0.040 +0.053 -0.067 +0.069 +0.164 -0.171 +0.048 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.070{{c}}, ~5/4 = 386.337{{c}}, ~7/4 = 969.013{{c}}
: error map: {{val| 0.000 +0.115 +0.023 +0.187 +0.304 -0.018 +0.237 }}
{{Optimal ET sequence|legend=0| 41, 53j, 87, 130j, 176, 183, 217, 224, 270, 311, 487, 494, 1205, 1699 }}
Badness (Sintel): 0.505
==== Commas 2080/2079, 3250/3249, 4096/4095 ====
This is a [[weak extension]] of olympic, requiring the split of the prime 7 in half; or alternatively, the septimal comma or the garischisma.
Subgroup: 2.3.5.7.11.13.19
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 }}
Optimal tunings:
* WE: ~2 = 1199.9639{{c}}, ~3/2 = 702.0643{{c}}, ~5/4 = 386.3218{{c}}, ~247/132 = 1084.4676{{c}}
: error map: {{val| -0.036 +0.073 -0.064 +0.073 +0.129 -0.166 -0.004 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0887{{c}}, ~5/4 = 386.3257{{c}}, ~247/132 = 1084.5006{{c}}
: error map: {{val| 0.000 +0.134 +0.012 +0.175 +0.236 -0.032 +0.135 }}
{{Optimal ET sequence|legend=0| 41, 53, 84, 94, 137, 176, 217, 270, 581, 851, 935, 1205, 2193, 2463, 2774e }}
Badness (Sintel): 0.618
===== 2.3.5.7.11.13.19.29 subgroup =====
Subgroup: 2.3.5.7.11.13.19.29
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 }}
Optimal tunings:
* WE: ~2 = 1199.9594{{c}}, ~3/2 = 702.0487{{c}}, ~5/4 = 386.3262{{c}}, ~247/132 = 1084.4677{{c}}
: error map: {{val| -0.041 +0.053 -0.069 +0.069 +0.163 -0.170 +0.005 +0.048 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0709{{c}}, ~5/4 = 386.3327{{c}}, ~247/132 = 1084.5064{{c}}
: error map: {{val| -0.000 +0.116 +0.019 +0.187 +0.302 -0.015 +0.171 +0.241 }}
{{Optimal ET sequence|legend=0| 41, 53j, 94, 137j, 176, 217, 270, 487, 581j, 718, 894, 935, 988, 1205, 2193, 3398dehj}}
Badness (Sintel): 1.10
==== Metaolympic ====
Metaolympic is the above extension with prime 17 added. 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}}
: error map: {{val| -0.047 +0.047 -0.154 +0.006 +0.208 -0.088 +0.184 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0798{{c}}, ~5/4 = 386.2658{{c}}, ~187/100 = 1084.4811{{c}}
: error map: {{val| 0.000 +0.125 -0.057 +0.136 +0.359 +0.094 +0.363 }}
{{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}}
: error map: {{val| -0.046 +0.045 -0.141 +0.009 +0.227 -0.088 0.192 -0.058 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0768{{c}}, ~5/4 = 386.2655{{c}}, ~187/100 = 1084.4813{{c}}
: error map: {{val| 0.000 +0.122 -0.048 +0.137 +0.373 +0.090 +0.367 +0.118 }}
{{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
===== 2.3.5.7.11.13.17.19.29 subgroup =====
Subgroup: 2.3.5.7.11.13.17.19.29
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 }}
Optimal tunings:
* WE: ~2 = 1199.9561{{c}}, ~3/2 = 702.0552{{c}}, ~5/4 = 386.2577{{c}}, ~187/100 = 1084.4387{{c}}
: error map: {{val| -0.044 +0.056 -0.144 +0.008 +0.212 -0.080 +0.205 -0.068 -0.031 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0795{{c}}, ~5/4 = 386.2631{{c}}, ~187/100 = 1084.4800{{c}}
: error map: {{val| 0.000 +0.124 -0.051 +0.134 +0.365 +0.090 +0.368 +0.111 +0.177 }}
{{Optimal ET sequence|legend=0| 53j, 94, 137j, 176g, 217, 270, 311, 487, 581j, 718, 935, 1205g, 1516e, 2234eg, 2721egg }}
Badness (Sintel): 1.27
== 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 1,040: Line 1,243:
[[Badness]] (Sintel): 0.456
[[Badness]] (Sintel): 0.456


[[Category:Temperaments]]
[[Category:Lists of temperaments]]
[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Regular temperament theory]]
[[Category:Catalogs of rank-4 temperaments]]
[[Category:Rank 4| ]] <!-- main article -->
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