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Catalog of rank-4 temperaments: Difference between revisions

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→Olympic (131072/130977): correct a ratio. Move part of the description of metaolympic to the no-17 extension
 
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=== Tridecimal olympic ===
=== Tridecimal olympic ===
==== 13-limit ====
[[Subgroup]]: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13


Line 997: Line 998:
[[Badness]] (Sintel): 0.292
[[Badness]] (Sintel): 0.292


==== Subgroup: 2.3.5.7.11.13.29 ====
===== 2.3.5.7.11.13.29 subgroup =====
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[5104/5103]]
Subgroup: 2.3.5.7.11.13.29


{{Mapping|legend=1| 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}}
Comma list: 2080/2079, 3510/3509, 4096/4095


[[Optimal tuning]]s:  
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 }}
* [[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}}
Optimal tunings:  
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.070{{c}}, ~5/4 = 386.337{{c}}, ~7/4 = 969.013{{c}}
* WE: ~2 = 1199.960{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.247{{c}}, ~7/4 = 968.895{{c}}
: error map: {{val|0.000, 0.115, 0.023, 0.187, 0.304, -0.018, 0.237}}
: 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


{{Optimal ET sequence|legend=1|TBA}}
Comma list: 2080/2079, 3250/3249, 4096/4095


[[Badness]] (Sintel): 0.505
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 }}


=== Metaolympic ===
Optimal tunings:
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.  
* 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 }}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=0| 41, 53, 84, 94, 137, 176, 217, 270, 581, 851, 935, 1205, 2193, 2463, 2774e }}


[[Comma list]]: [[2080/2079]], [[4096/4095]], [[35000/34969]]
Badness (Sintel): 0.618


[[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 }}
===== 2.3.5.7.11.13.19.29 subgroup =====
: mapping generators: ~2, ~3, ~5, ~187/100
Subgroup: 2.3.5.7.11.13.19.29


Optimal tunings:  
Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095
* [[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 }}
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 }}


[[Badness]] (Sintel): 1.59
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 }}


==== 19-limit ====
{{Optimal ET sequence|legend=0| 41, 53j, 94, 137j, 176, 217, 270, 487, 581j, 718, 894, 935, 988, 1205, 2193, 3398dehj}}
[[Subgroup]]: 2.3.5.7.11.13.17.19


[[Comma list]]: [[2080/2079]], [[3250/3249]], [[4096/4095]], [[4200/4199]]
Badness (Sintel): 1.10


[[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 }}
==== 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.


Optimal tunings:  
Subgroup: 2.3.5.7.11.13.17
* 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 }}
Comma list: 2080/2079, 4096/4095, 35000/34969


[[Badness]] (Sintel): 0.764
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


==== 2.3.5.7.11.13.17.19.29 subgroup ====
Optimal tunings:
[[Comma list]]: [[2080/2079]], [[3250/3249]], [[4096/4095]], [[4200/4199]], [[5104/5103]]
* 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 }}


[[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 ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2815eg }}


Optimal tunings:
Badness (Sintel): 1.59
* [[WE]]: ~2 = 1199.951{{c}}, ~3/2 = 701.968{{c}}, ~5/4 = 386.215{{c}}, ~187/100 = 1084.473 {{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 = 702.048{{c}}, ~5/4 = 386.316{{c}}, ~187/100 = 1084.517{{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|TBA}}
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


[[Badness]] (Sintel): 1.290
Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199


==== 2.3.5.7.11.13.19 subgroup ====
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 }}
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[5776/5775]]


[[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 tunings:
{{Optimal ET sequence|legend=0| 43, 53, 84, 94, 137, 176g, 217, 270, 311, 487, 571e, 581, 935, 1516e, 2234eg, 2451eg }}
* [[WE]]: ~2 = 1199.964{{c}}, ~3/2 = 702.028{{c}}, ~5/4 = 386.250{{c}}, ~187/100 = 1084.468{{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 = 702.089{{c}}, ~5/4 = 386.326{{c}}, ~187/100 = 1084.501{{c}}  [[error map]]: {{val|0.000, 0.134, 0.012, 0.175, 0.236, -0.032, 0.135}}


{{Optimal ET sequence|legend=0|TBA}}
Badness (Sintel): 0.764


[[Badness]] (Sintel): 0.618
===== 2.3.5.7.11.13.17.19.29 subgroup =====
Subgroup: 2.3.5.7.11.13.17.19.29


==== 2.3.5.7.11.13.19.29 subgroup ====
Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199
[[Comma list]]: [[2080/2079]], [[4096/4095]], [[5104/5103]], [[5776/5775]]


[[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: {{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:
Optimal tunings:
* [[WE]]: ~2 = 1199.959{{c}}, ~3/2 = 702.008{{c}}, ~5/4 = 386.245{{c}}, ~187/100 = 1084.468{{c}} [[error map]]: {{val|-0.041, 0.053, -0.069, 0.069, 0.163, -0.170, 0.005, 0.048}}
* WE: ~2 = 1199.9561{{c}}, ~3/2 = 702.0552{{c}}, ~5/4 = 386.2577{{c}}, ~187/100 = 1084.4387{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.071{{c}}, ~5/4 = 386.333{{c}}, ~187/100 = 1084.506{{c}} [[error map]]: {{val|-0.000, 0.116, 0.019, 0.187, 0.302, -0.015, 0.171, 0.241}}
: 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|TBA}}
{{Optimal ET sequence|legend=0| 53j, 94, 137j, 176g, 217, 270, 311, 487, 581j, 718, 935, 1205g, 1516e, 2234eg, 2721egg }}


[[Badness]] (Sintel): 1.098
Badness (Sintel): 1.27


== Alphaxenic (131769/131072) ==
== Alphaxenic (131769/131072) ==
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