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Catalog of rank-4 temperaments: Difference between revisions

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Line 407: Line 407:
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 578: Line 578:


[[Badness]] (Sintel): 1.04
[[Badness]] (Sintel): 1.04
== Moctdelismic (1375/1372) ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 1375/1372
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 0 | 0 0 1 0 -3 | 0 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0558{{c}}, ~3/2 = 701.8991{{c}}, ~5/4 = 385.7504{{c}}, ~7/4 = 969.3743{{c}}
: [[error map]]: {{val| +0.056 -0.000 -0.452 +0.660 -0.334 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9100{{c}}, ~5/4 = 385.7894{{c}}, ~7/4 = 969.4163{{c}}
: error map: {{val| 0.000 -0.045 -0.524 +0.590 -0.437 }}
{{Optimal ET sequence|legend=1| 12, 19e, 27, 29, 31, 41, 60e, 68, 72, 140, 152, 224, 364, 436, 547, 588, 619, 843, 1055ce }}
[[Badness]] (Sintel): 0.723


== Lehmerismic (3025/3024) ==
== Lehmerismic (3025/3024) ==
Line 613: Line 631:


[[Badness]] (Sintel): 0.272
[[Badness]] (Sintel): 0.272
== 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
[[Comma list]]: 5632/5625
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 0 0 0 | 0 0 1 0 -3 | 0 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8976{{c}}, ~3/2 = 702.1145{{c}}, ~5/4 = 386.7638{{c}}, ~7/4 = 969.0306{{c}}
: [[error map]]: {{val| -0.102 +0.057 +0.245 -0.000 -0.136 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1309{{c}}, ~5/4 = 386.7770{{c}}, ~7/4 = 969.0087{{c}}
: error map: {{val| 0.000 +0.176 +0.463 +0.183 +0.052 }}
{{Optimal ET sequence|legend=1| 12, 19e, 22, 31, 53, 84e, 87, 99e, 118, 130, 152, 239, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}
[[Badness]] (Sintel): 0.804


== Trimitone (8019/8000) ==
== Trimitone (8019/8000) ==
Line 875: Line 913:
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 986: Line 1,024:
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 1,005: Line 1,043:
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 1,022: Line 1,060:
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 1,041: Line 1,079:
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 1,059: Line 1,097:
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 1,076: Line 1,114:
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 1,166: Line 1,204:
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 1,181: Line 1,219:
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 1,190: Line 1,228:


Badness (Sintel): 2.63
Badness (Sintel): 2.63
== Nexus (1771561/1769472) ==
[[Subgroup]]: 2.3.5.7.11
[[Comma list]]: 1771561/1769472
{{Mapping|legend=1| 3 0 0 0 8 | 0 2 0 0 1 | 0 0 1 0 0 | 0 0 0 1 0 }}
: mapping generators: ~121/96, ~1331/768, ~5, ~7
[[Optimal tuning]]s:
* [[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 }}
* [[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 }}
{{Optimal ET sequence|legend=1| 9, 15, 24, 33cd, 39(d), 48, 57, 63, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}
[[Badness]] (Sintel): 10.7
=== Commas 676/675, 6656/6655 ===
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 6656/6655
{{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:
* 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}}, ~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 }}
Badness (Sintel): 1.69


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