Xenharmonic Wiki

Catalog of rank-4 temperaments: Difference between revisions

← Older edit
VisualWikitext
+ vishdelismic (5632/5625)
 
(5 intermediate revisions by 2 users not shown)
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) ==
[[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 729: Line 765:


[[Badness]] (Sintel): 0.164
[[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 1,014: Line 1,033:


Badness (Sintel): 0.505
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 ====
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.  
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
Subgroup: 2.3.5.7.11.13.17
Line 1,068: Line 1,123:


Badness (Sintel): 1.27
Badness (Sintel): 1.27
==== Commas 2080/2079, 3250/3249, 4096/4095 ====
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}}, ~187/100 = 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}}, ~187/100 = 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}}, ~187/100 = 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}}, ~187/100 = 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


== Alphaxenic (131769/131072) ==
== Alphaxenic (131769/131072) ==
Retrieved from "https://en.xen.wiki/w/Catalog_of_rank-4_temperaments"