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No-threes subgroup temperaments: Difference between revisions

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Ostara can also refer to a collection of temperaments which temper out 16807/16796.{{clarify}}
Ostara can also refer to a collection of temperaments which temper out 16807/16796.{{clarify}}


[[Subgroup]]: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200
Comma list: 8589934592/8544921875, 30691800524/30517578125


{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}
: mapping generators: ~2, ~5120/3773


[[Optimal tuning]]s:  
Optimal tunings:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~5120/3773 = 529.003{{c}}
* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5120/3773 = 529.004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}


[[Badness]] (Sintel): 11.731
Badness (Sintel): 11.7


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}
* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}
* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}


{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}


Badness (Sintel): 3.415
Badness (Sintel): 3.42


==== 2.5.7.11.13.17 subgroup ====
==== 2.5.7.11.13.17 subgroup ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}
* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}
* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}


{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


Badness (Sintel): 1.985
Badness (Sintel): 1.99


==== 2.5.7.11.13.17.19 subgroup ====
==== 2.5.7.11.13.17.19 subgroup ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~19/14 = 529.006{{c}}
* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}
* CWE: ~2 = 1200.000{{c}}, ~19/14 = 529.005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}


{{Optimal ET sequence|legend=0| 93, 338, 431 }}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}


Badness (Sintel): 1.285
Badness (Sintel): 1.29


=== Tricesimoprimal miracloid ===
=== Tricesimoprimal miracloid ===
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Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688
Comma list: 10241/10240, 5858783/5856400, 4093705/4090624, 15109493/15089800, 102942875/102834688


Subgroup-val mapping: {{mapping| 1 419 48 177 157 624 625 | 0 -461 -50 -192 -169 -685 -686 }}
Subgroup-val mapping: {{mapping| 1 -42 -2 -15 -12 -61 -61 | 0 461 50 192 169 685 686 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~58/31 = 1084.628{{c}}
Optimal tunings:
* WE: ~2 = 1200.0079{{c}}, ~31/29 = 115.3723{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/29 = 115.3716{{c}}


{{Optimal ET sequence|legend=1| 52, 1737, 1789, … }}
{{Optimal ET sequence|legend=0| 52, 1737, 1789 }}
 
Badness (Sintel): 4.51


=== Huntington ===
=== Huntington ===
{{See also| Chromatic pairs #Huntington }}
{{See also| Chromatic pairs #Huntington }}


Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.  
Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  


[[Subgroup]]: 2.5.7.13
Subgroup: 2.5.7.13


[[Comma list]]: [[640/637]], [[10985/10976]]
Comma list: [[640/637]], [[10985/10976]]


{{Mapping|legend=2| 1 5 4 4 | 0 -9 -4 -1 }}
Subgroup-val mapping: {{mapping| 1 -4 0 3 | 0 9 4 1 }}


{{Mapping|legend=3| 1 0 5 4 0 4 | 0 0 -9 -4 0 -1 }}
Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}
: mapping generators: ~2, ~16/13
: mapping generators: ~2, ~13/8


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/13 = 357.002{{c}}
Optimal tunings:
* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd }}
{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3452 cents
Badness (Sintel): 0.319


==== Silver ====
==== Silver ====
{{See also| Chromatic pairs #Silver }}
{{See also| Chromatic pairs #Silver }}


Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.  
Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.  


[[Subgroup]]: 2.5.7.13.17
Subgroup: 2.5.7.13.17


[[Comma list]]: [[170/169]], [[640/637]], [[5525/5488]]
Comma list: [[170/169]], [[640/637]], [[5525/5488]]


{{Mapping|legend=2| 1 5 4 4 2 | 0 -9 -4 -1 7 }}
Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}


{{Mapping|legend=3| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}
Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/8 = 842.711{{c}}
Optimal tunings:
* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}


{{Optimal ET sequence|legend=1| 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee }}
{{Optimal ET sequence|legend=1| 10, 27, 37, 47, 84, 131, 178g }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
Badness (Sintel): 0.504


=== Pakkanen ===
=== Pakkanen ===
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{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
{{Mapping|legend=2| 1 0 0 -3 | 0 1 0 4 | 0 0 1 -1 }}
: mapping generators: ~2, ~5, ~11
: mapping generators: ~2, ~5, ~7


[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.6544{{c}}, ~5/4 = 380.3004{{c}}, ~11/8 = 551.9653{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6499{{c}}, ~5/4 = 380.2990{{c}}, ~7/4 = 969.2326{{c}}
: [[error map]]: {{val| +0.650 -4.715 +1.706 +2.595 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 380.6249{{c}}, ~7/4 = 969.6520{{c}}
: error map: {{val| 0.000 -5.689 +0.826 +1.530 }}


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
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[[Comma list]]: 5021863/5000000
[[Comma list]]: 5021863/5000000


{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
{{Mapping|legend=2| 1 0 -2 3 | 0 1 1 1 | 0 0 4 -3 }}
: mapping generators: ~2, ~5, ~100/77
: mapping generators: ~2, ~5, ~77/50


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.080{{c}}, ~5 = 2786.820{{c}}, ~100/77 = 454.618{{c}}
* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5 = 2786.740{{c}}, ~100/77 = 454.590{{c}}
: [[error map]]: {{val| +0.080 +0.507 -1.318 -0.643 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.7404{{c}}, ~77/50 = 745.4102{{c}}
: error map: {{val| 0.000 +0.427 -0.445 -0.808 }}


{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}


[[Badness]] (Sintel): 1.862
[[Badness]] (Sintel): 1.86


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
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Comma list: 847/845, 1001/1000
Comma list: 847/845, 1001/1000


Subgroup-val mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 | 0 1 1 1 1 | 0 0 4 -3 -1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.034{{c}}, ~5 = 2786.678{{c}}, ~13/10 = 454.569{{c}}
* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5 = 2786.646{{c}}, ~13/10 = 454.557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}


{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}


Badness (Sintel): 0.179
Badness (Sintel): 0.179
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Comma list: 170/169, 221/220, 847/845
Comma list: 170/169, 221/220, 847/845


Subgroup-val mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
Subgroup-val mapping: {{mapping| 1 0 -2 3 2 3 | 0 1 1 1 1 1 | 0 0 4 -3 -1 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.407{{c}}, ~5 = 2787.484{{c}}, ~13/10 = 455.036{{c}}
* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}
* CWE: ~2 = 1200.000{{c}}, ~5 = 2787.107{{c}}, ~13/10 = 454.906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}


{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}


Badness (Sintel): 0.438
Badness (Sintel): 0.438
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