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17th-octave temperaments: Difference between revisions

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=== 7-limit ===
=== 7-limit ===
Gothic is identical to 17et in the no-5 subgroups, but has an independent generator for prime 5.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}
Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}


[[Optimal tuning]]s:  
Optimal tunings:  
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}
* CTE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}


{{Optimal ET sequence|legend=0| 289, 323, 612 }}
{{Optimal ET sequence|legend=0| 289, 323, 612 }}
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== Leaves ==
== Leaves ==
Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
Defined as the {{nowrap|323 & 2023}} temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 1729951171875/1727094849536, {{monzo|10 39 -14 -14}}
 
[[Mapping]]: [{{val|17 10 31 9}}, {{val|0 14 7 32}}]
 
: mapping generators: ~25/24, ~6125/5832
 
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.427{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.426{{c}}
 
{{Optimal ET sequence|legend=1| 323, 1700d, 2023, 2346}}
 
[[Badness]] (Sintel): 32.032
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 160083/160000, 198359290368/198165034375, 1729951171875/1727094849536
 
Mapping: [{{val|17 10 31 9 106}}, {{val|0 14 7 32 -39}}]
 
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.420{{c}}
 
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}
 
Badness (Sintel): 20.456


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
Comma list: 160083/160000, 928125/927472, 1990656/1990625, 4831530/4826809


Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]


Mapping generators: ~25/24, ~1024/975
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.420{{c}}


Optimal tuning (CTE): ~1024/975 = 85.421
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}


{{Optimal ET sequence|legend=1| 323, 1700, 2023}}
Badness (Sintel): 10.096


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
Comma list: 39325/39304, 57375/57344, 71874/71825, 111537/111475, 140800/140777


Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]


Mapping generators: ~25/24, ~765/728
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}


Optimal tuning (CTE): ~765/728 = 85.421
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}


{{Optimal ET sequence|legend=1| 323, 1700, 2023}}
Badness (Sintel): 6.678


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}


[[Category:17edo]]
[[Category:17edo]]
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