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17th-octave temperaments: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~256/243 = 1\17, ~5/4 = 386.3137 (~20480/19683 = 33.3725)
* [[CTE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 386.3137{{c}} (~20480/19683 = 33.3725{{c}})
* [[CWE]]: ~256/243 = 1\17, ~5/4 = 388.2316 (~20480/19683 = 35.2904)
* [[CWE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 388.2316{{c}} (~20480/19683 = 35.2904{{c}})


{{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}
{{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}


[[Badness]]: 0.541
[[Badness]] (Sintel): 12.7
 
== Leaves ==
Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
 
Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
 
Mapping generators: ~25/24, ~1024/975
 
Optimal tuning (CTE): ~1024/975 = 85.421
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
 
Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
 
Mapping generators: ~25/24, ~765/728
 
Optimal tuning (CTE): ~765/728 = 85.421
 
{{Optimal ET sequence|legend=1|323, 1700, 2023}}


== Chlorine ==
== Chlorine ==
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: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}


[[Optimal tuning]] ([[POTE]]): ~{{monzo| 26 9 -17 }} = 950.9746
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 26 9 -17 }} = 950.982{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 26 9 -17 }} = 950.978{{c}}


{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}


[[Badness]]: 0.077072
[[Badness]] (Sintel): 1.81


=== 7-limit ===
=== 7-limit ===
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{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}


[[Badness]]: 0.041658
[[Badness]] (Sintel): 1.05


=== 11-limit ===
=== 11-limit ===
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Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749
Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749


{{Optimal ET sequence|legend=1| 289, 323, 612 }}
{{Optimal ET sequence|legend=0| 289, 323, 612 }}
 
Badness (Sintel): 2.11
 
== Leaves ==
Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
 
Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
 
Mapping generators: ~25/24, ~1024/975
 
Optimal tuning (CTE): ~1024/975 = 85.421
 
{{Optimal ET sequence|legend=1| 323, 1700, 2023}}
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
 
Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
 
Mapping generators: ~25/24, ~765/728
 
Optimal tuning (CTE): ~765/728 = 85.421


Badness: 0.063706
{{Optimal ET sequence|legend=1| 323, 1700, 2023}}


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


[[Category:17edo]]
[[Category:17edo]]
Retrieved from "https://en.xen.wiki/w/17th-octave_temperaments"