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17th-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|17}}
{{Technical data page}}
17edo is a "wheel" for some fractional-octave temperaments. The most notable relationship is the tempering out of the [[septendecima]], the amount by which seventeen [[25/24]] chromatic semitones exceed an octave.
{{Infobox fractional-octave|17}}
[[17edo]] is a "wheel" for some [[fractional-octave temperaments]]. The most notable relationship is the tempering out of the [[septendecima]], the amount by which seventeen [[25/24]] chromatic semitones exceed an octave.


== Gothic ==
== Gothic ==
The gothic temperament is associated with the [[17-comma]].
The gothic temperament is associated with the [[17-comma]]. It used to be known as '''septendecic'''.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 14: Line 15:


[[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==
=== 7-limit ===
Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
Gothic is identical to 17et in the no-5 subgroups, but has an independent generator for prime 5.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 64/63, 17496/16807
 
{{Mapping|legend=1| 17 27 0 48 | 0 0 1 0 }}
 
[[Optimal tuning]]s:
* [[CTE]]: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* [[CWE]]: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 391.765{{c}}
 
{{Optimal ET sequence|legend=1| 17c, 34d }}
 
[[Badness]] (Sintel): 3.44


Subgroup: 2.3.5.7.11.13
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 64/63, 99/98, 243/242
 
Mapping: {{Mapping| 17 27 0 48 59 | 0 0 1 0 0 }}


Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
Optimal tunings:  
* CTE: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* CWE: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 392.472{{c}}


Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
{{Optimal ET sequence|legend=0| 17c, 34d }}


Mapping generators: ~25/24, ~1024/975
Badness (Sintel): 1.77


Optimal tuning (CTE): ~1024/975 = 85.421
=== 13-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11.13.17
Comma list: 64/63, 78/77, 99/98, 144/143


Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
Mapping: {{Mapping| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}


Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
Optimal tunings:  
* CTE: ~27/26 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* CWE: ~27/26 = 70.588{{c}} (1\17), ~5/4 = 392.129{{c}}


Mapping generators: ~25/24, ~765/728
{{Optimal ET sequence|legend=0| 17c, 34d }}


Optimal tuning (CTE): ~765/728 = 85.421
Badness (Sintel): 1.35


== Chlorine ==
== Chlorine ==
The name of chlorine temperament comes from Chlorine, the 17th element.
The name of chlorine temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].


Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  
Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  
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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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{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}


{{Multival|legend=1| 34 17 170 -52 174 347 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.998{{c}}
[[Optimal tuning]] ([[POTE]]): ~{{monzo| 24 -5 -9 2 }} = 950.9995
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.999{{c}}


{{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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Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192


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 tunings:
* 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}}
 
{{Optimal ET sequence|legend=0| 289, 323, 612 }}
 
Badness (Sintel): 2.11
 
== Leaves ==
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
 
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}}]
 
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 ET sequence|legend=0| 323, 1700d, 2023, 2346e}}
 
Badness (Sintel): 10.096


Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
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}}]


{{Optimal ET sequence|legend=1| 289, 323, 612 }}
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}}


Badness: 0.063706
{{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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