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17th-octave temperaments: Difference between revisions

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== Gothic ==
== Gothic ==
The gothic temperament is associated with the [[17-comma]]. It used to be known as '''septendecic'''.  
The gothic temperament is associated with the [[17-comma]]. It used to be known as ''septendecic''.
 
It can be extended to the higher limits using the identical mapping to 17et in the no-5 subgroups. While this comes at the cost of greater error for primes [[7/1|7]] and [[11/1|11]], it is the only [[strong extension]] for these primes that makes practical sense.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 11: Line 13:


{{mapping|legend=1| 17 27 0 | 0 0 1 }}
{{mapping|legend=1| 17 27 0 | 0 0 1 }}
: mapping generators: ~256/243, ~5
: mapping generators: ~256/243, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~256/243 = 1\17, ~5/4 = 386.3137 (~20480/19683 = 33.3725)
* [[WE]]: ~256/243 = 70.5152{{c}}, ~5/4 = 388.7908{{c}}
* [[CWE]]: ~256/243 = 1\17, ~5/4 = 388.2316 (~20480/19683 = 35.2904)
: [[error map]]: {{val| -1.241 +1.956 -0.006 }}
* [[CWE]]: ~256/243 = 70.5882{{c}}, ~5/4 = 388.2315{{c}}
: error map: {{val| 0.000 +3.927 +1.918 }}


{{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]].
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5.7.11.13
[[Comma list]]: 64/63, 17496/16807


Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
{{Mapping|legend=1| 17 27 0 48 | 0 0 1 0 }}


Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
[[Optimal tuning]]s:  
* [[WE]]: ~28/27 = 70.4030{{c}}, ~5/4 = 392.5654{{c}}
: [[error map]]: {{val| -3.149 -1.075 -0.047 +10.517 }}
* [[CWE]]: ~28/27 = 70.5882{{c}}, ~5/4 = 391.7654{{c}}
: error map: {{val| 0.000 +3.927 +5.452 +19.409 }}


Mapping generators: ~25/24, ~1024/975
{{Optimal ET sequence|legend=1| 17c, 34d, 85cdd }}


Optimal tuning (CTE): ~1024/975 = 85.421
[[Badness]] (Sintel): 3.44


=== 17-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11


Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
Comma list: 64/63, 99/98, 243/242


Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
{{Mapping|legend=0| 17 27 0 48 59 | 0 0 1 0 0 }}


Mapping generators: ~25/24, ~765/728
Optimal tunings:
* WE: ~28/27 = 70.3922{{c}}, ~5/4 = 392.9298{{c}}
* CWE: ~28/27 = 70.5882{{c}}, ~5/4 = 392.4722{{c}}


Optimal tuning (CTE): ~765/728 = 85.421
{{Optimal ET sequence|legend=0| 17c, 34d, 85cdde }}


{{Optimal ET sequence|legend=1|323, 1700, 2023}}
Badness (Sintel): 1.77


== Chlorine ==
=== 13-limit ===
The name of chlorine temperament comes from Chlorine, the 17th element.
Subgroup: 2.3.5.7.11.13
 
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.  


[[Subgroup]]: 2.3.5
Comma list: 64/63, 78/77, 99/98, 144/143


[[Comma list]]: {{monzo| -52 -17 34 }}
{{Mapping|legend=0| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}


{{Mapping|legend=1| 17 0 26 | 0 2 1 }}
Optimal tunings:
* WE: ~27/26 = 70.4103{{c}}, ~5/4 = 392.3109{{c}}
* CWE: ~27/26 = 70.5882{{c}}, ~5/4 = 392.1288{{c}}


: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
{{Optimal ET sequence|legend=0| 17c, 34d }}


[[Optimal tuning]] ([[POTE]]): ~{{monzo| 26 9 -17 }} = 950.9746
Badness (Sintel): 1.35


{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
== Chlorine ==
: ''For extensions, see [[Ragismic microtemperaments #Chlorine]].''


[[Badness]]: 0.077072
Chlorine has a period of 1/17 octave. It tempers out {{monzo| -52 -17 34 }}, the [[septendecima]], by which 17 classical chromatic semitones ((25/24)<sup>17</sup>) exceed an octave.  


=== 7-limit ===
The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
[[Subgroup]]: 2.3.5


{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
[[Comma list]]: {{monzo| -52 -17 34 }}


[[Optimal tuning]] ([[POTE]]): ~{{monzo| 24 -5 -9 2 }} = 950.9995
{{Mapping|legend=1| 17 0 26 | 0 2 1 }}
: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}


{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
[[Optimal tuning]]s:
* [[WE]]: ~25/24 = 70.5887{{c}}, ~{{monzo| 26 9 -17 }} = 950.9807{{c}}
: [[error map]]: {{val| +0.008 +0.006 -0.027 }}
* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 26 9 -17 }} = 950.9776{{c}}
: error map: {{val| 0.000 -0.000 -0.042 }}


[[Badness]]: 0.041658
{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
 
Mapping: {{mapping|  17 0 26 -87 207 | 0 2 1 10 -11  }}
 
Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749
 
{{Optimal ET sequence|legend=1| 289, 323, 612 }}


Badness: 0.063706
[[Badness]] (Sintel): 1.81


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


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