17th-octave temperaments: Difference between revisions

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-{{Fractional-octave navigation|17}}
+{{Technical data page}}
-edo is a "wheel" for some fractional-octave temperaments. The most notable relationship is the tempering out of [[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.
-Temperaments discussed elsewhere include:
+== Gothic ==
+The gothic temperament is associated with the [[17-comma]]. It used to be known as ''septendecic''.
-* Chlorine, - [[Ragismic microtemperaments#Chlorine]]
+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.
-== Gothic ==
-The gothic temperament is associated with the [[17-comma]].
 [[Subgroup]]: 2.3.5
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 {{mapping|legend=1| 17 27 0 | 0 0 1 }}
 : mapping generators: ~256/243, ~5
 [[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 }}
-[[Badness]]: 0.541
+[[Badness]] (Sintel): 12.7
+=== 7-limit ===
+[[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:
+* [[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 }}
+{{Optimal ET sequence|legend=1| 17c, 34d, 85cdd }}
+[[Badness]] (Sintel): 3.44
-==Leaves==
+=== 11-limit ===
-Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].
+Subgroup: 2.3.5.7.11
+Comma list: 64/63, 99/98, 243/242
+{{Mapping|legend=0| 17 27 0 48 59 | 0 0 1 0 0 }}
+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 ET sequence|legend=0| 17c, 34d, 85cdde }}
+Badness (Sintel): 1.77
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
+Comma list: 64/63, 78/77, 99/98, 144/143
+{{Mapping|legend=0| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}
-Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
+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, ~1024/975
+{{Optimal ET sequence|legend=0| 17c, 34d }}
-Optimal tuning (CTE): ~1024/975 = 85.421
+Badness (Sintel): 1.35
-=== 17-limit ===
-Subgroup: 2.3.5.7.11.13.17
+== Chlorine ==
+: ''For extensions, see [[Ragismic microtemperaments #Chlorine]].''
-Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
+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.
+The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].
+[[Subgroup]]: 2.3.5
+[[Comma list]]: {{monzo| -52 -17 34 }}
+{{Mapping|legend=1| 17 0 26 | 0 2 1 }}
+: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}
+[[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 }}
-Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
+{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}
-Mapping generators: ~25/24, ~765/728
+[[Badness]] (Sintel): 1.81
-Optimal tuning (CTE): ~765/728 = 85.421
+{{Navbox fractional-octave}}
-{{Optimal ET sequence|legend=1|323, 1700, 2023}}
 [[Category:17edo]]
-[[Category:Temperament collections]]