17th-octave temperaments: Difference between revisions
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17edo 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. | 17edo 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. | ||
== | Temperaments discussed elsewhere include: | ||
* Chlorine, - [[Ragismic microtemperaments#Chlorine]] | |||
==Gothic== | |||
The gothic temperament is associated with the [[17-comma]]. | |||
Subgroup: 2.3.5 | |||
Comma list: 134217728/129140163 | |||
{{mapping|legend=1|17 27 34|0 0 1}} | |||
: mapping generators: ~256/243 = 1\17, ~5/4 = 386.314 | |||
[[Support]]ing [[ET]]s: {{EDOs|34, 17[+5], 51[+5], 85[+5], 119[+5], 68[+5]}} | |||
==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 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]]. | ||
Revision as of 22:24, 16 June 2023
17edo 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. Temperaments discussed elsewhere include:
- Chlorine, - Ragismic microtemperaments#Chlorine
Gothic
The gothic temperament is associated with the 17-comma.
Subgroup: 2.3.5
Comma list: 134217728/129140163
Mapping: [⟨17 27 34], ⟨0 0 1]]
- mapping generators: ~256/243 = 1\17, ~5/4 = 386.314
Supporting ETs: 34, 17[+5], 51[+5], 85[+5], 119[+5], 68[+5]
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: [⟨17 10 31 9 106 98], ⟨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: [⟨17 10 31 9 106 98 107], ⟨0 14 7 32 -39 -29 -31]]
Mapping generators: ~25/24, ~765/728
Optimal tuning (CTE): ~765/728 = 85.421