17th-octave temperaments: Difference between revisions
No edit summary |
→Gothic: complete |
||
| Line 5: | Line 5: | ||
* Chlorine, - [[Ragismic microtemperaments#Chlorine]] | * Chlorine, - [[Ragismic microtemperaments#Chlorine]] | ||
==Gothic== | == Gothic == | ||
The gothic temperament is associated with the [[17-comma]]. | The gothic temperament is associated with the [[17-comma]]. | ||
Subgroup: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Comma list: 134217728/129140163 | [[Comma list]]: 134217728/129140163 | ||
{{mapping|legend=1|17 27 | {{mapping|legend=1| 17 27 0 | 0 0 1 }} | ||
: mapping generators: ~256/243 = 1\17, ~5/4 = 386. | : mapping generators: ~256/243, ~5 | ||
[[Optimal tuning]]s: | |||
* [[CTE]]: ~256/243 = 1\17, ~5/4 = 386.3137 (~20480/19683 = 33.3725) | |||
* [[CWE]]: ~256/243 = 1\17, ~5/4 = 388.2316 (~20480/19683 = 35.2904) | |||
{{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }} | |||
[[Badness]]: 0.541 | |||
==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 14:57, 7 July 2024
Template:Fractional-octave navigation 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 0], ⟨0 0 1]]
- mapping generators: ~256/243, ~5
- CTE: ~256/243 = 1\17, ~5/4 = 386.3137 (~20480/19683 = 33.3725)
- CWE: ~256/243 = 1\17, ~5/4 = 388.2316 (~20480/19683 = 35.2904)
Optimal ET sequence: 17c, 34, 323bbcc, 357bbcc, 391bbcc
Badness: 0.541
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