17th-octave temperaments: Difference between revisions
→Chlorine: Mentioned the lack of connection with the chlorisma. |
No edit summary Tags: Mobile edit Mobile web edit |
||
| Line 15: | 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]] | [[Badness]] (Sintel): 12.7 | ||
== Chlorine == | == Chlorine == | ||
| Line 61: | Line 35: | ||
: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }} | : mapping generators: ~25/24, ~{{monzo| 26 9 -17 }} | ||
[[Optimal tuning]] ([[ | [[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]]: | [[Badness]] (Sintel): 1.81 | ||
=== 7-limit === | === 7-limit === | ||
| Line 78: | Line 54: | ||
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }} | {{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.05 | ||
=== 11-limit === | === 11-limit === | ||
| Line 89: | Line 65: | ||
Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749 | Optimal tuning (POTE): ~{{monzo| 24 -5 -9 2 }} = 950.9749 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 289, 323, 612 }} | ||
Badness (Sintel): 2.11 | |||
== 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: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}] | |||
Mapping generators: ~25/24, ~1024/975 | |||
Optimal tuning (CTE): ~1024/975 = 85.421 | |||
{{Optimal ET sequence|legend=1| 323, 1700, 2023}} | |||
=== 17-limit === | |||
Subgroup: 2.3.5.7.11.13.17 | |||
Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256 | |||
Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}] | |||
Mapping generators: ~25/24, ~765/728 | |||
Optimal tuning (CTE): ~765/728 = 85.421 | |||
{{Optimal ET sequence|legend=1| 323, 1700, 2023}} | |||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
[[Category:17edo]] | [[Category:17edo]] | ||
Revision as of 23:57, 20 February 2026
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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
The gothic temperament is associated with the 17-comma. It used to be known as septendecic.
Subgroup: 2.3.5
Comma list: 134217728/129140163
Mapping: [⟨17 27 0], ⟨0 0 1]]
- mapping generators: ~256/243, ~5
- CTE: ~256/243 = 70.5882 ¢ (1\17), ~5/4 = 386.3137 ¢ (~20480/19683 = 33.3725 ¢)
- CWE: ~256/243 = 70.5882 ¢ (1\17), ~5/4 = 388.2316 ¢ (~20480/19683 = 35.2904 ¢)
Optimal ET sequence: 17c, 34, 323bbcc, 357bbcc, 391bbcc
Badness (Sintel): 12.7
Chlorine
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, [-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 [-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: [-52 -17 34⟩
Mapping: [⟨17 0 26], ⟨0 2 1]]
- mapping generators: ~25/24, ~[26 9 -17⟩
- CTE: ~25/24 = 70.588 ¢ (1\17), ~[26 9 -17⟩ = 950.982 ¢
- CWE: ~25/24 = 70.588 ¢ (1\17), ~[26 9 -17⟩ = 950.978 ¢
Optimal ET sequence: 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797
Badness (Sintel): 1.81
7-limit
Subgroup: 2.3.5.7
Comma list: 4375/4374, [-49 4 22 -3⟩
Mapping: [⟨17 0 26 -87], ⟨0 2 1 10]]
Optimal tuning (POTE): ~[24 -5 -9 2⟩ = 950.9995
Optimal ET sequence: 289, 323, 612, 935, 1547
Badness (Sintel): 1.05
11-limit
Subgroup: 2.3.5.7.11
Comma list: 4375/4374, 41503/41472, 1879453125/1879048192
Mapping: [⟨17 0 26 -87 207], ⟨0 2 1 10 -11]]
Optimal tuning (POTE): ~[24 -5 -9 2⟩ = 950.9749
Optimal ET sequence: 289, 323, 612
Badness (Sintel): 2.11
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
Optimal ET sequence: 323, 1700, 2023
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