17th-octave temperaments

Fractional-octave temperaments

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← 16th-octave • 17th-octave • 18th-octave →

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.

Subgroup: 2.3.5

Comma list: 134217728/129140163

Mapping: [⟨17 27 0], ⟨0 0 1]]

mapping generators: ~256/243, ~5

Optimal tunings:

• 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

Chlorine

The name of chlorine temperament comes from Chlorine, the 17th element.

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⟩

Optimal tuning (POTE): ~[26 9 -17⟩ = 950.9746

Optimal ET sequence: 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797

Badness: 0.077072

7-limit

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-49 4 22 -3⟩

Mapping: [⟨17 0 26 -87], ⟨0 2 1 10]]

Wedgie: ⟨⟨ 34 17 170 -52 174 347 ]]

Optimal tuning (POTE): ~[24 -5 -9 2⟩ = 950.9995

Optimal ET sequence: 289, 323, 612, 935, 1547

Badness: 0.041658

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: 0.063706

Optimal ET sequence: 323, 1700, 2023