17th-octave temperaments: Difference between revisions
Delete leaves |
|||
| (One intermediate revision by the same user not shown) | |||
| Line 73: | Line 73: | ||
== Chlorine == | == Chlorine == | ||
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 | : ''For extensions, see [[Ragismic microtemperaments #Chlorine]].'' | ||
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]]. | The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]]. | ||
| Line 93: | Line 95: | ||
[[Badness]] (Sintel): 1.81 | [[Badness]] (Sintel): 1.81 | ||
{{Navbox fractional-octave}} | {{Navbox fractional-octave}} | ||
[[Category:17edo]] | [[Category:17edo]] | ||
Latest revision as of 09:34, 20 August 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.
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 and 11, it is the only strong extension for these primes that makes practical sense.
Subgroup: 2.3.5
Comma list: 134217728/129140163
Mapping: [⟨17 27 0], ⟨0 0 1]]
- mapping generators: ~256/243, ~5
- WE: ~256/243 = 70.5152 ¢, ~5/4 = 388.7908 ¢
- error map: ⟨-1.241 +1.956 -0.006]
- CWE: ~256/243 = 70.5882 ¢, ~5/4 = 388.2315 ¢
- error map: ⟨0.000 +3.927 +1.918]
Optimal ET sequence: 17c, 34, 323bbcc, 357bbcc, 391bbcc
Badness (Sintel): 12.7
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 17496/16807
Mapping: [⟨17 27 0 48], ⟨0 0 1 0]]
- WE: ~28/27 = 70.4030 ¢, ~5/4 = 392.5654 ¢
- error map: ⟨-3.149 -1.075 -0.047 +10.517]
- CWE: ~28/27 = 70.5882 ¢, ~5/4 = 391.7654 ¢
- error map: ⟨0.000 +3.927 +5.452 +19.409]
Optimal ET sequence: 17c, 34d, 85cdd
Badness (Sintel): 3.44
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 99/98, 243/242
Mapping: [⟨17 27 0 48 59], ⟨0 0 1 0 0]]
Optimal tunings:
- WE: ~28/27 = 70.3922 ¢, ~5/4 = 392.9298 ¢
- CWE: ~28/27 = 70.5882 ¢, ~5/4 = 392.4722 ¢
Optimal ET sequence: 17c, 34d, 85cdde
Badness (Sintel): 1.77
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 99/98, 144/143
Mapping: [⟨17 27 0 48 59 63], ⟨0 0 1 0 0 0]]
Optimal tunings:
- WE: ~27/26 = 70.4103 ¢, ~5/4 = 392.3109 ¢
- CWE: ~27/26 = 70.5882 ¢, ~5/4 = 392.1288 ¢
Badness (Sintel): 1.35
Chlorine
- For extensions, see Ragismic microtemperaments #Chlorine.
Chlorine has a period of 1/17 octave. It tempers out [-52 -17 34⟩, the septendecima, by which 17 classical chromatic semitones ((25/24)17) 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: [-52 -17 34⟩
Mapping: [⟨17 0 26], ⟨0 2 1]]
- mapping generators: ~25/24, ~[26 9 -17⟩
- WE: ~25/24 = 70.5887 ¢, ~[26 9 -17⟩ = 950.9807 ¢
- error map: ⟨+0.008 +0.006 -0.027]
- CWE: ~25/24 = 70.5882 ¢, ~[26 9 -17⟩ = 950.9776 ¢
- error map: ⟨0.000 -0.000 -0.042]
Optimal ET sequence: 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797
Badness (Sintel): 1.81