17th-octave temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,171 edits
"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
m Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "
 
(21 intermediate revisions by 8 users not shown)
Line 1: Line 1:
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.
{{Technical data page}}
==Chlorine==
{{Infobox fractional-octave|17}}
See [[Ragismic microtemperaments#Chlorine]]
[[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.
==Leaves==
Defined as the 323 & 2023 temperament. 2 generators reach [[17/13]], 7 generators reach [[5/4]], 10 generators produce [[13/11]].


== 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 [[7/1|7]] and [[11/1|11]], it is the only [[strong extension]] for these primes that makes practical sense.
[[Subgroup]]: 2.3.5
[[Comma list]]: 134217728/129140163
{{mapping|legend=1| 17 27 0 | 0 0 1 }}
: mapping generators: ~256/243, ~5
[[Optimal tuning]]s:
* [[WE]]: ~256/243 = 70.5152{{c}}, ~5/4 = 388.7908{{c}}
: [[error map]]: {{val| -1.241 +1.956 -0.006 }}
* [[CWE]]: ~256/243 = 70.5882{{c}}, ~5/4 = 388.2315{{c}}
: error map: {{val| 0.000 +3.927 +1.918 }}
{{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}
[[Badness]] (Sintel): 12.7
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64/63, 17496/16807
{{Mapping|legend=1| 17 27 0 48 | 0 0 1 0 }}
[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 70.4030{{c}}, ~5/4 = 392.5654{{c}}
: [[error map]]: {{val| -3.149 -1.075 -0.047 +10.517 }}
* [[CWE]]: ~28/27 = 70.5882{{c}}, ~5/4 = 391.7654{{c}}
: error map: {{val| 0.000 +3.927 +5.452 +19.409 }}
{{Optimal ET sequence|legend=1| 17c, 34d, 85cdd }}
[[Badness]] (Sintel): 3.44
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 64/63, 99/98, 243/242
{{Mapping|legend=0| 17 27 0 48 59 | 0 0 1 0 0 }}
Optimal tunings:
* WE: ~28/27 = 70.3922{{c}}, ~5/4 = 392.9298{{c}}
* CWE: ~28/27 = 70.5882{{c}}, ~5/4 = 392.4722{{c}}
{{Optimal ET sequence|legend=0| 17c, 34d, 85cdde }}
Badness (Sintel): 1.77
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 160083/160000, 928125/927472, 1990656/1990625, 20726199/20706224
Comma list: 64/63, 78/77, 99/98, 144/143
 
{{Mapping|legend=0| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}
 
Optimal tunings:
* WE: ~27/26 = 70.4103{{c}}, ~5/4 = 392.3109{{c}}
* CWE: ~27/26 = 70.5882{{c}}, ~5/4 = 392.1288{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 34d }}
 
Badness (Sintel): 1.35
 
== Chlorine ==
: ''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.


Mapping: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].


Mapping generators: ~25/24, ~1024/975
[[Subgroup]]: 2.3.5


Optimal tuning (CTE): ~1024/975 = 85.421
[[Comma list]]: {{monzo| -52 -17 34 }}
=== 17-limit ===


Subgroup: 2.3.5.7.11.13.17
{{Mapping|legend=1| 17 0 26 | 0 2 1 }}
: mapping generators: ~25/24, ~{{monzo| 26 9 -17 }}


Comma list: 57375/57344, 111537/111475, 140800/140777, 111537/111475, 1026675/1026256
[[Optimal tuning]]s:  
* [[WE]]: ~25/24 = 70.5887{{c}}, ~{{monzo| 26 9 -17 }} = 950.9807{{c}}
: [[error map]]: {{val| +0.008 +0.006 -0.027 }}
* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 26 9 -17 }} = 950.9776{{c}}
: error map: {{val| 0.000 -0.000 -0.042 }}


Mapping: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
{{Optimal ET sequence|legend=1| 34, 153, 187, 221, 255, 289, 323, 612, 3349, 3961, 4573, 5185, 5797 }}


Mapping generators: ~25/24, ~765/728
[[Badness]] (Sintel): 1.81


Optimal tuning (CTE): ~765/728 = 85.421
{{Navbox fractional-octave}}


{{Optimal ET sequence|legend=1|323, 1700, 2023}}
[[Category:17edo]]
[[Category:17edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]

Latest revision as of 12:32, 24 September 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.

← 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. 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

Optimal tunings:

  • 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]]

Optimal tunings:

  • 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 ¢

Optimal ET sequence: 17c, 34d

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⟩

Optimal tunings:

  • 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

View • Talk • EditFractional-octave temperaments 
← 12th • 13th • 14th • 15th • 16th • 17th-octave • 18th • 19th • 20th • 21st • 22nd →
Retrieved from "https://en.xen.wiki/index.php?title=17th-octave_temperaments&oldid=237705"