17th-octave temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Switch to Sintel's badness, WE & CWE tunings (1/)
Switch to Sintel's badness, WE & CWE tunings (2/)
Line 73: Line 73:


== Chlorine ==
== Chlorine ==
The name of chlorine temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].
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. This temperament can be described as 289 & 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  


Chlorine temperament has a period of 1/17 octave. It tempers out the [[septendecima]], {{monzo| -52 -17 34 }}, by which 17 chromatic semitones (25/24) exceed an octave. This temperament can be described as 289 &amp; 323 temperament, which tempers out {{monzo| -49 4 22 -3 }} as well as the ragisma. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  
The name of this temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 82: Line 82:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 26 9 -17 }} = 950.982{{c}}
* [[WE]]: ~25/24 = 70.5887{{c}}, ~{{monzo| 26 9 -17 }} = 950.9807{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 26 9 -17 }} = 950.978{{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 }}


{{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 }}
Line 101: Line 102:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.998{{c}}
* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.999{{c}}
: [[error map]]: {{val| -0.004 +0.037 -0.030 -0.019 }}
* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}
: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}


{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547 }}
{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}


[[Badness]] (Sintel): 1.05
[[Badness]] (Sintel): 1.05
Line 116: Line 119:


Optimal tunings:  
Optimal tunings:  
* CTE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}
* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}
* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}


{{Optimal ET sequence|legend=0| 289, 323, 612 }}
{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}


Badness (Sintel): 2.11
Badness (Sintel): 2.11

Revision as of 13:49, 17 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.

← 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

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. 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.

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

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 tunings:

  • WE: ~25/24 = 70.5880 ¢, ~[24 -5 -9 2⟩ = 950.9962 ¢
error map: ⟨-0.004 +0.037 -0.030 -0.019]
  • CWE: ~25/24 = 70.5882 ¢, ~[24 -5 -9 2⟩ = 950.9990 ¢
error map: ⟨0.000 +0.043 -0.021 -0.013]

Optimal ET sequence: 289, 323, 612, 935, 1547, 3706, 5253

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 tunings:

  • WE: ~25/24 = 70.5905 ¢, ~693/400 = 951.0054 ¢
  • CWE: ~25/24 = 70.5882 ¢, ~693/400 = 950.9754 ¢

Optimal ET sequence: 289, 323, 612, 3349de, 3961de, …, 5797ddee

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

Comma list: 1729951171875/1727094849536, [10 39 -14 -14⟩

Mapping: [⟨17 10 31 9], ⟨0 14 7 32]]

mapping generators: ~25/24, ~6125/5832

Optimal tunings:

  • CTE: ~25/24 = 70.588 ¢ (1\17), ~6125/5832 = 85.427 ¢
  • CWE: ~25/24 = 70.588 ¢ (1\17), ~6125/5832 = 85.426 ¢

Optimal ET sequence: 323, 1700d, 2023, 2346

Badness (Sintel): 32.032

11-limit

Subgroup: 2.3.5.7.11

Comma list: 160083/160000, 198359290368/198165034375, 1729951171875/1727094849536

Mapping: [⟨17 10 31 9 106], ⟨0 14 7 32 -39]]

Optimal tunings:

  • CTE: ~25/24 = 70.588 ¢ (1\17), ~6125/5832 = 85.421 ¢
  • CWE: ~25/24 = 70.588 ¢ (1\17), ~6125/5832 = 85.420 ¢

Optimal ET sequence: 323, 1700d, 2023, 2346e

Badness (Sintel): 20.456

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 160083/160000, 928125/927472, 1990656/1990625, 4831530/4826809

Mapping: [⟨17 10 31 9 106 98], ⟨0 14 7 32 -39 -29]]

Optimal tunings:

  • CTE: ~25/24 = 70.588 ¢ (1\17), ~1024/975 = 85.421 ¢
  • CWE: ~25/24 = 70.588 ¢ (1\17), ~1024/975 = 85.420 ¢

Optimal ET sequence: 323, 1700d, 2023, 2346e

Badness (Sintel): 10.096

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 39325/39304, 57375/57344, 71874/71825, 111537/111475, 140800/140777

Mapping: [⟨17 10 31 9 106 98 107], ⟨0 14 7 32 -39 -29 -31]]

Optimal tunings:

  • CTE: ~25/24 = 70.588 ¢ (1\17), ~765/728 = 85.421 ¢
  • CWE: ~25/24 = 70.588 ¢ (1\17), ~765/728 = 85.421 ¢

Optimal ET sequence: 323, 1700d, 2023, 2346e

Badness (Sintel): 6.678

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=236001"