17th-octave temperaments: Difference between revisions

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== Gothic ==
== Gothic ==
The gothic temperament is associated with the [[17-comma]]. It used to be known as '''septendecic'''.  
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
[[Subgroup]]: 2.3.5
Line 11: Line 13:


{{mapping|legend=1| 17 27 0 | 0 0 1 }}
{{mapping|legend=1| 17 27 0 | 0 0 1 }}
: mapping generators: ~256/243, ~5
: mapping generators: ~256/243, ~5


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 386.3137{{c}} (~20480/19683 = 33.3725{{c}})
* [[WE]]: ~256/243 = 70.5152{{c}}, ~5/4 = 388.7908{{c}}
* [[CWE]]: ~256/243 = 70.5882{{c}} (1\17), ~5/4 = 388.2316{{c}} (~20480/19683 = 35.2904{{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 }}
{{Optimal ET sequence|legend=1| 17c, 34, 323bbcc, 357bbcc, 391bbcc }}
Line 23: Line 26:


=== 7-limit ===
=== 7-limit ===
Gothic is identical to 17et in the no-5 subgroups, but has an independent generator for prime 5.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 32: Line 33:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* [[WE]]: ~28/27 = 70.4030{{c}}, ~5/4 = 392.5654{{c}}
* [[CWE]]: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 391.765{{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 }}
{{Optimal ET sequence|legend=1| 17c, 34d, 85cdd }}


[[Badness]] (Sintel): 3.44
[[Badness]] (Sintel): 3.44
Line 44: Line 47:
Comma list: 64/63, 99/98, 243/242
Comma list: 64/63, 99/98, 243/242


Mapping: {{Mapping| 17 27 0 48 59 | 0 0 1 0 0 }}
Mapping: {{mapping| 17 27 0 48 59 | 0 0 1 0 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* WE: ~28/27 = 70.3922{{c}}, ~5/4 = 392.9298{{c}}
* CWE: ~28/27 = 70.588{{c}} (1\17), ~5/4 = 392.472{{c}}
* CWE: ~28/27 = 70.5882{{c}}, ~5/4 = 392.4722{{c}}


{{Optimal ET sequence|legend=0| 17c, 34d }}
{{Optimal ET sequence|legend=0| 17c, 34d, 85cdde }}


Badness (Sintel): 1.77
Badness (Sintel): 1.77
Line 59: Line 62:
Comma list: 64/63, 78/77, 99/98, 144/143
Comma list: 64/63, 78/77, 99/98, 144/143


Mapping: {{Mapping| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}
Mapping: {{mapping| 17 27 0 48 59 63 | 0 0 1 0 0 0 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/26 = 70.588{{c}} (1\17), ~5/4 = 386.314{{c}}
* WE: ~27/26 = 70.4103{{c}}, ~5/4 = 392.3109{{c}}
* CWE: ~27/26 = 70.588{{c}} (1\17), ~5/4 = 392.129{{c}}
* CWE: ~27/26 = 70.5882{{c}}, ~5/4 = 392.1288{{c}}


{{Optimal ET sequence|legend=0| 17c, 34d }}
{{Optimal ET sequence|legend=0| 17c, 34d }}
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== Chlorine ==
== Chlorine ==
The name of chlorine temperament comes from chlorine, the 17th element, and has no relation to the [[chlorisma]].
: ''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.  


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 79: Line 84:


{{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 }}


[[Badness]] (Sintel): 1.81
[[Badness]] (Sintel): 1.81
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}
{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.998{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~{{monzo| 24 -5 -9 2 }} = 950.999{{c}}
{{Optimal ET sequence|legend=1| 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: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.978{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~693/400 = 950.975{{c}}
{{Optimal ET sequence|legend=0| 289, 323, 612 }}
Badness (Sintel): 2.11
== Leaves ==
Defined as the {{nowrap|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, {{monzo|10 39 -14 -14}}
[[Mapping]]: [{{val|17 10 31 9}}, {{val|0 14 7 32}}]
: mapping generators: ~25/24, ~6125/5832
[[Optimal tuning]]s:
* [[CTE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.427{{c}}
* [[CWE]]: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.426{{c}}
{{Optimal ET sequence|legend=1| 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: [{{val|17 10 31 9 106}}, {{val|0 14 7 32 -39}}]
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~6125/5832 = 85.420{{c}}
{{Optimal ET sequence|legend=0| 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: [{{val|17 10 31 9 106 98}}, {{val|0 14 7 32 -39 -29}}]
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~1024/975 = 85.420{{c}}
{{Optimal ET sequence|legend=0| 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: [{{val|17 10 31 9 106 98 107}}, {{val|0 14 7 32 -39 -29 -31}}]
Optimal tunings:
* CTE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}
* CWE: ~25/24 = 70.588{{c}} (1\17), ~765/728 = 85.421{{c}}
{{Optimal ET sequence|legend=0| 323, 1700d, 2023, 2346e}}
Badness (Sintel): 6.678


{{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.

← 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

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