17th-octave temperaments: Difference between revisions

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== 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

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: mapping generators: ~256/243, ~5

[[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 }}

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=== 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

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[[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 }}

[[Badness]] (Sintel): 3.44

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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:

* ~~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}}

Badness (Sintel): 1.77

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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:

* ~~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}}

@@ Line 4: / Line 4: @@
 == 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
@@ Line 11: / Line 13: @@
 {{mapping|legend=1| 17 27 0 | 0 0 1 }}
 : mapping generators: ~256/243, ~5
 [[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 }}
@@ Line 23: / Line 26: @@
 === 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
@@ Line 32: / Line 33: @@
 [[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
@@ Line 44: / Line 47: @@
 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:
-* 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
@@ Line 59: / Line 62: @@
 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:
-* 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 }}