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16th-octave temperaments: Difference between revisions

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{{Fractional-octave navigation|16}}
{{Infobox fractional-octave|16}}
16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.
16edo is an interesting system when it comes to fractional-octave temperaments, as it has no straightforward JI approximation on its own, but some of its multiples do.


A temperament discussed elsewhere is [[hexadecoid]], a weak extension of octopus / octoid to the 19-limit with slightly different mappings.
Temperaments discussed elsewhere include [[hexadecoid]] and [[Jubilismic clan|sedecic]].


== Sulfur ==
== Sulfur ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Comma list: {{monzo| 115 96 -16 }}
[[Comma list]]: {{monzo| -115 96 -16 }}


{{Mapping|legend=1| 16 0 -115 | 0 1 6 }}
{{Mapping|legend=1| 16 0 -115 | 0 1 6 }}


: mapping generators: ~214748364800000/205891132094649 = 1\16, ~3
: Mapping generators: ~214748364800000/205891132094649 = 1＼16, ~3


[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.980
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.8895


[[Support]]ing [[ET]]s: {{EDOs|48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472}}
[[Support]]ing [[ET]]s: {{EDOs|48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472}}
=== 7-limit ===
Subgroup: 2.3.5.7
Comma list: 14348907/14336000, 2147483648/2144153025
{{Mapping|legend=1| 16 0 -115 121 | 0 1 6 -3 }}
: Mapping generators: ~256/245 = 1＼16, ~3
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.9129
[[Optimal ET sequence]]: {{EDOs|48, 128c, 176, 224, 400, 624 }}
[[Badness]]: 0.166051
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 46656/46585, 131072/130977
Mapping: {{Mapping| 16 0 -115 121 30 | 0 1 6 -3 1 }}
: Mapping generators: ~256/245 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9070
Optimal ET sequence: {{EDOs|48, 128c, 176, 224, 400, 624 }}
Badness: 0.041764
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 4096/4095, 39366/39325
Mapping: {{Mapping| 16 0 -115 121 30 186 | 0 1 6 -3 1 -5 }}
: Mapping generators: ~117/112 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9047
Optimal ET sequence: {{EDOs|48, 128cf, 176, 224, 400, 624, 848 }}
Badness: 0.025967
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 936/935, 1701/1700, 1716/1715, 4096/4095, 11016/11011
Mapping: {{Mapping| 16 0 -115 121 30 186 319 | 0 1 6 -3 1 -5 -10 }}
: Mapping generators: ~117/112 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9418
Optimal ET sequence: {{EDOs|176, 224, 400, 624 }}
Badness: 0.023704


=== 19-limit ===
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1521/1520, 1701/1700, 1716/1715, 4096/4095, 11016/11011
Mapping: {{Mapping| 16 0 -115 121 30 186 319 423 | 0 1 6 -3 1 -5 -10 -14 }}
: Mapping generators: ~117/112 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.9505
Optimal ET sequence: {{EDOs|176h, 224, 400, 624 }}


Subgroup: 2.3.5.7.11.13.17.19
Badness: 0.020421
 
== Ntiscifer ==
{{Main| Ntiscifer }}
 
Ntiscifer tempers out the Pythagorean double-augmented second, and is equivalent to the 16edo [[circle of fifths]] with an added dimension for [[5/4]]. In 16edo, this maps 5/4 to 375 cents, as in mavila temperament. Tunings with a separate, more accurate third include [[64edo]], [[80edo]], and [[96edo]]; 96edo is a particularly accurate tuning, though 64edo might be considered more practical.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: [[43046721/33554432]]
 
{{Mapping|legend=1| 16 25 0 | 0 0 1 }}
 
: mapping generators: ~2048/2187, ~5
 
[[Optimal tuning]]s:
* [[CTE]]: ~2048/2187 = 1\16, ~5/4 = 386.3137 (~135/128 = 11.3137)
* [[CWE]]: ~2048/2187 = 1\16, ~5/4 = 373.1508 (~128/135 = 1.8492)
 
{{Optimal ET sequence|legend=1| 16 }}
 
[[Badness]]: 3.05
 
== Hilarity ==
Hilarity, the [[80edo|80]] & [[128edo|128]] temperament, tempers [[40/27]] to 9\16.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 39 -48 16 }}
 
{{Mapping|legend=1| 16 0 -39 | 0 1 3 }}
 
: mapping generators: ~10460353203/10240000000, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~10460353203/1024000000 = 1\16, ~3/2 = 703.422
* [[CWE]]: ~10460353203/1024000000 = 1\16, ~3/2 = 703.621
 
{{Optimal ET sequence|legend=1| 16cc, 32c, 48, 80, 128, 208 }}
 
[[Badness]]: 4.22
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 179200/177147, 390625/388962
 
{{Mapping|legend=1| 16 0 -39 -82 | 0 1 3 5 }}
 
: mapping generators: ~392/375, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.630
* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.817
 
{{Optimal ET sequence|legend=1| 32cdd, 48d, 80, 128, 208, 496bd }}
 
[[Badness]]: 0.302
 
=== 11-limit ===
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 896/891, 2200/2187, 125000/124509
 
{{Mapping|legend=1| 16 0 -39 -82 30 | 0 1 3 5 1 }}
 
: mapping generators: ~392/375, ~3
 
[[Optimal tuning]]s:
* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.593
* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.882
 
{{Optimal ET sequence|legend=1| 48d, 80, 128, 208, 288be, 496bdee }}
 
[[Badness]]: 0.0886
 
=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 325/324, 352/351, 896/891, 540000/539539
 
{{Mapping|legend=1| 16 0 -39 -82 30 110 | 0 1 3 5 1 -2 }}


Comma list: 936/935, 1716/1715, 2080/2079, 2376/2375, 4096/4095, 11016/11011
: mapping generators: ~392/375, ~3


{{Mapping|legend=1| 16 0 -115 121 30 186 319 423 | 0 1 6 -3 1 -5 -10 -14 }}
[[Optimal tuning]]s:
* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.653
* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.902


: mapping generators: ~117/112 = 1\16, ~3
{{Optimal ET sequence|legend=1| 80, 128, 208, 496bdeef }}


[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.950
[[Badness]]: 0.0611


[[Support]]ing [[ET]]s: {{EDOs|48ghh, 176h, 224, 400, 624, 848gh, 1024e}}
{{Navbox fractional-octave}}
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