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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


Subgroup: 2.3.5.7.11.13.17.19
Comma list: 936/935, 1521/1520, 1701/1700, 1716/1715, 4096/4095, 11016/11011


Comma list: 936/935, 1716/1715, 2080/2079, 2376/2375, 4096/4095, 11016/11011
Mapping: {{Mapping| 16 0 -115 121 30 186 319 423 | 0 1 6 -3 1 -5 -10 -14 }}


{{Mapping|legend=1| 16 0 -115 121 30 186 319 423 | 0 1 6 -3 1 -5 -10 -14 }}
: Mapping generators: ~117/112 = 1＼16, ~3


: mapping generators: ~117/112 = 1\16, ~3
Optimal tuning (CTE): ~3/2 = 701.9505


[[Optimal tuning]] ([[CTE]]): ~3/2 = 701.950
Optimal ET sequence: {{EDOs|176h, 224, 400, 624 }}


[[Support]]ing [[ET]]s: {{EDOs|48ghh, 176h, 224, 400, 624, 848gh, 1024e}}
Badness: 0.020421


== Ntiscifer ==
== Ntiscifer ==
Line 51: Line 112:


[[Badness]]: 3.05
[[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 }}
: mapping generators: ~392/375, ~3
[[Optimal tuning]]s:
* [[CTE]]: ~392/375 = 1\16, ~3/2 = 703.653
* [[CWE]]: ~392/375 = 1\16, ~3/2 = 703.902
{{Optimal ET sequence|legend=1| 80, 128, 208, 496bdeef }}
[[Badness]]: 0.0611


{{Navbox fractional-octave}}
{{Navbox fractional-octave}}
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