16th-octave temperaments: Difference between revisions
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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. | ||
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 | ||
[[Optimal tuning]] ([[CTE]]): ~3/2 = 701. | [[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 }} | |||
Badness: 0.020421 | |||
== Ntiscifer == | == Ntiscifer == | ||
Revision as of 12:56, 30 December 2024
Template:Fractional-octave navigation 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.
Temperaments discussed elsewhere include hexadecoid and sedecic.
Sulfur
Subgroup: 2.3.5
Comma list: [-115 96 -16⟩
Mapping: [⟨16 0 -115], ⟨0 1 6]]
- Mapping generators: ~214748364800000/205891132094649 = 1＼16, ~3
Optimal tuning (CTE): ~3/2 = 701.8895
Supporting ETs: 48, 176, 224, 400, 624, 848, 1024, 1072, 1296, 1472
7-limit
Subgroup: 2.3.5.7
Comma list: 14348907/14336000, 2147483648/2144153025
Mapping: [⟨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: 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: [⟨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: 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: [⟨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: 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: [⟨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: 176, 224, 400, 624
Badness: 0.023704
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: [⟨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: 176h, 224, 400, 624
Badness: 0.020421
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
Mapping: [⟨16 25 0], ⟨0 0 1]]
- mapping generators: ~2048/2187, ~5
- 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)
Badness: 3.05