Xenharmonic Wiki

16th-octave temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
BudjarnLambeth (talk | contribs)
autopatrolled, emailconfirmed
21,282 edits
m navbox
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,031 edits
No edit summary
Line 2: Line 2:
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 ==
Retrieved from "https://en.xen.wiki/w/16th-octave_temperaments"