56th-octave temperaments: Difference between revisions
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==Barium== | {{Infobox fractional-octave|56}}{{todo|intro}} | ||
One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the {{monzo| -225 224 -56 }} | |||
== Barium == | |||
One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the [[barium comma]] ({{monzo| -225 224 -56 }}), which sets 56 syntonic commas equal to the octave. It can be expressed as the 224 & 2072 temperament. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 15: | Line 17: | ||
[[Badness]]: 4.70 | [[Badness]]: 4.70 | ||
====7-limit==== | |||
==== 7-limit ==== | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 27: | Line 30: | ||
[[Badness]]: 0.227 | [[Badness]]: 0.227 | ||
====11-limit==== | |||
==== 11-limit ==== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 39: | Line 43: | ||
Badness: 0.0345 | Badness: 0.0345 | ||
====13-limit==== | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 48: | Line 53: | ||
Optimal tuning (CTE): ~3/2 = 701.9431 | Optimal tuning (CTE): ~3/2 = 701.9431 | ||
{{Optimal ET sequence|legend=1| 224, 1848, 2072}} | {{Optimal ET sequence|legend=1| 224, 1848, 2072}} | ||
{{Navbox fractional-octave}} | |||
[[Category:56edo]] | [[Category:56edo]] | ||
Latest revision as of 23:31, 23 November 2025
Barium
One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the barium comma ([-225 224 -56⟩), which sets 56 syntonic commas equal to the octave. It can be expressed as the 224 & 2072 temperament.
Subgroup: 2.3.5
Comma list: [-225 224 -56⟩
Mapping: [⟨56 0 -225], ⟨0 1 4]]
Mapping generators: ~81/80, ~3
Optimal tuning (CTE): ~3/2 = 701.9379
Optimal ET sequence: 224, 1176, 1400, 1624, 1848, 2072, 5992, 8064, 26264, 34328b, 42392b
Badness: 4.70
7-limit
Subgroup: 2.3.5.7
Comma list: [-12 29 -11 -3⟩, [47 -7 -7 -7⟩
Mapping: [⟨56 0 -225 601], ⟨0 1 4 -5]]
Optimal tuning (CTE): ~3/2 = 701.9433
Optimal ET sequence: 224, 1176, 1400, 1624, 1848, 2072, 5768, 7616, 17080, 24696cd
Badness: 0.227
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 1019215872/1019046875, 14765025303/14763950080
Mapping: [⟨56 0 -225 601 460], ⟨0 1 4 -5 -3]]
Optimal tuning (CTE): ~3/2 = 701.9431
Optimal ET sequence: 224, 1176, 1400, 1624, 1848, 3920, 5768, 7616, 21000cd, 28616cd
Badness: 0.0345
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 9801/9800, 67392/67375, 26802913280/26795786661
Mapping: [⟨56 0 -225 601 460 651], ⟨0 1 4 -5 -3 -5]]
Optimal tuning (CTE): ~3/2 = 701.9431