56th-octave temperaments: Difference between revisions
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==Barium== | {{Infobox fractional-octave|56}} | ||
One step of 56edo is close to a syntonic comma. Named after the 56th element, barium tempers out the {{monzo| -225 224 -56 }} | |||
56th-octave temperaments occur naturally through an edo join between any two equal divisions whose greatest common divisor is 56. Some of these are [[224edo]], a zeta edo, and [[1848edo]], an extremely precise 11-limit system. | |||
56edo is also useful for fractional-octave augmentation because its step is close to the syntonic comma, [[81/80]]. Hence considered below is barium. | |||
== 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. [[2072edo]] is the best tuning for the 13-limit, while [[1848edo]] is useful if restricting to 11-limit is necessary. | |||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
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[[Badness]]: 4.70 | [[Badness]]: 4.70 | ||
====7-limit==== | |||
==== 7-limit ==== | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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[[Badness]]: 0.227 | [[Badness]]: 0.227 | ||
====11-limit==== | |||
==== 11-limit ==== | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.0345 | Badness: 0.0345 | ||
====13-limit==== | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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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]] | |||