56th-octave temperaments: Difference between revisions
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{{Infobox fractional-octave|56}} | {{Infobox fractional-octave|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 == | == 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. | 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 | ||
Latest revision as of 23:31, 2 September 2026
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 ([-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
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