56edo: Difference between revisions
→Approximation to JI: -zeta peak index |
→Music: Add Bryan Deister's ''Waltz in 56edo'' (2025) |
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== Theory == | == Theory == | ||
56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. | 56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. | ||
One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo. | |||
56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara. | 56edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 56 89 130 158 }} (56d) as the equal temperament val, for [[pajara]]. It provides the [[optimal patent val]] for 7-, 11- and 13-limit [[sycamore]], and the 11-limit 56d val is close to the [[POTE tuning]] for 11-limit pajara. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 56 factors into {{nowrap|2<sup>3</sup> × 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}. | Since 56 factors into {{nowrap|2<sup>3</sup> × 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 28 }}. | ||
== Intervals == | == Intervals == | ||
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* [[Supra12]] | * [[Supra12]] | ||
* Subsets of [[echidnic]][16] (6u8d): | * Subsets of [[echidnic]][16] (6u8d): | ||
** Frankincense (this is the original/default tuning): 364.3 - 492.9 - 707.1 - 835.7 - 1200.0 | ** Frankincense{{idio}} (this is the original/default tuning): 364.3 - 492.9 - 707.1 - 835.7 - 1200.0 | ||
** Quasi-[[equipentatonic]]: 257.1 - 492.9 - 707.1 - 964.3 - 1200.0 | ** Quasi-[[equipentatonic]]: 257.1 - 492.9 - 707.1 - 964.3 - 1200.0 | ||
** Sakura-like scale containing [[phi]]: 107.1 - 492.9 - 707.1 - 835.7 - 1200.0 | ** Sakura-like scale containing [[phi]]: 107.1 - 492.9 - 707.1 - 835.7 - 1200.0 | ||
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** Evened minor pentatonic (approximated from [[72edo]]): 321.4 - 492.9 - 685.7 - 1028.6 - 1200.0 | ** Evened minor pentatonic (approximated from [[72edo]]): 321.4 - 492.9 - 685.7 - 1028.6 - 1200.0 | ||
== Instruments == | |||
[[Lumatone mapping for 56edo|Lumatone mappings for 56edo]] are available. | |||
== Music == | == Music == | ||
; [[Bryan Deister]] | ; [[Bryan Deister]] | ||
* [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023) | * [https://www.youtube.com/shorts/o0imqFPDh9k ''56edo''] (2023) | ||
* [https://www.youtube.com/watch?v=xkfao6yGKGE ''Curious Light - DOORS (microtonal cover in 56edo)''] (2025) | |||
* [https://www.youtube.com/watch?v=qzMOnS-lgWs ''Waltz in 56edo''] (2025) | |||
; [[Budjarn Lambeth]] | ; [[Budjarn Lambeth]] | ||
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* [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020) | * [https://www.youtube.com/watch?v=s1h083BRWXU ''Canon 3-in-1 on a Ground''] (2020) | ||
[[Category:Hemithirds]] | [[Category:Hemithirds]] | ||
[[Category:Keen]] | [[Category:Keen]] | ||