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= 56edo = | = 56edo = | ||
== Theory == | == Theory == | ||
56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th | 56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. 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. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo. | ||
56edo has unambiguous approximations to prime harmonics up to [[19/1|19 | 56edo has unambiguous approximations to prime harmonics up to [[19/1|19]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]]. | ||
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. | 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. | ||
Revision as of 05:34, 18 January 2026
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56edo
Theory
56edo shares its near perfect quality of classical major third with 28edo, which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of 17edo and 22edo. 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. However, this interval represents the pythagorean major third consistently in 224edo, which is the quadruple of 56edo.
56edo has unambiguous approximations to prime harmonics up to 19. However, the harmonic 3 is quite sharp, leading harmonic 9 to be even more so, and causing intervals like 10/9, 9/7, and 13/9 to be inconsistent. Therefore, 56edo is not very popular compared to edos like 53 and 58.
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 ⟨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 undecimal pajara.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +5.19 | -0.60 | -4.54 | +5.82 | -4.81 | +2.19 | +2.49 | -6.85 | -1.01 | -9.32 |
| Relative (%) | +0.0 | +24.2 | -2.8 | -21.2 | +27.2 | -22.5 | +10.2 | +11.6 | -31.9 | -4.7 | -43.5 | |
| Steps (reduced) |
56 (0) |
89 (33) |
130 (18) |
157 (45) |
194 (26) |
207 (39) |
229 (5) |
238 (14) |
253 (29) |
272 (48) |
277 (53) | |
Subsets and supersets
Since 56 factors into 23 × 7, 56edo has subset edos 2, 4, 7, 8, 14, 28.
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