User:Overthink/Draft edits
This is a page where I will draft edits before making them on the actual page. This may possibly include drafting a new page to be created. If you have something to add to any of them, or any concerns, please suggest them on the talk page. If a template is set to debug, make sure to remove that setting when editing the target page.
56edo
Theory
56edo shares its near perfect quality of the classical major third with 28edo, which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of 17edo and 22edo. It has decent approximations to prime harmonics up to 19, but due to the sharpness of its harmonic 3, several intervals of 9 are inconsistent. Therefore, 56edo is not very popular compared to edos like 53edo or 58edo.
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) | |
As a tuning of other temperaments
In the 5-limit, 56et most notably tempers out the diaschisma, as well as the shibboleth comma. Using the patent val, it tempers out 686/675, 875/864, and 1029/1024 in the 7-limit, 100/99, 245/242, and 385/384 in the 11-limit, and 91/90 and 169/168 in the 13-limit. It supports the diaschismic extension keen in the 7- and 11-limit, and its 13- and 17-limit extension keenic. It also supports hemithirds, superkleismic, and sycamore in various limits, being an especially optimal tuning for sycamore in the 11-, and 13-limits. It is also a very sharp tuning of slendric, mapping 7/6 to an inframinor third of 257.1 ¢, and mapping 9/7 inconsistently to an ultramajor third of 450 ¢.
Another interesting val to consider is 56d (⟨56 89 130 158]), which maps 7/4 sharply to around 986 ¢. This mapping tempers out 50/49 and 64/63 in the 7-limit, providing an alternative to 22edo for pajara. It improves accuracy of the 3rd harmonic and makes the 5th harmonic basically just, especially improving 6/5 and 10/9, which are quite out of tune in 22edo. Its approximated 7th harmonic is sharper than 22edo's, and combined with the fact that the 3rd harmonic is sharp, one may want to compress the octave, using tunings such as 145ed6 or 201ed12. It is also an excellent tuning for the 11-limit version of pajara, which additionally tempers out 99/98, 100/99, and 176/175. Finally, it gives an excellent tuning for the no-fives supra temperament tempering out 64/63 and 99/98.
Miscellaneous properties
One step of 56edo is the closest to the syntonic comma, 81/80, of any integer edo's step size by direct approximation, with the number of directly approximated syntonic commas per octave being 55.7976. 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. Because it contains 28edo's major third and has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third 5/4 at 18\56, and the Pythagorean major third 81/64 at 19\56. Unfortunately, 56edo does not map the Pythagorean major third 19\56, but instead inconsistently to 20\56, a supermajor third of 428.6 ¢. However, the Pythagorean major third is mapped to 19\56 consistently in 224edo, which is the quadruple of 56edo.
The perfect fifth generates a diatonic scale with a step ratio that is a convergent towards the bronze metallic mean, following 17edo and preceding 185edo.
Subsets and supersets
Since 56 factors into 23 × 7, 56edo has subset edos 2, 4, 7, 8, 14, and 28.
Main page
Welcome to the Xenharmonic Wiki!
The Xenharmonic Wiki is an open resource dedicated to musical tuning systems, focusing on xenharmonic music while also documenting historical tunings and tuning practices from world traditions. It covers the theory and practical applications of these systems.
For a lengthier introduction, see Xenharmonic Wiki: Introduction.
If you are new to musical tuning
- Why use alternative tunings?
- What are microtonal and xenharmonic music?
- Listen to alternatively tuned music, in case you're wondering what it all sounds like.
- Discover approaches to musical tuning
- Explore links to xenharmonic websites
- Browse the library of published works about microtonal/xenharmonic music
- Learn about the Xenharmonic Alliance, a social group of xenharmonic musicians
Popular topics
- Just intonation – Tuning based on intervals with rational number frequency ratios
- Equal divisions of the octave and other equal-step tunings
- Moment of Symmetry (MOS) scales – Scales with at most two distinct sizes (e.g. major and minor) for each interval class, among many other things
- Regular temperaments – Tuning systems that appear the same everywhere, excellent for free modulation; equal temperaments are a basic example
- Historical temperaments – such as Pythagorean tuning, meantone temperaments, and well temperaments
Practical xenharmonics
Contributing to the Xenharmonic Wiki
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to sign up and contribute!