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= 24edo =
= Pajara =
== Theory ==
There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only [[11-odd-limit]] [[diamond monotone]] tuning, where primes [[3/1|3]] and [[5/1|5]] are less accurate than in optimal tunings of canonical 11-limit pajara.
The [[5-limit]] approximations in 24edo are the same as those in 12edo, so 24edo offers nothing new as far as approximating the 5-limit is concerned.
The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth. 24edo is also good at the 13th harmonic, which makes it a good 2.3.5.11.13 system. Specifically, intervals of 13/5 are particularly well approximated. And of course, 24edo shares its 17 and 19 tunings with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.
While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically: a supermajor chord is available at [0 9 14] and a subminor chord at [0 5 14], though they're more ultramajor and inframinor.
The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.
Its step, at 50 cents, is notable for being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant.
= Main page =
== Welcome to the Xenharmonic Wiki! ==
The [[Xenharmonic Wiki]] is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.
=== Prime harmonics ===
For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
{{Harmonics in equal|24|prec=2}}
=== Subsets and supersets ===
[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}.
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|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 [[interval]]s with {{W|rational number}} [[frequency ratio]]s
* [[EDO|Equal divisions of the octave]] and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]] – Scales with at most two distinct sizes (e.g. {{w|major and minor}}) for each interval class, [[MOS scale#Equivalent definitions and generalizations|among many other things]]
* [[Regular temperaments]] – Tuning systems that appear the same everywhere, excellent for free modulation; [[equal temperament]]s are a basic example
* [[Historical temperaments]] – such as [[Pythagorean tuning]], [[meantone]] temperaments, and [[well temperament]]s
== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]
== 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 [[Help: How to get your Xenharmonic Wiki account|sign up and contribute]]!
Latest revision as of 23:34, 18 February 2026
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.
There are two different mappings of the 11-limit. One is just called pajara and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called pajarous to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only 11-odd-limitdiamond monotone tuning, where primes 3 and 5 are less accurate than in optimal tunings of canonical 11-limit pajara.
In the following tables, odd harmonics 1–11 and their inverses are in bold.
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!
