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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.
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 =
= 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.
56edo shares its near perfect quality of the [[5/4|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 harmonic]]s up to [[19/1|19]], but due to the sharpness of its harmonic [[3/1|3]], several intervals of [[9/1|9]] are [[consistency|inconsistent]]. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].


=== Prime harmonics ===
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.
{{Harmonics in equal|56}}


=== As a tuning of other temperaments ===
{| class="wikitable center-1 right-2 right-4"
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 [[Ultramajor and inframinor|inframinor]] third of 257.1[[Cent|{{c}}]], and mapping 9/7 inconsistently to an ultramajor third of 450{{c}}.
|+ style="font-size: 105%;" | Pajara ({{nowrap| 12 & 22 }})
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
|-
! Cents*
! Approximate ratios
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
| 600.0
| 7/5, 10/7
|-
| 1
| 707.2
| '''3/2'''
| 107.2
| 15/14, 16/15, 21/20
|-
| 2
| 214.4
| '''8/7''', '''9/8'''
| 814.4
| '''8/5'''
|-
| 3
| 921.5
| 12/7
| 321.5
| 6/5
|-
| 4
| 428.7
| 9/7, 14/11
| 1028.7
| 9/5, 20/11
|-
| 5
| 1135.9
| 21/11, 27/14, 48/25, <br>64/33, 96/49
| 535.9
| 15/11, 27/20
|-
| 6
| 643.1
| '''16/11'''
| 43.1
| 45/44, 56/55, 81/80
|}


Another interesting val to consider is 56d ({{Val|56 89 130 '''158'''}}), which maps 7/4 sharply to around 986{{c}}. 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 [[Octave stretch|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]].
{| class="wikitable center-1 right-2 right-4"
 
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
=== Miscellaneous properties ===
|-
 
! rowspan="2" | #
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{{c}}. However, the Pythagorean major third is mapped to 19\56 consistently in [[224edo]], which is the quadruple of 56edo.
! colspan="2" | Period 0
 
! colspan="2" | Period 1
The perfect fifth generates a [[5L 2s|diatonic]] scale with a [[step ratio]] that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]].
|-
 
! Cents*
=== Subsets and supersets ===
! Approximate ratios
Since 56 factors into {{nowrap|2<sup>3</sup> &times; 7}}, 56edo has subset edos {{EDOs| 2, 4, 7, 8, 14, and 28}}.
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
| 600.0
| 7/5, 10/7
|-
| 1
| 709.6
| '''3/2'''
| 109.6
| 15/14, 16/15, 21/20
|-
| 2
| 219.1
| '''8/7''', '''9/8'''
| 819.1
| '''8/5'''
|-
| 3
| 928.7
| 12/7
| 328.7
| 6/5, 11/9
|-
| 4
| 438.2
| 9/7
| 1038.2
| 9/5, 11/6
|-
| 5
| 1147.8
| 27/14, 48/25, 55/28, <br>88/45, 96/49
| 547.8
| '''11/8''', 27/20
|-
| 6
| 657.3
| 22/15
| 57.3
| 22/21, 33/32, 81/80
|}
<nowiki/>* In 11-limit CWE tuning, octave-reduced


= Main page =
= Main page =
== Welcome to the Xenharmonic Wiki! ==
== 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.
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.


For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
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