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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 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 [[19/1|19]], but due to the sharpness of its harmonic [[3/1|3]], several intervals of [[9/1|9]] are inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo]] or [[58edo]].  


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. [[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 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, which is in fact a supermajor third of 428.6 [[cent]]s. However, the Pythagorean major third is represented as such consistently in [[224edo]], which is the quadruple of 56edo.
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.  


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]].
{| class="wikitable center-1 right-2 right-4"
|+ 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
|}


=== Prime harmonics ===
{| class="wikitable center-1 right-2 right-4"
{{Harmonics in equal|56}}
|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
 
|-
=== As a tuning of other temperaments ===
! rowspan="2" | #
As an equal temperament, 56et most notably tempers out [[2048/2025]], the diaschisma, as well as the [[shibboleth comma]] in the [[5-limit]]. 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 7- and 11-limit keen, 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 a wide semifourth of 257.1{{c}}, and 9/7 inconsistently to a 450{{c}} [[Interseptimal interval|naiadic]].
! colspan="2" | Period 0
 
! colspan="2" | Period 1
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]], 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 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 tempers out [[99/98]] and [[100/99]]. Finally, it gives an excellent tuning for the no-fives [[supra]] temperament tempering out [[64/63]] and [[99/98]].
|-
 
! 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, 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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