User:Overthink/Draft edits: Difference between revisions

← Older edit

VisualWikitext

@@ Line 1: / Line 1: @@
 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.
-edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. 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.
-edo has unambiguous approximations to prime harmonics up to [[19/1|19]], and possibly up to [[29/1|29]]. However 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]].
+In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.
-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.
+{| 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
+|}
-edo can be used to tune [[hemithirds]], [[superkleismic]], [[sycamore]] and [[keen]] temperaments, and using {{val| 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.
+{| class="wikitable center-1 right-2 right-4"
+|+ style="font-size: 105%;" | Pajarous ({{nowrap| 10 & 22 }})
-=== Prime harmonics ===
+|-
-{{Harmonics in equal|56}}
+! rowspan="2" | #
+! colspan="2" | Period 0
-=== Subsets and supersets ===
+! colspan="2" | Period 1
-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
+! 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 =
 == 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]].