Kestrel chords: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
m Recategorize
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,013 edits
Update the article
Line 1: Line 1:
The ''[[kestrel]] [[tetrad]]'' is the [[13-limit]], [[Dyadic chord|essentially tempered tetrad]] consisting, in close position, of the steps 13/11-7/6-13/11-11/9 in a [[temperament]] tempering out the kestrel comma, [[1188/1183]], leading to the tempering of 1-13/11-18/13-18/11. Contained in the kestrel tetrad are the two [[Kestrel triad|kestrel triads]], 1-7/6-18/13 and 1-13/11-18/13.  
'''Kestrel chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the kestrel comma, [[1188/1183]].


Closely related tetrads require the tempering of both 1188/1183 and [[540/539]], with steps of 7/6-13/11-13/11-11/9, leading to the tempered version of 1-7/6-18/13-18/11, and 11/9-13/11-13/11-7/6, leading to the 1-11/9-13/9-12/7 chord. [[EDO|Equal temperaments]] tempering out both 1188/1183 and 540/539 include [[41edo|41]], [[53edo|53]], [[58edo|58]], [[94edo|94]], [[103edo|103]], [[111edo|111]], [[161edo|161]], [[205edo|205]], [[214edo|214]], [[255edo|255]].  
Kestrel chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the 2.3.7.11.13 subgroup [[13-odd-limit]], meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.


255edo is the [[optimal patent val]] for both {1188/1183, 540/539}-tempering and 1188/1183-tempering, so nothing significant is lost in tuning accuracy by also tempering out 540/539. Kestrel tetrads are essentially tempered tetrads of the 2.7/3.11/9.13/9 [[just intonation subgroup]].
For triads, there are three pairs of chords in inverse relationship:
* 1-7/6-18/13 with steps of 7/6-13/11-13/9, and its inverse
* 1-13/11-18/13 with steps of 13/11-7/6-13/9;
* 1-14/11-18/13 with steps of 14/11-13/12-13/9, and its inverse
* 1-13/12-18/13 with steps of 13/12-14/11-13/9;
* 1-13/11-9/7 with steps of 13/11-13/12-14/9, and its inverse
* 1-13/12-9/7 with steps of 13/12-13/11-14/9.
 
For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are
* 1-13/11-18/13-18/11 with steps of 13/11-7/6-13/11-11/9;
* 1-13/12-18/13-3/2 with steps of 13/12-14/11-13/12-4/3;
* 1-13/12-13/11-9/7 with steps of 13/12-12/11-13/12-14/9.
 
The inversely related pairs of chords are
* 1-11/9-13/9-11/7 with steps of 11/9-13/11-13/12-14/11, and its inverse
* 1-13/12-9/7-11/7 with steps of 13/12-13/11-11/9-14/11;
* 1-7/6-18/13-3/2 with steps of 7/6-13/11-13/12-4/3, and its inverse
* 1-13/12-9/7-3/2 with steps of 13/12-13/11-7/6-4/3;
* 1-13/11-9/7-18/13 with steps of 13/11-13/12-14/13-13/9, and its inverse
* 1-14/13-7/6-18/13 with steps of 14/13-13/12-13/11-13/9;
* 1-13/11-14/11-18/13 with steps of 13/11-14/13-13/12-13/9, and its inverse
* 1-13/12-7/6-18/13 with steps of 13/12-14/13-13/11-13/9;
* 1-7/6-14/11-18/13 with steps of 7/6-12/11-13/12-13/9, and its inverse
* 1-13/12-13/11-18/13 with steps of 13/12-12/11-7/6-13/9;
* 1-13/12-9/7-18/13 with steps of 13/12-13/11-14/13-13/9, and its inverse
* 1-14/13-14/11-18/13 with steps of 14/13-13/11-13/12-13/9.
 
For pentads, there are three pairs of chords in inverse relationship:
* 1-13/11-9/7-18/13-18/11 with steps of 13/11-13/12-14/13-13/11-11/9, and its inverse
* 1-13/11-14/11-18/13-18/11 with steps of 13/11-14/13-13/12-13/11-11/9;
* 1-13/12-7/6-18/13-3/2 with steps of 13/12-14/13-13/11-13/12-4/3, and its inverse
* 1-13/12-9/7-18/13-3/2 with steps of 13/12-13/11-14/13-13/12-4/3;
* 1-13/12-13/11-9/7-18/13 with steps of 13/12-12/11-13/12-14/13-13/9, and its inverse
* 1-14/13-7/6-14/11-18/13 with steps of 14/13-13/12-12/11-13/12-13/9.
 
Equal temperaments with kestrel chords include {{Optimal ET sequence|17, 24, 26, 41, 53, 58, 77, 94, 103, 111, 161, 205, 214 and 255}}, with 255edo giving the [[optimal patent val]].


[[Category:13-odd-limit]]
[[Category:13-odd-limit]]
Line 9: Line 44:
[[Category:Triads]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Kestrel]]
[[Category:Kestrel]]

Revision as of 13:48, 3 November 2023

Kestrel chords are essentially tempered chords tempered by the kestrel comma, 1188/1183.

Kestrel chords are of pattern 2 in the 2.3.7.11.13 subgroup 13-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

For triads, there are three pairs of chords in inverse relationship:

  • 1-7/6-18/13 with steps of 7/6-13/11-13/9, and its inverse
  • 1-13/11-18/13 with steps of 13/11-7/6-13/9;
  • 1-14/11-18/13 with steps of 14/11-13/12-13/9, and its inverse
  • 1-13/12-18/13 with steps of 13/12-14/11-13/9;
  • 1-13/11-9/7 with steps of 13/11-13/12-14/9, and its inverse
  • 1-13/12-9/7 with steps of 13/12-13/11-14/9.

For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are

  • 1-13/11-18/13-18/11 with steps of 13/11-7/6-13/11-11/9;
  • 1-13/12-18/13-3/2 with steps of 13/12-14/11-13/12-4/3;
  • 1-13/12-13/11-9/7 with steps of 13/12-12/11-13/12-14/9.

The inversely related pairs of chords are

  • 1-11/9-13/9-11/7 with steps of 11/9-13/11-13/12-14/11, and its inverse
  • 1-13/12-9/7-11/7 with steps of 13/12-13/11-11/9-14/11;
  • 1-7/6-18/13-3/2 with steps of 7/6-13/11-13/12-4/3, and its inverse
  • 1-13/12-9/7-3/2 with steps of 13/12-13/11-7/6-4/3;
  • 1-13/11-9/7-18/13 with steps of 13/11-13/12-14/13-13/9, and its inverse
  • 1-14/13-7/6-18/13 with steps of 14/13-13/12-13/11-13/9;
  • 1-13/11-14/11-18/13 with steps of 13/11-14/13-13/12-13/9, and its inverse
  • 1-13/12-7/6-18/13 with steps of 13/12-14/13-13/11-13/9;
  • 1-7/6-14/11-18/13 with steps of 7/6-12/11-13/12-13/9, and its inverse
  • 1-13/12-13/11-18/13 with steps of 13/12-12/11-7/6-13/9;
  • 1-13/12-9/7-18/13 with steps of 13/12-13/11-14/13-13/9, and its inverse
  • 1-14/13-14/11-18/13 with steps of 14/13-13/11-13/12-13/9.

For pentads, there are three pairs of chords in inverse relationship:

  • 1-13/11-9/7-18/13-18/11 with steps of 13/11-13/12-14/13-13/11-11/9, and its inverse
  • 1-13/11-14/11-18/13-18/11 with steps of 13/11-14/13-13/12-13/11-11/9;
  • 1-13/12-7/6-18/13-3/2 with steps of 13/12-14/13-13/11-13/12-4/3, and its inverse
  • 1-13/12-9/7-18/13-3/2 with steps of 13/12-13/11-14/13-13/12-4/3;
  • 1-13/12-13/11-9/7-18/13 with steps of 13/12-12/11-13/12-14/13-13/9, and its inverse
  • 1-14/13-7/6-14/11-18/13 with steps of 14/13-13/12-12/11-13/12-13/9.

Equal temperaments with kestrel chords include 17, 24, 26, 41, 53, 58, 77, 94, 103, 111, 161, 205, 214 and 255, with 255edo giving the optimal patent val.

Retrieved from "https://en.xen.wiki/index.php?title=Kestrel_chords&oldid=126835"