Kestrel chords: Difference between revisions

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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 triads, 1-7/6-18/13 and 1-13/11-18/13. 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. Equal temperaments tempering out both 1188/1183 and 540/539 include 41, 53, 58, 94, 103, 111, 161, 205, 214, 255. 255edo is the [[Optimal_patent_val|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.
'''Kestrel chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the kestrel comma, [[1188/1183]].
[[Category:13-limit]]
 
[[Category:chord]]
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.
[[Category:dyadic]]
 
[[Category:kestrel]]
For triads, there are three pairs of chords in inverse relationship:
[[Category:swetisma]]
* 1–7/6–18/13 with steps of 7/6, 13/11, 13/9, and its inverse
[[Category:todo:add_links]]
* 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 chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Kestrel]]

Latest revision as of 13:53, 19 March 2025

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.

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