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 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]]

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[[Category:Triads]]

[[Category:Tetrads]]

[[Category:Pentads]]

[[Category:Kestrel]]

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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 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.
-edo 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]]
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 [[Category:Triads]]
 [[Category:Tetrads]]
+[[Category:Pentads]]
 [[Category:Kestrel]]