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.

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 [[41edo|41]], [[53edo|53]], [[58edo|58]], [[94edo|94]], [[103edo|103]], [[111edo|111]], [[161edo|161]], [[205edo|205]], [[214edo|214]], [[255edo|255]].

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]].

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]].

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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.
-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 [[41edo|41]], [[53edo|53]], [[58edo|58]], [[94edo|94]], [[103edo|103]], [[111edo|111]], [[161edo|161]], [[205edo|205]], [[214edo|214]], [[255edo|255]].
+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]].
 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]].