User:Unque/Buzzard: Difference between revisions

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=== Canonical Buzzard ===

The interval at five generators up falls short of the octave by a small interval (namely [[64/63]], the septimal comma), which in Buzzard functions as an [[aberrisma]]. In 2.3.7 Buzzard, this aberrisma can be used to inflect between 3-limit and 7-limit consonances, such as between 21/16 and 4/3, or between 9/8 and 8/7. The canonical extension to include prime 5 is created by using this ~~amerrisma~~ as an all-purpose inflection, allowing 5-limit consonances to be accessed by inflecting in the opposite direction; for instance, 9/8 can be inflected upwards to reach 8/7, or downwards to reach 10/9. To do this, it tempers out [[5120/5103]], equating 64/63 with [[81/80]].

The interval at five generators up falls short of the octave by a small interval (namely [[64/63]], the septimal comma), which in Buzzard functions as an [[aberrisma]]. In 2.3.7 Buzzard, this aberrisma can be used to inflect between 3-limit and 7-limit consonances, such as between 21/16 and 4/3, or between 9/8 and 8/7. The canonical extension to include prime 5 is created by using this aberrisma as an all-purpose inflection, allowing 5-limit consonances to be accessed by inflecting in the opposite direction; for instance, 9/8 can be inflected upwards to reach 8/7, or downwards to reach 10/9. To do this, it tempers out [[5120/5103]], equating 64/63 with [[81/80]].

The canonical extension to 11-limit further equates this aberrisma with [[100/99]] (thus tempering out [[176/175]]), allowing 11-limit consonances to be reached via similar inflections; for instance, 11/9 can be found as an inflection downwards from 5/4 (or equivalently, two inflections below 81/64, or three below 9/7), and [[15/11]] can be found as an inflection upwards from 27/20 (or two inflections above 4/3, or three above 21/16).

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While it makes little sense from the perspective of a target tuning, Meantone occurs quite often in low EDO tunings of 2.3.7 Buzzard, such as 38edo and 43edo, making it a reasonable extension to the Buzzard structure. This extension finds 5-limit consonances by equating them with 3-limit consonances, or equivalently by equating the aberrisma with [[36/35]] (thus tempering out [[81/80]]). 5/4, for instance, can be reached at one inflection below 9/7, and 5/3 at one inflection below 12/7.

Another possibility is to split the aberrisma into two "semi-aberrisma" intervals; this allows the 5-limit consonances to be reached as a midpoint between existing 3-limit and 7-limit consonances (thus tempering out [[~~413343~~/~~409600~~]]). 5/4, for instance, can be reached as the midpoint of 81/64 and 9/7; and 5/3 as the midpoint of 27/16 and 12/7. While this extension is of lower accuracy than Buteo or Canonical, it is supported by many low-EDO tunings of 2.3.7 Buzzard, such as 28edo and 33edo.

Another possibility is to split the aberrisma into two "semi-aberrisma" intervals; this allows the 5-limit consonances to be reached as a midpoint between existing 3-limit and 7-limit consonances (thus tempering out [[729/700]]). 5/4, for instance, can be reached as the midpoint of 81/64 and 9/7; and 5/3 as the midpoint of 27/16 and 12/7. While this extension is of lower accuracy than Buteo or Canonical, it is supported by many low-EDO tunings of 2.3.7 Buzzard, such as 28edo and 33edo.

== Interval Chain ==

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 === Canonical Buzzard ===
-The interval at five generators up falls short of the octave by a small interval (namely [[64/63]], the septimal comma), which in Buzzard functions as an [[aberrisma]].  In 2.3.7 Buzzard, this aberrisma can be used to inflect between 3-limit and 7-limit consonances, such as between 21/16 and 4/3, or between 9/8 and 8/7.  The canonical extension to include prime 5 is created by using this amerrisma as an all-purpose inflection, allowing 5-limit consonances to be accessed by inflecting in the opposite direction; for instance, 9/8 can be inflected upwards to reach 8/7, or downwards to reach 10/9.  To do this, it tempers out [[5120/5103]], equating 64/63 with [[81/80]].
+The interval at five generators up falls short of the octave by a small interval (namely [[64/63]], the septimal comma), which in Buzzard functions as an [[aberrisma]].  In 2.3.7 Buzzard, this aberrisma can be used to inflect between 3-limit and 7-limit consonances, such as between 21/16 and 4/3, or between 9/8 and 8/7.  The canonical extension to include prime 5 is created by using this aberrisma as an all-purpose inflection, allowing 5-limit consonances to be accessed by inflecting in the opposite direction; for instance, 9/8 can be inflected upwards to reach 8/7, or downwards to reach 10/9.  To do this, it tempers out [[5120/5103]], equating 64/63 with [[81/80]].
 The canonical extension to 11-limit further equates this aberrisma with [[100/99]] (thus tempering out [[176/175]]), allowing 11-limit consonances to be reached via similar inflections; for instance, 11/9 can be found as an inflection downwards from 5/4 (or equivalently, two inflections below 81/64, or three below 9/7), and [[15/11]] can be found as an inflection upwards from 27/20 (or two inflections above 4/3, or three above 21/16).
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 While it makes little sense from the perspective of a target tuning, Meantone occurs quite often in low EDO tunings of 2.3.7 Buzzard, such as 38edo and 43edo, making it a reasonable extension to the Buzzard structure.  This extension finds 5-limit consonances by equating them with 3-limit consonances, or equivalently by equating the aberrisma with [[36/35]] (thus tempering out [[81/80]]).  5/4, for instance, can be reached at one inflection below 9/7, and 5/3 at one inflection below 12/7.
-Another possibility is to split the aberrisma into two "semi-aberrisma" intervals; this allows the 5-limit consonances to be reached as a midpoint between existing 3-limit and 7-limit consonances (thus tempering out [[413343/409600]]).  5/4, for instance, can be reached as the midpoint of 81/64 and 9/7; and 5/3 as the midpoint of 27/16 and 12/7.  While this extension is of lower accuracy than Buteo or Canonical, it is supported by many low-EDO tunings of 2.3.7 Buzzard, such as 28edo and 33edo.
+Another possibility is to split the aberrisma into two "semi-aberrisma" intervals; this allows the 5-limit consonances to be reached as a midpoint between existing 3-limit and 7-limit consonances (thus tempering out [[729/700]]).  5/4, for instance, can be reached as the midpoint of 81/64 and 9/7; and 5/3 as the midpoint of 27/16 and 12/7.  While this extension is of lower accuracy than Buteo or Canonical, it is supported by many low-EDO tunings of 2.3.7 Buzzard, such as 28edo and 33edo.
 == Interval Chain ==