Buzzard: Difference between revisions

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| Odd limit 2 = 15 | Mistuning 2 = 4.09 | Complexity 2 = 43

}}

'''Buzzard''' is a [[regular temperament|temperament]] that splits a tempered [[3/1|perfect twelfth (3/1)]] into four [[generator]]s of [[21/16]] subfourths, tempering out [[65536/64827]]. If [[harmonic]] [[5/1|5]] ~~is desired, it is found by~~ twenty-one generators ~~octave-reduced~~, [[~~tempering out~~]] [[1728/1715]] and [[5120/5103]]. It extends to the [[13-limit]] by ~~tempering out~~ [[176/175]], [[351/350]], [[540/539]], and [[676/675]].

'''Buzzard''' is a [[regular temperament|temperament]] that splits a tempered [[3/1|perfect twelfth (3/1)]] into four [[generator]]s of [[21/16]] subfourths, [[tempering out]] the [[buzzardsma]] ([[ratio]]: 65536/64827). Two generators therefore give us a [[semitwelfth]], and five give us a sub-octave just short of the [[2/1|octave]] by a [[64/63|septimal comma]]. Bending the semitwelfth up by a septimal comma results in ~[[7/4]], and down results in ~[[12/7]], with the implication that the septimal diesis of [[49/48]] is equated to two septimal commas. In fact, buzzard slices the [[256/243|Pythagorean limma]] into four, one for 64/63, two for 49/48, and three for [[28/27]].

By finding [[harmonic]] [[5/1|5]] twenty-one generators away, buzzard is [[extension and restriction|extended]] to the full [[7-limit]], where it tempers out [[1728/1715]] and [[5120/5103]]. This equates the [[81/80|syntonic comma]] with the septimal comma and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. Buzzard then extends naturally to the [[13-limit]] by identifying the semitwelfth as [[26/15]], and identifying the comma step as the [[100/99|ptolemisma]] (100/99, {{S|10}}). This means [[176/175]], [[351/350]], [[540/539]], and [[676/675]] all vanish.

Finally, it is possible to extend buzzard to the [[19-limit]], where it merges [[17/16]] and [[16/15]], tempering out [[256/255]] ({{S|16}}), and merges [[26/15]] and [[19/11]], tempering out [[286/285]].

Buzzard can be tuned to [[53edo]], [[58edo]], or [[111edo]]. [[Mos scale]]s of buzzard cluster strongly around [[5edo]], similar to those of [[rodan]]. Rather than directly using mos scales, which are either extremely imbalanced or overly large, an approach to buzzard may involve picking and choosing which intervals from each pentatonic category to keep in the scale.

Alternative extensions of [[2.3.7 subgroup|2.3.7-]][[subgroup]] buzzard include [[subfourth]] (58 & 63) and [[lemongrass]] (63 & 68).

Buzzard was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html#10551 Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.

See [[Buzzardsmic clan #Buzzard]] for technical data.

~~__TOC__~~

~~{{Todo|improve synopsis}}~~

~~{{Clear}}~~

== Interval chain ==

In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.

@@ Line 15: / Line 15: @@
 | Odd limit 2 = 15 | Mistuning 2 = 4.09 | Complexity 2 = 43
 }}
-'''Buzzard''' is a [[regular temperament|temperament]] that splits a tempered [[3/1|perfect twelfth (3/1)]] into four [[generator]]s of [[21/16]] subfourths, tempering out [[65536/64827]]. If [[harmonic]] [[5/1|5]] is desired, it is found by twenty-one generators octave-reduced, [[tempering out]] [[1728/1715]] and [[5120/5103]]. It extends to the [[13-limit]] by tempering out [[176/175]], [[351/350]], [[540/539]], and [[676/675]].
+'''Buzzard''' is a [[regular temperament|temperament]] that splits a tempered [[3/1|perfect twelfth (3/1)]] into four [[generator]]s of [[21/16]] subfourths, [[tempering out]] the [[buzzardsma]] ([[ratio]]: 65536/64827). Two generators therefore give us a [[semitwelfth]], and five give us a sub-octave just short of the [[2/1|octave]] by a [[64/63|septimal comma]]. Bending the semitwelfth up by a septimal comma results in ~[[7/4]], and down results in ~[[12/7]], with the implication that the septimal diesis of [[49/48]] is equated to two septimal commas. In fact, buzzard slices the [[256/243|Pythagorean limma]] into four, one for 64/63, two for 49/48, and three for [[28/27]].
+By finding [[harmonic]] [[5/1|5]] twenty-one generators away, buzzard is [[extension and restriction|extended]] to the full [[7-limit]], where it tempers out [[1728/1715]] and [[5120/5103]]. This equates the [[81/80|syntonic comma]] with the septimal comma and turns it into a generic comma step that can be used to bridge Pythagorean intervals with both classical and septimal intervals. Buzzard then extends naturally to the [[13-limit]] by identifying the semitwelfth as [[26/15]], and identifying the comma step as the [[100/99|ptolemisma]] (100/99, {{S|10}}). This means [[176/175]], [[351/350]], [[540/539]], and [[676/675]] all vanish.
+Finally, it is possible to extend buzzard to the [[19-limit]], where it merges [[17/16]] and [[16/15]], tempering out [[256/255]] ({{S|16}}), and merges [[26/15]] and [[19/11]], tempering out [[286/285]].
+Buzzard can be tuned to [[53edo]], [[58edo]], or [[111edo]]. [[Mos scale]]s of buzzard cluster strongly around [[5edo]], similar to those of [[rodan]]. Rather than directly using mos scales, which are either extremely imbalanced or overly large, an approach to buzzard may involve picking and choosing which intervals from each pentatonic category to keep in the scale.
+Alternative extensions of [[2.3.7 subgroup|2.3.7-]][[subgroup]] buzzard include [[subfourth]] (58 & 63) and [[lemongrass]] (63 & 68).
 Buzzard was named by [[Herman Miller]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html#10551 Yahoo! Tuning Group (Archive) | ''Names for important high-complexity temperaments'']</ref>.
 See [[Buzzardsmic clan #Buzzard]] for technical data.
-__TOC__
-{{Todo|improve synopsis}}
-{{Clear}}
 == Interval chain ==
 In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.