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]] 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]].

'''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 [[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.

@@ 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]] 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]].
+'''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 [[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.