Porcupine: Difference between revisions

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[[File:porcupine.png|thumb|Porcupine equates three minor thirds ([[6/5]], in red) with two perfect fourths ([[4/3]], in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]

[[File:porcupine.png|thumb|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]

[[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]

'''Porcupine''' is a [[linear temperament]] that equates a stack of three 6/5s to a stack of two 4/3s, [[tempering out]] [[250/243]], the porcupine [[comma]]. As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. Each of these parts is the [[generator]] of porcupine, which represents the difference between 4/3 and 6/5, a [[10/9|minor whole tone (10/9)]], that is tuned flat to around 160–165 [[cent]]s. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where the 10/9 interval is sharpened to merge with [[9/8]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

'''Porcupine''' is a [[linear temperament]] that equates a stack of three [[6/5]]s to a stack of two [[4/3]]s, [[tempering out]] [[250/243]], the porcupine [[comma]]. As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. Each of these parts is the [[generator]] of porcupine, which represents the difference between 4/3 and 6/5, a [[10/9|minor whole tone (10/9)]], that is tuned flat to around 160–165 [[cent]]s. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where the 10/9 interval is sharpened to merge with [[9/8]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

One may also note that in [[just intonation]], a stack of three 6/5s is flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s is the Pythagorean minor seventh 16/9, which is flat of 9/5 by 81/80. Thus, it can be determined that porcupine equates the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as meantone and schismic). Its [[pergen]] is (P8, P4/3).

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-[[File:porcupine.png|thumb|Porcupine equates three minor thirds ([[6/5]], in red) with two perfect fourths ([[4/3]], in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]
+[[File:porcupine.png|thumb|Porcupine equates three minor thirds (6/5, in red) with two perfect fourths (4/3, in green). To do so, it tempers out 250/243, which implies a generator of a flat 10/9.|600x600px]]
 [[File:porcupinesymmetricminor22edo.mp3|thumb|Symmetric minor mode of the porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in [[22edo]] tuning.]]
-'''Porcupine''' is a [[linear temperament]] that equates a stack of three 6/5s to a stack of two 4/3s, [[tempering out]] [[250/243]], the porcupine [[comma]]. As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. Each of these parts is the [[generator]] of porcupine, which represents the difference between 4/3 and 6/5, a [[10/9|minor whole tone (10/9)]], that is tuned flat to around 160–165 [[cent]]s. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where the 10/9 interval is sharpened to merge with [[9/8]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.
+'''Porcupine''' is a [[linear temperament]] that equates a stack of three [[6/5]]s to a stack of two [[4/3]]s, [[tempering out]] [[250/243]], the porcupine [[comma]]. As a consequence of this, 4/3 is divided into 3 equal parts, and 6/5 is divided into 2 of those same equal parts. Each of these parts is the [[generator]] of porcupine, which represents the difference between 4/3 and 6/5, a [[10/9|minor whole tone (10/9)]], that is tuned flat to around 160–165 [[cent]]s. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where the 10/9 interval is sharpened to merge with [[9/8]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.
 One may also note that in [[just intonation]], a stack of three 6/5s is flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s is the Pythagorean minor seventh 16/9, which is flat of 9/5 by 81/80. Thus, it can be determined that porcupine equates the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]], which simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to a strong extension of [[pythagorean]] (such as meantone and schismic). Its [[pergen]] is (P8, P4/3).