Porcupine: Difference between revisions

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[[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 [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]], is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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 [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine or Triyo [[comma]], is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's 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]]).

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 [[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 [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]], is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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 [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine or Triyo [[comma]], is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's 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]]).