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.]]

[[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 the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]]. This simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to an extension of [[pythagorean]] (i.e. meantone, schismic), by [[tempering out]] [[250/243]], the porcupine [[comma]]. Its [[pergen]] is (P8, P4/3). Porcupine's basic 5-limit harmonic structure can be understood by noting that tempering out 250/243 also makes (4/3)2 equivalent to (6/5)3; or, in other words, two "perfect fourths" are equivalent to three "minor thirds". 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. As a result, the [[generator]] of porcupine is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–165 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. This is obviously in stark contrast to [[12edo]]. 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 the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]]. This 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]] (i.e. meantone, schismic), by [[tempering out]] [[250/243]], the porcupine [[comma]]. Its [[pergen]] is (P8, P4/3). Porcupine's basic 5-limit harmonic structure can be understood by noting that tempering out 250/243 also makes (4/3)2 equivalent to (6/5)3; or, in other words, two "perfect fourths" are equivalent to three "minor thirds". 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. As a result, the [[generator]] of porcupine is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–165 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. This is obviously in stark contrast to [[12edo]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain, with 7 found at +6 generators (tuned to about 960-990 cents), and 11 found at -4 generators (tuned to about 540-560 cents). Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called ''porkypine''). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy.

In the 7-limit, porcupine can be seen as as an extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to 9/7, and its fifth is tuned sharp, ranging from around 705-720 cents.

In the 7-limit, porcupine can be seen as as a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to 9/7, and its fifth is tuned sharp, ranging from around 705-720 cents.

See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.

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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.]]
 [[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 the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]]. This simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to an extension of [[pythagorean]] (i.e. meantone, schismic), by [[tempering out]] [[250/243]], the porcupine [[comma]].  Its [[pergen]] is (P8, P4/3). Porcupine's basic 5-limit harmonic structure can be understood by noting that tempering out 250/243 also makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>; or, in other words, two "perfect fourths" are equivalent to three "minor thirds". 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. As a result, the [[generator]] of porcupine is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–165 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. This is obviously in stark contrast to [[12edo]]. 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 the syntonic comma [[81/80]] with the 5-limit chromatic semitone [[25/24]]. This 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]] (i.e. meantone, schismic), by [[tempering out]] [[250/243]], the porcupine [[comma]].  Its [[pergen]] is (P8, P4/3). Porcupine's basic 5-limit harmonic structure can be understood by noting that tempering out 250/243 also makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>; or, in other words, two "perfect fourths" are equivalent to three "minor thirds". 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. As a result, the [[generator]] of porcupine is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–165 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. This is obviously in stark contrast to [[12edo]]. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.
 To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain, with 7 found at +6 generators (tuned to about 960-990 cents), and 11 found at -4 generators (tuned to about 540-560 cents). Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called ''porkypine''). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy.
-In the 7-limit, porcupine can be seen as as an extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to 9/7, and its fifth is tuned sharp, ranging from around 705-720 cents.
+In the 7-limit, porcupine can be seen as as a weak extension of [[archy]], splitting its generator into three parts; its Pythagorean major third is mapped to 9/7, and its fifth is tuned sharp, ranging from around 705-720 cents.
 See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.