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 [[tempering ~~out|tempers~~ out]] [[250/243]], the porcupine [[~~comma~~]], and ~~whose~~ [[generator]] is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around ~~160–170~~ [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. ~~Its~~ [[~~pergen~~]] ~~is (P8, P4~~/3~~). It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.~~3~~.5.11 [[subgroup temperament]] (sometimes known as ''porkypine''). It~~ is ~~one~~ of ~~the most efficient temperaments in the 2.3.5.11 subgroup~~ of ~~reasonable accuracy~~.

'''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'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~~. ~~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.

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

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The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". This means that the [[27/20]] "acute fourth" of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone).

The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". An interval in the neutral third range is not found for a long time (until 17 generators up), and as a result that interval varies drastically depending on the tuning. This also means that the [[27/20]] "acute fourth" of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone).

The ~~characteristic small~~ interval ~~of porcupine, which is 60.3 cents in this tuning but can range from < 50 to 80 cents in general, represents~~ both [[25/24]] and [[81/80]].

The interval representing both [[25/24]] and [[81/80]] can be found in this interval chain at -7 steps, and ranges from about 45 to 80 cents depending on the tuning.

== Chords ==

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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 [[tempering out|tempers out]] [[250/243]], the porcupine  [[comma]], and whose [[generator]] is a [[10/9|minor whole tone (10/9)]] which is tuned flat to around 160–170 [[cent]]s such that two of them stack to a [[6/5|classic minor third (6/5)]]. Its [[pergen]] is (P8, P4/3). It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.3.5.11 [[subgroup temperament]] (sometimes known as ''porkypine''). It is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy.
+'''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'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. 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.
 See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.
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 |}
-The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". This means that the [[27/20]] "acute fourth" of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone).
+The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third". An interval in the neutral third range is not found for a long time (until 17 generators up), and as a result that interval varies drastically depending on the tuning. This also means that the [[27/20]] "acute fourth" of the JI diatonic scale is equivalent to [[11/8]] (rather than becoming 4/3 as in meantone).
-The characteristic small interval of porcupine, which is 60.3 cents in this tuning but can range from < 50 to 80 cents in general, represents both [[25/24]] and [[81/80]].
+The interval representing both [[25/24]] and [[81/80]] can be found in this interval chain at -7 steps, and ranges from about 45 to 80 cents depending on the tuning.
 == Chords ==