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~~]] in the [[porcupine family~~]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is ~~usually~~ around ~~160–165~~ [[cent]]s. It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

'''Porcupine''' is a [[linear temperament]] that 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]]. It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

~~The~~ basic 5-limit harmonic structure ~~of porcupine~~ can be understood ~~simply~~ by noting that tempering out 250/243 makes (4/3)2 equivalent to (6/5)3~~. In perhaps more familiar musical terms~~, this ~~means~~ two "perfect fourths" equals 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~~]], and to [[meantone~~]], in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine 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, this measn that two "perfect fourths" equals 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]], in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

== Interval chain ==

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== See also ==

* [[Porcupine Notation]]

* [[Porcupine family]]

* [[Porcupine modes]]

* [[Porcupine Album Project]]

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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]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is usually around 160–165 [[cent]]s. It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
+'''Porcupine''' is a [[linear temperament]] that 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]]. It can be thought of as a [[5-limit]], [[7-limit]], or [[11-limit]] temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
-The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>. In perhaps more familiar musical terms, this means two "perfect fourths" equals 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]], and to [[meantone]], in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine 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, this measn that two "perfect fourths" equals 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]], in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.
 == Interval chain ==
@@ Line 533: / Line 533: @@
 == See also ==
 * [[Porcupine Notation]]
+* [[Porcupine family]]
 * [[Porcupine modes]]
 * [[Porcupine Album Project]]