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)]]. 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 best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

'''Porcupine''' is a [[linear temperament]] that [[tempering out|tempers out]] [[250/243]], the porcupine or Triyo [[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 best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

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]]~~, 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.

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.

See [[Porcupine family #Porcupine]] for technical data.

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{| class="wikitable center-all right-2 left-3 right-6 left-7"

! colspan="4" |up from the tonic, aka fourthward

! colspan="4" |down from the octave, aka fifthward

|-

! #

! Cents

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! Ups and downs notation

! #

! ~~2/1 inverse~~

! Cents

! Ratios

! Ups and downs notation

@@ Line 7: / Line 7: @@
 [[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)]]. 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 best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
+'''Porcupine''' is a [[linear temperament]] that [[tempering out|tempers out]] [[250/243]], the porcupine or Triyo [[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 best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
-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]], 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.
+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.
 See [[Porcupine family #Porcupine]] for technical data.
@@ Line 17: / Line 17: @@
 {| class="wikitable center-all right-2 left-3 right-6 left-7"
+! colspan="4" |up from the tonic, aka fourthward
+! colspan="4" |down from the octave, aka fifthward
+|-
 ! #
 ! Cents
@@ Line 22: / Line 25: @@
 ! Ups and downs<br />notation
 ! #
-! 2/1 inverse
+! Cents
 ! Ratios
 ! Ups and downs<br />notation