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.|600x600px]]

[[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 a stack of three 6/5s to a stack of two 4/3s. 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 a stack of three 6/5s to a stack of two 4/3s, [[tempering out]] [[250/243]], the porcupine [[comma]]. 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. Each of these parts is the [[generator]] of porcupine, which represents the difference between 4/3 and 6/5, a [[10/9|minor whole tone (10/9)]], that is tuned flat to around 160–165 [[cent]]s. 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.

In just intonation, a stack of three 6/5s is ~~a classical diminished seventh,~~ flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s 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]]~~. 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).

One may also note that in [[just intonation]], a stack of three 6/5s is flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s 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). Its [[pergen]] is (P8, P4/3).

~~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:

Both primes 7 and 11 are also found naturally in simple positions along the porcupine generator chain:

* 7 is found at +6 generators (~~tuned~~ to ~~about 960~~-~~990 cents~~), ~~because~~ the 16/9 ~~has already been flattened~~ to ~~merge it with (~~6/5~~)<sup>3<~~/~~sup>~~. This ~~makes porcupine an~~ [[~~archy~~]] ~~temperament.~~

Porcupine can be thought of as a [[2.3.5.11 subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third", due to the extreme flatness of 10/9. 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), found at -4 generators (tuned to about 540-560 cents); as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.

* 11 is found at -4 generators (tuned to about 540-560 cents)~~, corresponding with the acute fourth of 27/20~~; as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. ~~This can also be understood as~~ the ~~generator 10/9 being flattened~~.

~~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.~~

It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960-990 cents), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine 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, with the best tunings around 711-712 cents, which roughly splits the damage on 7/4 and 9/7.

~~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, with the best tunings around 711-712 cents, which roughly splits the damage on 7/4 and 9/7.

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

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| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]]. <br> Except the first, the intervals are the same.

|}

The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third", due to the extreme flatness of 10/9. 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 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. This can be considered the "chroma" of porcupine temperament.

@@ 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.|600x600px]]
 [[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 a stack of three 6/5s to a stack of two 4/3s. 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 a stack of three 6/5s to a stack of two 4/3s, [[tempering out]] [[250/243]], the porcupine [[comma]]. 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. Each of these parts is the [[generator]] of porcupine, which represents the difference between 4/3 and 6/5, a [[10/9|minor whole tone (10/9)]], that is tuned flat to around 160–165 [[cent]]s. 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.
-In just intonation, a stack of three 6/5s is a classical diminished seventh, flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s 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]]. 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).
+One may also note that in [[just intonation]], a stack of three 6/5s is flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3s 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). Its [[pergen]] is (P8, P4/3).
-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:
+Both primes 7 and 11 are also found naturally in simple positions along the porcupine generator chain:
-* 7 is found at +6 generators (tuned to about 960-990 cents), because the 16/9 has already been flattened to merge it with (6/5)<sup>3</sup>. This makes porcupine an [[archy]] temperament.
+Porcupine can be thought of as a [[2.3.5.11 subgroup]] temperament (sometimes called ''porkypine'') without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also [[11/10]] and [[12/11]] (equivalently, [[55/54]], [[100/99]], and [[121/120]] are tempered out), with the consequence that the [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third", due to the extreme flatness of 10/9. 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), found at -4 generators (tuned to about 540-560 cents); as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.
-* 11 is found at -4 generators (tuned to about 540-560 cents), corresponding with the acute fourth of 27/20; as the syntonic comma has been expanded, sharpening a fourth by a comma now leads to a significantly sharp interval close to the 11th harmonic. This can also be understood as the generator 10/9 being flattened.
-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.
+It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960-990 cents), has already been flattened to merge it with (6/5)<sup>3</sup>, and therefore can be equated to [[7/4]]. This makes porcupine 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, with the best tunings around 711-712 cents, which roughly splits the damage on 7/4 and 9/7.
-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, with the best tunings around 711-712 cents, which roughly splits the damage on 7/4 and 9/7.
 See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.
@@ Line 199: / Line 196: @@
 | Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]]. <br> Except the first, the intervals are the same.
 |}
-The [[11/9]] interval, usually considered a "neutral third", is in porcupine identical to the [[6/5]] "minor third", due to the extreme flatness of 10/9. 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 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. This can be considered the "chroma" of porcupine temperament.