Porcupine: Difference between revisions

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}}{{Infobox regtemp|Title=Porcupine; porkypine|Subgroups=2.3.5, 2.3.5.11, 2.3.5.7.11|Comma basis=[[250/243]] (2.3.5); <br> [[100/99]], [[55/54]] (2.3.5.11)|Edo join 1=22|Edo join 2=7|Generator=10/9|Generator tuning=164|Optimization method=CWE|MOS scales=[[1L 6s]], [[7L 1s]], [[8L 7s]]|Mapping=1; -3 -5 -4|Pergen=(P8, P4/3)|Color name=Triyo|Odd limit 1=5|Mistuning 1=?|Complexity 1=15|Odd limit 2=(2.3.5.11) 15|Mistuning 2=?|Complexity 2=22}}[[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: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 [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]] ~~(which may, in color notation, be referred to as the Triyo comma)~~, is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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.

'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]], is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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.

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

Porcupine can be thought of as a [[2.3.5.11 subgroup|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). This is because 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.

Porcupine can be thought of as a [[2.3.5.11 subgroup|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). This is because 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.

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.

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+}}{{Infobox regtemp|Title=Porcupine; porkypine|Subgroups=2.3.5, 2.3.5.11, 2.3.5.7.11|Comma basis=[[250/243]] (2.3.5); <br> [[100/99]], [[55/54]] (2.3.5.11)|Edo join 1=22|Edo join 2=7|Generator=10/9|Generator tuning=164|Optimization method=CWE|MOS scales=[[1L 6s]], [[7L 1s]], [[8L 7s]]|Mapping=1; -3 -5 -4|Pergen=(P8, P4/3)|Color name=Triyo|Odd limit 1=5|Mistuning 1=?|Complexity 1=15|Odd limit 2=(2.3.5.11) 15|Mistuning 2=?|Complexity 2=22}}[[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: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 [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]] (which may, in color notation, be referred to as the Triyo comma), is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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.
+'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a minor whole tone, tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], so that the generator represents [[10/9]], the difference between the two, and [[250/243]], the porcupine [[comma]], is [[tempered out]]. 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. Its [[pergen]] is (P8, P4/3). 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.
 One may also note that in [[just intonation]], a stack of three 6/5's is flat of the classical minor seventh [[9/5]] by [[25/24]], and a stack of two 4/3's 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]]).
-Porcupine can be thought of as a [[2.3.5.11 subgroup|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). This is because 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.
+Porcupine can be thought of as a [[2.3.5.11 subgroup|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). This is because 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.
 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.