Porcupine: Difference between revisions

Line 8:

| Title = Porcupine

| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.7.11

| Comma basis = [[250/243]] (2.3.5); [[55/54]], [[100/99]] (2.3.5.11)

| Comma basis = [[250/243]] (2.3.5); [[55/54]], [[100/99]] (2.3.5.11); [[55/54]], [[64/63]], [[100/99]] (2.3.5.7.11)

| Mapping = 1; -3 -5 6 -4

| Edo join 1 = 7 | Edo join 2 = 15

| Edo join 1 = 15 | Edo join 2 = 22

| Generator = 11/10

| Generator tuning = 164

| Optimization method = CWE

| MOS scales = [[1L~~ ~~6s]], [[7L~~ ~~1s]], [[7L~~ ~~8s]]

| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]], [[15L 7s]]

| Ploidacot = omega-tricot

| Pergen = (P8, P4/3)

| Color name = Triyoti

| Odd limit 1 = 5 | Mistuning 1 = ? | Complexity 1 = 15

| Odd limit 2 = (~~2.3.5.~~11) 15 | Mistuning 2 = ? | Complexity 2 = 22

| Odd limit 2 = (11-limit) 15 | Mistuning 2 = ? | Complexity 2 = 37

}}

[[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 which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]]. One generator represents [[10/9]], two (stacked) represent [[6/5]], and three represent [[4/3]]. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] and 6/5 is (in most cases) tuned flat or approximately just instead of sharp. 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 which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]]. One generator represents a severely flattened [[10/9]], two (stacked) represent [[6/5]], and three represent [[4/3]]. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]], and 6/5 is (in most cases) tuned flat or approximately just instead of sharp. 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 [[27/20]], the 5-limit "acute fourth", 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 [[27/20]], the 5-limit "acute fourth", 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)3, 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.

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

See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s and alternative 7-limit extensions.

== Interval chain ==

@@ Line 8: / Line 8: @@
 | Title = Porcupine
 | Subgroups = 2.3.5, 2.3.5.11, 2.3.5.7.11
-| Comma basis = [[250/243]] (2.3.5);<br>[[55/54]], [[100/99]] (2.3.5.11)
+| Comma basis = [[250/243]] (2.3.5);<br>[[55/54]], [[100/99]] (2.3.5.11);<br>[[55/54]], [[64/63]], [[100/99]] (2.3.5.7.11)
 | Mapping = 1; -3 -5 6 -4
-| Edo join 1 = 7 | Edo join 2 = 15
+| Edo join 1 = 15 | Edo join 2 = 22
 | Generator = 11/10
 | Generator tuning = 164
 | Optimization method = CWE
-| MOS scales = [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]]
+| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]], [[15L 7s]]
 | Ploidacot = omega-tricot
 | Pergen = (P8, P4/3)
 | Color name = Triyoti
 | Odd limit 1 = 5 | Mistuning 1 = ? | Complexity 1 = 15
-| Odd limit 2 = (2.3.5.11) 15 | Mistuning 2 = ? | Complexity 2 = 22
+| Odd limit 2 = (11-limit) 15 | Mistuning 2 = ? | Complexity 2 = 37
 }}
 [[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 which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]]. One generator represents [[10/9]], two (stacked) represent [[6/5]], and three represent [[4/3]]. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] and 6/5 is (in most cases) tuned flat or approximately just instead of sharp. 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 which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]]. One generator represents a severely flattened [[10/9]], two (stacked) represent [[6/5]], and three represent [[4/3]]. This is obviously in stark contrast to [[meantone]] temperaments, including [[12edo]], where 10/9 is tuned sharp and equated with [[9/8]] so that two of them reach a ''major'' third of [[5/4]], and 6/5 is (in most cases) tuned flat or approximately just instead of sharp. 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 [[27/20]], the 5-limit "acute fourth", 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 [[27/20]], the 5-limit "acute fourth", 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.
-See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s.
+See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s and alternative 7-limit extensions.
 == Interval chain ==