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.]]

[[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~~ [[6/5]]~~s to a stack~~ of ~~two~~ [[4/3]]s, [[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.

'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a [[10/9|minor whole tone (10/9)]], tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], [[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. 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 6/5s 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~~).

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]]).

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

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); 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]] 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.

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.

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.

== Interval chain ==

In the following table, odd harmonics 1–11 are in '''bold'''.

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! Cents

! Ratios

!Porcupine notation

! Porcupine notation

! Ups and downs notation

! #

! Cents

! Ratios

!Porcupine notation

! Porcupine notation

! Ups and downs notation

|-

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| 0.0

| '''1/1'''

|P1

| P1

| 0

| 1200.0

| '''2/1'''

|P8

| P8

|-

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| 162.8

| 10/9, 11/10, 12/11

|P2

| P2

| vM2 = ^^m2

| -1

| 1037.2

| 9/5, 11/6, 20/11

|P7

| P7

| ^m7 = vvM7

|-

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| 325.6

| 6/5, 11/9

|m3

| m3

| ^m3 = vvM3

| -2

| 874.4

| 5/3, 18/11

|M6

| M6

| vM6 = ^^m6

|-

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| 488.4

| 4/3

|m4

| m4

| P4

| -3

| 711.6

| '''3/2'''

|M5

| M5

| P5

|-

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| 651.3

| 16/11, 22/15

|m5

| m5

| v5 = ^^d5

| -4

| 548.7

| '''11/8''', 15/11

|M4

| M4

| ^4 = vvA4

|-

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| 814.1

| 8/5

|m6

| m6

| ^m6 = vvM6

| -5

| 385.9

| '''5/4'''

|M3

| M3

| vM3 = ^^m3

|-

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| 976.9

| '''7/4''', 16/9

|d7

| d7

| m7

| -6

| 223.1

| 8/7, '''9/8'''

|A2

| A2

| M2

|-

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| 1139.7

| 48/25, 160/81

|d8

| d8

| v8 = ^^d8

| -7

| 60.3

| 25/24, 81/80

|A1

| A1

| ^1 = vvA1

|-

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| 102.5

| 16/15, 21/20

|d2

| d2

| ^m2 = vvM2

| -8

| 1097.5

| 15/8, 40/21

|A7

| A7

| vM7 = ^^m7

|-

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| 265.3

| 7/6

|d3

| d3

| m3

| -9

| 934.7

| 12/7

|A6

| A6

| M6

|-

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| 428.2

| 14/11

|d4

| d4

| v4 = ^^d4

| -10

| 771.8

| 11/7

|A5

| A5

| ^5 = vvA5

|-

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| 591.0

| 7/5

|d5

| d5

| ^d5 = vv5

| -11

| 609.0

| 10/7

|A4

| A4

| vA4 = ^^4

|-

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| 753.8

| 14/9

|d6

| d6

| m6

| -12

| 446.2

| 9/7

|A3

| A3

| M3

|}

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== History ==

Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his Mizarian Porcupine Overture in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.

Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.

== See also ==

* [[Porcupine ~~Notation~~]]

* [[Porcupine notation]]

* [[Porcupine modes]]

* [[Porcupine Album Project]]

@@ Line 6: / Line 6: @@
 }}
 [[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.]]
+[[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 [[6/5]]s to a stack of two [[4/3]]s, [[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.
+'''Porcupine''' is a [[regular temperament|temperament]] that is [[generator|generated]] by a [[10/9|minor whole tone (10/9)]], tuned flat to around 160–165 [[cent]]s, two of which represent [[6/5]] and three of which represent [[4/3]], [[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. 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 6/5s 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).
+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]]).
-Both primes 7 and 11 are also found naturally in simple positions along the porcupine generator chain:
+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); 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]] 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.
+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.
-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.
 == Interval chain ==
-{{Main|Porcupine intervals}}
+{{Main| Porcupine intervals }}
 In the following table, odd harmonics 1–11 are in '''bold'''.
@@ Line 32: / Line 30: @@
 ! Cents
 ! Ratios
-!Porcupine notation
+! Porcupine notation
 ! Ups and downs<br>notation
 ! #
 ! Cents
 ! Ratios
-!Porcupine notation
+! Porcupine notation
 ! Ups and downs<br>notation
 |-
@@ Line 43: / Line 41: @@
 | 0.0
 | '''1/1'''
-|P1
+| P1
 | P1
 | 0
 | 1200.0
 | '''2/1'''
-|P8
+| P8
 | P8
 |-
@@ Line 54: / Line 52: @@
 | 162.8
 | 10/9, 11/10, 12/11
-|P2
+| P2
 | vM2 = ^^m2
 | -1
 | 1037.2
 | 9/5, 11/6, 20/11
-|P7
+| P7
 | ^m7 = vvM7
 |-
@@ Line 65: / Line 63: @@
 | 325.6
 | 6/5, 11/9
-|m3
+| m3
 | ^m3 = vvM3
 | -2
 | 874.4
 | 5/3, 18/11
-|M6
+| M6
 | vM6 = ^^m6
 |-
@@ Line 76: / Line 74: @@
 | 488.4
 | 4/3
-|m4
+| m4
 | P4
 | -3
 | 711.6
 | '''3/2'''
-|M5
+| M5
 | P5
 |-
@@ Line 87: / Line 85: @@
 | 651.3
 | 16/11, 22/15
-|m5
+| m5
 | v5 = ^^d5
 | -4
 | 548.7
 | '''11/8''', 15/11
-|M4
+| M4
 | ^4 = vvA4
 |-
@@ Line 98: / Line 96: @@
 | 814.1
 | 8/5
-|m6
+| m6
 | ^m6 = vvM6
 | -5
 | 385.9
 | '''5/4'''
-|M3
+| M3
 | vM3 = ^^m3
 |-
@@ Line 109: / Line 107: @@
 | 976.9
 | '''7/4''', 16/9
-|d7
+| d7
 | m7
 | -6
 | 223.1
 | 8/7, '''9/8'''
-|A2
+| A2
 | M2
 |-
@@ Line 120: / Line 118: @@
 | 1139.7
 | 48/25, 160/81
-|d8
+| d8
 | v8 = ^^d8
 | -7
 | 60.3
 | 25/24, 81/80
-|A1
+| A1
 | ^1 = vvA1
 |-
@@ Line 131: / Line 129: @@
 | 102.5
 | 16/15, 21/20
-|d2
+| d2
 | ^m2 = vvM2
 | -8
 | 1097.5
 | 15/8, 40/21
-|A7
+| A7
 | vM7 = ^^m7
 |-
@@ Line 142: / Line 140: @@
 | 265.3
 | 7/6
-|d3
+| d3
 | m3
 | -9
 | 934.7
 | 12/7
-|A6
+| A6
 | M6
 |-
@@ Line 153: / Line 151: @@
 | 428.2
 | 14/11
-|d4
+| d4
 | v4 = ^^d4
 | -10
 | 771.8
 | 11/7
-|A5
+| A5
 | ^5 = vvA5
 |-
@@ Line 164: / Line 162: @@
 | 591.0
 | 7/5
-|d5
+| d5
 | ^d5 = vv5
 | -11
 | 609.0
 | 10/7
-|A4
+| A4
 | vA4 = ^^4
 |-
@@ Line 175: / Line 173: @@
 | 753.8
 | 14/9
-|d6
+| d6
 | m6
 | -12
 | 446.2
 | 9/7
-|A3
+| A3
 | M3
 |}
@@ Line 386: / Line 384: @@
 == History ==
-Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his Mizarian Porcupine Overture in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.
+Porcupine temperament/scales were discovered by [[Dave Keenan]], but did not have a name until [[Herman Miller]] mentioned that his ''Mizarian Porcupine Overture'' in 15et had a section that pumps the 250/243 comma. Although this music did not use a porcupine mos or [[modmos]] (which would have 7 or 8 notes), the name was adopted for such scales as well, once the essentially one-to-one relationship between vanishing commas and sequences of [[DE]] scales was fully evident. It was clear that even though Herman's piece was in 15edo, 22edo was a porcupine tuning par excellence, and that was an interesting development in itself.
 == See also ==
-* [[Porcupine Notation]]
+* [[Porcupine notation]]
 * [[Porcupine modes]]
 * [[Porcupine Album Project]]