Porcupine: Difference between revisions

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| 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)

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

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

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| Generator tuning = 164

| Optimization method = CWE

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

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

| Ploidacot = omega-tricot

| Pergen = (P8, P4/3)

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[[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]], 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 which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]], such that the generator represents [[10/9]], two of which represents [[6/5]], and three of which represent [[4/3]]. 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 10/9 interval is tuned sharp and equated 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 [[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.

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{| class="wikitable center-all right-2 left-3 right-7 left-8"

|-

! colspan="5" | Up from the tonic, aka fourthward

! colspan="5" | Down from the octave, aka fifthward

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! Cents*

! Ratios

! Porcupine notation

! Porcupine notation

! Ups and downs notation

! Ups and downs notation

! #

! Cents*

! Ratios

! Porcupine notation

! Porcupine notation

! Ups and downs notation

! Ups and downs notation

|-

| 0

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{| class="wikitable"

|-

| [[File:OtonalPentad_JI.mp3]]

| [[File:OtonalPentad_22edo.mp3]]

| [[File:OtonalPentad_29edo.mp3]]

|-

| 8:9:10:11:12 chord, in just intonation. All intervals are slightly different.

| 8:9:10:11:12 chord, in just intonation. All intervals are slightly different.

| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]]. Except the first, the intervals are the same.

| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]]. Except the first, the intervals are the same.

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

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

|}

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| POTE: ~10/9 = 163.9504{{c}}

|-

! Benedetti, Wilson

! Benedetti, Wilson

| CBE: ~10/9 = 164.3761{{c}}

| CSBE: ~10/9 = 164.3761{{c}}

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| POTE: ~11/10 = 164.0777{{c}}

|-

! Benedetti, Wilson

! Benedetti, Wilson

| CBE: ~11/10 = 164.2393{{c}}

| CSBE: ~11/10 = 164.4623{{c}}

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| POTE: ~11/10 = 162.7474{{c}}

|-

! Benedetti, Wilson

! Benedetti, Wilson

| CBE: ~11/10 = 163.5299{{c}}

| CSBE: ~11/10 = 163.2310{{c}}

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{| class="wikitable center-all left-4"

|-

! Edo generator

! Edo generator

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]

! Generator (¢)

! Comments

@@ 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)
 | Mapping = 1; -3 -5 6 -4
 | Edo join 1 = 7 | Edo join 2 = 15
@@ Line 14: / Line 14: @@
 | Generator tuning = 164
 | Optimization method = CWE
-| MOS scales = [[1L 6s]], [[7L 1s]], [[7L 8s]]
+| MOS scales = [[1L&nbsp;6s]], [[7L&nbsp;1s]], [[7L&nbsp;8s]]
 | Ploidacot = omega-tricot
 | Pergen = (P8, P4/3)
@@ Line 24: / Line 24: @@
 [[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]], 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 which is tuned flat to around 160–165 [[cent]]s, and the porcupine [[comma]] ([[250/243]]) is [[tempering out|tempered out]], such that the generator represents [[10/9]], two of which represents [[6/5]], and three of which represent [[4/3]]. 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 10/9 interval is tuned sharp and equated 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 [[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.
@@ Line 40: / Line 40: @@
 {| class="wikitable center-all right-2 left-3 right-7 left-8"
+|-
 ! colspan="5" | Up from the tonic, aka fourthward
 ! colspan="5" | Down from the octave, aka fifthward
@@ Line 46: / Line 47: @@
 ! Cents*
 ! Ratios
-! Porcupine<br>notation
+! Porcupine<br />notation
-! Ups and downs<br>notation
+! Ups and downs<br />notation
 ! #
 ! Cents*
 ! Ratios
-! Porcupine<br>notation
+! Porcupine<br />notation
-! Ups and downs<br>notation
+! Ups and downs<br />notation
 |-
 | 0
@@ Line 207: / Line 208: @@
 {| class="wikitable"
+|-
 | [[File:OtonalPentad_JI.mp3]]
 | [[File:OtonalPentad_22edo.mp3]]
 | [[File:OtonalPentad_29edo.mp3]]
 |-
-| 8:9:10:11:12 chord, in just intonation. <br> All intervals are slightly different.
+| 8:9:10:11:12 chord, in just intonation.<br />All intervals are slightly different.
-| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]]. <br> Except the first, the intervals are the same.
+| Porcupine-tempered 8:9:10:11:12 chord, in [[22edo]].<br />Except the first, the intervals are the same.
-| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]]. <br> Except the first, the intervals are the same.
+| Porcupine-tempered 8:9:10:11:12 chord, in [[29edo]].<br />Except the first, the intervals are the same.
 |}
@@ Line 249: / Line 251: @@
 | POTE: ~10/9 = 163.9504{{c}}
 |-
-! Benedetti, <br>Wilson
+! Benedetti, <br />Wilson
 | CBE: ~10/9 = 164.3761{{c}}
 | CSBE: ~10/9 = 164.3761{{c}}
@@ Line 273: / Line 275: @@
 | POTE: ~11/10 = 164.0777{{c}}
 |-
-! Benedetti, <br>Wilson
+! Benedetti, <br />Wilson
 | CBE: ~11/10 = 164.2393{{c}}
 | CSBE: ~11/10 = 164.4623{{c}}
@@ Line 297: / Line 299: @@
 | POTE: ~11/10 = 162.7474{{c}}
 |-
-! Benedetti, <br>Wilson
+! Benedetti, <br />Wilson
 | CBE: ~11/10 = 163.5299{{c}}
 | CSBE: ~11/10 = 163.2310{{c}}
@@ Line 306: / Line 308: @@
 {| class="wikitable center-all left-4"
 |-
-! Edo<br>generator
+! Edo<br />generator
-! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
+! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]
 ! Generator (¢)
 ! Comments