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)

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

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

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 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]]. 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 [[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 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]]).

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.

Line 41:

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

|-

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

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

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

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

|-

! #

! 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

Line 213:

| [[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.

|}

Line 251:

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

|-

! Benedetti, Wilson

! Benedetti, Wilson

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

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

Line 275:

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

|-

! Benedetti, Wilson

! Benedetti, Wilson

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

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

Line 299:

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

|-

! Benedetti, Wilson

! Benedetti, Wilson

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

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

Line 308:

{| 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 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 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]].  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 [[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 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]]).
+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.
@@ Line 41: / Line 41: @@
 {| class="wikitable center-all right-2 left-3 right-7 left-8"
 |-
-! colspan="5" | Up from the tonic, aka fourthward
+! colspan="5" | Up from the tonic, and fourthward
-! colspan="5" | Down from the octave, aka fifthward
+! colspan="5" | Down from the octave, and fifthward
 |-
 ! #
 ! 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 213: / Line 213: @@
 | [[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 251: / 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 275: / 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 299: / 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 308: / 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