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


== Interval chain ==
== Interval chain ==
{{Main|Porcupine intervals}}
{{Main| Porcupine intervals }}


In the following table, odd harmonics 1–11 are in '''bold'''.  
In the following table, odd harmonics 1–11 are in '''bold'''.  
Line 32: Line 30:
! Cents
! Cents
! Ratios
! Ratios
!Porcupine notation
! Porcupine notation
! Ups and downs<br>notation
! Ups and downs<br>notation
! #
! #
! Cents
! Cents
! Ratios
! Ratios
!Porcupine notation
! Porcupine notation
! Ups and downs<br>notation
! Ups and downs<br>notation
|-
|-
Line 43: Line 41:
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|P1
| P1
| P1
| P1
| 0
| 0
| 1200.0
| 1200.0
| '''2/1'''
| '''2/1'''
|P8
| P8
| P8
| P8
|-
|-
Line 54: Line 52:
| 162.8
| 162.8
| 10/9, 11/10, 12/11
| 10/9, 11/10, 12/11
|P2
| P2
| vM2 = ^^m2
| vM2 = ^^m2
| -1
| -1
| 1037.2
| 1037.2
| 9/5, 11/6, 20/11
| 9/5, 11/6, 20/11
|P7
| P7
| ^m7 = vvM7
| ^m7 = vvM7
|-
|-
Line 65: Line 63:
| 325.6
| 325.6
| 6/5, 11/9
| 6/5, 11/9
|m3
| m3
| ^m3 = vvM3
| ^m3 = vvM3
| -2
| -2
| 874.4
| 874.4
| 5/3, 18/11
| 5/3, 18/11
|M6
| M6
| vM6 = ^^m6
| vM6 = ^^m6
|-
|-
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| 488.4
| 488.4
| 4/3
| 4/3
|m4
| m4
| P4
| P4
| -3
| -3
| 711.6
| 711.6
| '''3/2'''
| '''3/2'''
|M5
| M5
| P5
| P5
|-
|-
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| 651.3
| 651.3
| 16/11, 22/15
| 16/11, 22/15
|m5
| m5
| v5 = ^^d5
| v5 = ^^d5
| -4
| -4
| 548.7
| 548.7
| '''11/8''', 15/11
| '''11/8''', 15/11
|M4
| M4
| ^4 = vvA4
| ^4 = vvA4
|-
|-
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| 814.1
| 814.1
| 8/5
| 8/5
|m6
| m6
| ^m6 = vvM6
| ^m6 = vvM6
| -5
| -5
| 385.9
| 385.9
| '''5/4'''
| '''5/4'''
|M3
| M3
| vM3 = ^^m3
| vM3 = ^^m3
|-
|-
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| 976.9
| 976.9
| '''7/4''', 16/9
| '''7/4''', 16/9
|d7
| d7
| m7
| m7
| -6
| -6
| 223.1
| 223.1
| 8/7, '''9/8'''
| 8/7, '''9/8'''
|A2
| A2
| M2
| M2
|-
|-
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| 1139.7
| 1139.7
| 48/25, 160/81
| 48/25, 160/81
|d8
| d8
| v8 = ^^d8
| v8 = ^^d8
| -7
| -7
| 60.3
| 60.3
| 25/24, 81/80
| 25/24, 81/80
|A1
| A1
| ^1 = vvA1
| ^1 = vvA1
|-
|-
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| 102.5
| 102.5
| 16/15, 21/20
| 16/15, 21/20
|d2
| d2
| ^m2 = vvM2
| ^m2 = vvM2
| -8
| -8
| 1097.5
| 1097.5
| 15/8, 40/21
| 15/8, 40/21
|A7
| A7
| vM7 = ^^m7
| vM7 = ^^m7
|-
|-
Line 142: Line 140:
| 265.3
| 265.3
| 7/6
| 7/6
|d3
| d3
| m3
| m3
| -9
| -9
| 934.7
| 934.7
| 12/7
| 12/7
|A6
| A6
| M6
| M6
|-
|-
Line 153: Line 151:
| 428.2
| 428.2
| 14/11
| 14/11
|d4
| d4
| v4 = ^^d4
| v4 = ^^d4
| -10
| -10
| 771.8
| 771.8
| 11/7
| 11/7
|A5
| A5
| ^5 = vvA5
| ^5 = vvA5
|-
|-
Line 164: Line 162:
| 591.0
| 591.0
| 7/5
| 7/5
|d5
| d5
| ^d5 = vv5
| ^d5 = vv5
| -11
| -11
| 609.0
| 609.0
| 10/7
| 10/7
|A4
| A4
| vA4 = ^^4
| vA4 = ^^4
|-
|-
Line 175: Line 173:
| 753.8
| 753.8
| 14/9
| 14/9
|d6
| d6
| m6
| m6
| -12
| -12
| 446.2
| 446.2
| 9/7
| 9/7
|A3
| A3
| M3
| M3
|}
|}
Line 386: Line 384:


== History ==
== 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 ==
== See also ==
* [[Porcupine Notation]]
* [[Porcupine notation]]
* [[Porcupine modes]]
* [[Porcupine modes]]
* [[Porcupine Album Project]]
* [[Porcupine Album Project]]

Revision as of 09:11, 21 April 2025

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.
Symmetric minor mode of the Porcupine[7] scale, containing two equal tetrachords with a major wholetone between them, in 22edo tuning.

Porcupine is a temperament that is generated by a minor whole tone (10/9), tuned flat to around 160–165 cents, 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 6/5'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 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.

See Porcupine family #Porcupine for technical data. See Porcupine extensions for a discussion on 13-limit extensions.

Interval chain

Main article: Porcupine intervals

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

Up from the tonic, aka fourthward Down from the octave, aka fifthward
# Cents Ratios Porcupine notation Ups and downs
notation 		# Cents Ratios Porcupine notation Ups and downs
notation
0 0.0 1/1 P1 P1 0 1200.0 2/1 P8 P8
1 162.8 10/9, 11/10, 12/11 P2 vM2 = ^^m2 -1 1037.2 9/5, 11/6, 20/11 P7 ^m7 = vvM7
2 325.6 6/5, 11/9 m3 ^m3 = vvM3 -2 874.4 5/3, 18/11 M6 vM6 = ^^m6
3 488.4 4/3 m4 P4 -3 711.6 3/2 M5 P5
4 651.3 16/11, 22/15 m5 v5 = ^^d5 -4 548.7 11/8, 15/11 M4 ^4 = vvA4
5 814.1 8/5 m6 ^m6 = vvM6 -5 385.9 5/4 M3 vM3 = ^^m3
6 976.9 7/4, 16/9 d7 m7 -6 223.1 8/7, 9/8 A2 M2
7 1139.7 48/25, 160/81 d8 v8 = ^^d8 -7 60.3 25/24, 81/80 A1 ^1 = vvA1
8 102.5 16/15, 21/20 d2 ^m2 = vvM2 -8 1097.5 15/8, 40/21 A7 vM7 = ^^m7
9 265.3 7/6 d3 m3 -9 934.7 12/7 A6 M6
10 428.2 14/11 d4 v4 = ^^d4 -10 771.8 11/7 A5 ^5 = vvA5
11 591.0 7/5 d5 ^d5 = vv5 -11 609.0 10/7 A4 vA4 = ^^4
12 753.8 14/9 d6 m6 -12 446.2 9/7 A3 M3

The specific tuning shown is the full 11-limit CWE tuning, but of course there is a range of acceptable porcupine tunings that includes generators as small as 160 cents (15edo) and as large as 165.5 cents (29edo). However, the 29edo patent val does not support 11-limit porcupine proper, since it does not temper out 64/63.

12/11, 11/10, and 10/9 are all represented by the same interval, the generator. This makes chords such as 8:9:10:11:12 exceptionally common and easy to find.

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 29edo.
Except the first, the intervals are the same.

The interval representing both 25/24 and 81/80 can be found in this interval chain at -7 steps, and ranges from about 45 to 80 cents depending on the tuning. This can be considered the "chroma" of porcupine temperament.

Chords

Main article: Chords of porcupine

Scales

Mos scales, tuning optimized on the 2.3.5.11 subgroup
Mos scales, 8/5.12/7 eigenmonzo (unchanged-interval) tuning

Tunings

5-limit Prime-optimized tunings
Euclidean
Constrained Constrained & skewed
Equilateral CEE: ~10/9 = 163.6049 ¢ CSEE: ~10/9 = 163.2835 ¢
Tenney CTE: ~10/9 = 164.1659 ¢ CWE: ~10/9 = 164.0621 ¢
Benedetti,
Wilson 		CBE: ~10/9 = 164.3761 ¢ CSBE: ~10/9 = 164.3761 ¢
2.3.5.11-subgroup prime-optimized tunings
Euclidean
Constrained Constrained & skewed
Equilateral CEE: ~11/10 = 163.1459 ¢ CSEE: ~11/10 = 162.8445 ¢
Tenney CTE: ~11/10 = 163.8867 ¢ CWE: ~11/10 = 163.9951 ¢
Benedetti,
Wilson 		CBE: ~11/10 = 164.2393 ¢ CSBE: ~11/10 = 164.4623 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(Unchanged-interval) 		Generator (¢) Comments
1\8 150.000 Lower bound of 5-odd-limit diamond monotone
12/11 150.637 Lower bound of 11-odd-limit diamond tradeoff
6/5 157.821 Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
2\15 160.000 Lower bound of 7-, 9-, and 11-odd-limit diamond monotone
8/7 161.471
14/11 161.751
7/5 162.047
5\37 162.162
11/8 162.171
8\59 162.712
5/4 162.737 5- and 7-odd-limit minimax
15/14 162.897
7/6 162.986
3\22 163.636 Upper bound of 7-, 9-, and 11-odd-limit diamond monotone
9/7 163.743 9- and 11-odd-limit minimax
16/15 163.966
7\51 164.706
11/10 165.004
4\29 165.517
15/11 165.762
4/3 166.015 Upper bound of 5- and 7-odd-limit diamond tradeoff
1\7 171.429 Upper bound of 5-odd-limit diamond monotone
11/9 173.704
10/9 182.404 Upper bound of 9- and 11-odd-limit diamond tradeoff

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.

See also

Music

20th century

Herman Miller

21st century

Flora Canou
CellularAutomaton
Paul Erlich
  • Glassic – in 22edo tuning (at least the beginning part is in porcupine.)
Jake Freivald
Cody Hallenbeck
Lillian Hearne
Andrew Heathwaite
  • being a (2010) – in Porcupine[8], mode 1|6, 22edo tuning
Jollybard
Igliashon Jones
Löis Lancaster
John Moriarty
Omega9
Petr Pařízek
Ray Perlner
Gene Ward Smith and Modest Mussorgsky
Chris Vaisvil
Nick Vuci
Well-Tempered Fox

Diagrams

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