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'''Porcupine''' is a [[linear temperament]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-[[Harmonic Limit|limit]], 7-limit, or 11-limit temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
'''Porcupine''' is a [[linear temperament]] in the [[porcupine family]] that tempers out [[250/243]], the porcupine [[comma]], and whose generator is somewhere around 160–165 cents. It can be thought of as a 5-[[Harmonic Limit|limit]], 7-limit, or 11-limit temperament, or a 2.3.5.11 [[subgroup temperament]]. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.


The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)^2 equivalent to (6/5)^3. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". 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. This is obviously in stark contrast to [[12edo]], and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.
The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)<sup>2</sup> equivalent to (6/5)<sup>3</sup>. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". 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. This is obviously in stark contrast to [[12edo]], and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.


[[File:porcupinesymmetricminor22edo.mp3]]
[[File:porcupinesymmetricminor22edo.mp3]]
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{{Main|Porcupine intervals}}
{{Main|Porcupine intervals}}


{| class="wikitable center-all right-2 right-6"
{| class="wikitable center-all right-2 left-3 right-6 left-7"
! Generators
! #
! Cents
! Cents
! Ratios
! Ratios
! Ups and Downs <br> notation
! Ups and Downs <br> notation
! Generators
! #
! 2/1 inverse
! 2/1 inverse
! Ratios
! Ratios
Line 40: Line 40:
| 1
| 1
| 162.75
| 162.75
| 12/11~11/10~10/9
| 12/11, 11/10, 10/9
| vM2 = ^^m2
| vM2 = ^^m2
| -1
| -1
| 1037.25
| 1037.25
| 9/5~20/11~11/6
| 9/5, 20/11, 11/6
| ^m7 = vvM7
| ^m7 = vvM7
|-
|-
| 2
| 2
| 325.50
| 325.50
| 6/5~11/9
| 6/5, 11/9
| ^m3 = vvM3
| ^m3 = vvM3
| -2
| -2
| 874.50
| 874.50
| 18/11~5/3
| 18/11, 5/3
| vM6 = ^^m6
| vM6 = ^^m6
|-
|-
Line 67: Line 67:
| 4
| 4
| 651.00
| 651.00
| 16/11~22/15
| 16/11, 22/15
| v5 = ^^d5
| v5 = ^^d5
| -4
| -4
| 549.00
| 549.00
| 15/11~11/8
| 15/11, 11/8
| ^4 = vvA4
| ^4 = vvA4
|-
|-
Line 85: Line 85:
| 6
| 6
| 976.50
| 976.50
| 7/4~16/9
| 7/4, 16/9
| m7
| m7
| -6
| -6
| 223.50
| 223.50
| 9/8~8/7
| 9/8, 8/7
| M2
| M2
|-
|-
| 7
| 7
| 1139.25
| 1139.25
| 48/25~160/81
| 48/25, 160/81
| v8 = ^^d8
| v8 = ^^d8
| -7
| -7
| 60.75
| 60.75
| 81/80~25/24
| 81/80, 25/24
| ^1 = vvA1
| ^1 = vvA1
|-
|-
| 8
| 8
| 102.00
| 102.00
| 16/15~21/20
| 16/15, 21/20
| ^m2 = vvM2
| ^m2 = vvM2
| -8
| -8
| 1098.00
| 1098.00
| 40/21~15/8
| 40/21, 15/8
| vM7 = ^^m7
| vM7 = ^^m7
|-
|-
Line 164: Line 164:
The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from &lt;50 to 80 cents in general, represents both [[25/24]] and [[81/80]].
The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from &lt;50 to 80 cents in general, represents both [[25/24]] and [[81/80]].


== Spectrum of Porcupine Tunings by Eigenmonzos ==
== Chords ==
{{main| Chords of porcupine }}


{| class="wikitable"
== Scales ==
; [8/5 12/7] eigenmonzos:
* [[porcupinewoo15]]
* [[porcupinewoo22]]
 
== Spectrum of porcupine tunings by eigenmonzos ==
 
{| class="wikitable center-1 center-2"
! Eigenmonzo
! Eigenmonzo
! Neutral Second
! Neutral Second
! Comments
|-
|-
| 13/12
| 13/12
| 138.573
| 138.573
|
|-
|-
| 13/11
| 13/11
| 144.605
| 144.605
|
|-
|-
| 12/11
| 12/11
| 150.637
| 150.637
|
|-
|-
| 13/10
| 13/10
| 151.405
| 151.405
|
|-
|-
| 6/5
| 6/5
| 157.821
| 157.821
|
|-
|-
| 15/13
| 15/13
| 158.710
| 158.710
|
|-
|-
| 18/13
| 18/13
| 159.154
| 159.154
|
|-
|-
| 2\15
| (2\15)
| 160.000
| 160.000
|
|-
|-
| 8/7
| 8/7
| 161.471
| 161.471
|
|-
|-
| 14/11
| 14/11
| 161.751
| 161.751
|
|-
|-
| 7/5
| 7/5
| 162.047
| 162.047
|
|-
|-
| 5\37
| (5\37)
| 162.162
| 162.162
|
|-
|-
| 11/8
| 11/8
| 162.171 13- and 15-limit minimax
| 162.171
| 13- and 15-odd-limit minimax
|-
|-
| 8\59
| (8\59)
| 162.712
| 162.712
|-
|-
| 5/4
| 5/4
| 162.737 5-limit minimax
| 162.737
| 5-odd-limit minimax
|-
|-
| 15/14
| 15/14
| 162.897
| 162.897
|
|-
|-
| 7/6
| 7/6
| 162.986
| 162.986
|
|-
|-
| 3\22
| (3\22)
| 163.636
| 163.636
|
|-
|-
| 9/7
| 9/7
| 163.743 7- 9- and 11-limit minimax
| 163.743
| 7-, 9- and 11-odd-limit minimax
|-
|-
| 16/15
| 16/15
| 163.966
| 163.966
|
|-
|-
| 7\51
| (7\51)
| 164.706
| 164.706
|
|-
|-
| 11/10
| 11/10
| 165.004
| 165.004
|
|-
|-
| 4\29
| (4\29)
| 165.517
| 165.517
|
|-
|-
| 15/11
| 15/11
| 165.762
| 165.762
|
|-
|-
| 4/3
| 4/3
| 166.015
| 166.015
|
|-
|-
| 14/13
| 14/13
| 166.037
| 166.037
|
|-
|-
| 11/9
| 11/9
| 173.704
| 173.704
|
|-
|-
| 16/13
| 16/13
| 179.736
| 179.736
|
|-
|-
| 10/9
| 10/9
| 182.404
| 182.404
|
|}
|}


[8/5 12/7] eigenmonzos: [[porcupinewoo15|porcupinewoo15]] [[porcupinewoo22|porcupinewoo22]]
=== Spectrum of porcupinefish tunings ===


=== Spectrum of Porcupinefish Tunings ===
{| class="wikitable center-1 center-2"
 
! Eigenmonzo
{| class="wikitable"
! Neutral Second
! Comments
|-
| 12/11
| 12/11
| 150.637
| 150.637
|
|-
|-
| 6/5
| 6/5
| 157.821
| 157.821
|
|-
|-
| 2\15
| (2\15)
| 160.000
| 160.000
|
|-
|-
| 18/13
| 18/13
| 160.307
| 160.307
|
|-
|-
| 15/13
| 15/13
| 160.860
| 160.860
|
|-
|-
| 8/7
| 8/7
| 161.471
| 161.471
|
|-
|-
| 13/12
| 13/12
| 161.531
| 161.531
|
|-
|-
| 14/11
| 14/11
| 161.751
| 161.751
|
|-
|-
| 7/5
| 7/5
| 162.047
| 162.047
|
|-
|-
| 14/13
| 14/13
| 162.100
| 162.100
|
|-
|-
| 13/10
| 13/10
| 162.149
| 162.149
|
|-
|-
| 5\37
| (5\37)
| 162.162
| 162.162
|
|-
|-
| 11/8
| 11/8
| 162.171
| 162.171
|
|-
|-
| 16/13
| 16/13
| 162.322
| 162.322
|
|-
|-
| 13/11
| 13/11
| 162.368 13- and 15-limit minimax
| 162.368
| 13- and 15-odd-limit minimax
|-
|-
| 8\59
| (8\59)
| 162.712
| 162.712
|
|-
|-
| 5/4
| 5/4
| 162.737
| 162.737
|
|-
|-
| 15/14
| 15/14
| 162.897
| 162.897
|
|-
|-
| 7/6
| 7/6
| 162.986
| 162.986
|
|-
|-
| 3\22
| (3\22)
| 163.636
| 163.636
|
|-
|-
| 9/7
| 9/7
| 163.743
| 163.743
|
|-
|-
| 16/15
| 16/15
| 163.966
| 163.966
|
|-
|-
| 7\51
| (7\51)
| 164.706
| 164.706
|
|-
|-
| 11/10
| 11/10
| 165.004
| 165.004
|
|-
|-
| 4\29
| (4\29)
| 165.517
| 165.517
|
|-
|-
| 15/11
| 15/11
| 165.762
| 165.762
|
|-
|-
| 4/3
| 4/3
| 166.015
| 166.015
|
|-
|-
| 11/9
| 11/9
| 173.704
| 173.704
|
|-
|-
| 10/9
| 10/9
| 182.404
| 182.404
|
|}
|}


Line 356: Line 424:


== See also ==
== See also ==
* [[Chords of porcupine]]
* [[Porcupine Notation]]
* [[Porcupine Notation]]
* [[Porcupine modes]]
* [[Porcupine modes]]
Line 391: Line 457:


[[Category:Temperaments]]
[[Category:Temperaments]]
[[Category:Porcupine| ]] <!-- main article -->
[[Category:Porcupine family]]
[[Category:Porcupine family]]
[[Category:Porcupine| ]] <!-- main article -->
[[Category:Archytas]]

Revision as of 14:56, 27 April 2021

Porcupine is a linear temperament in the porcupine family that tempers out 250/243, the porcupine comma, and whose generator is somewhere around 160–165 cents. It can be thought of as a 5-limit, 7-limit, or 11-limit temperament, or a 2.3.5.11 subgroup temperament. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)2 equivalent to (6/5)3. In perhaps more familiar musical terms, this means two "perfect fourths" equals three "minor thirds". 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. This is obviously in stark contrast to 12edo, and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.

Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them. (Tuning in 22edo)

Interval chain

Main article: Porcupine intervals
# Cents Ratios Ups and Downs
notation 		# 2/1 inverse Ratios Ups and Downs
notation
0 0.00 1/1 P1 0 1200.00 2/1 P8
1 162.75 12/11, 11/10, 10/9 vM2 = ^^m2 -1 1037.25 9/5, 20/11, 11/6 ^m7 = vvM7
2 325.50 6/5, 11/9 ^m3 = vvM3 -2 874.50 18/11, 5/3 vM6 = ^^m6
3 488.25 4/3 P4 -3 711.75 3/2 P5
4 651.00 16/11, 22/15 v5 = ^^d5 -4 549.00 15/11, 11/8 ^4 = vvA4
5 813.75 8/5 ^m6 = vvM6 -5 386.25 5/4 vM3 = ^^m3
6 976.50 7/4, 16/9 m7 -6 223.50 9/8, 8/7 M2
7 1139.25 48/25, 160/81 v8 = ^^d8 -7 60.75 81/80, 25/24 ^1 = vvA1
8 102.00 16/15, 21/20 ^m2 = vvM2 -8 1098.00 40/21, 15/8 vM7 = ^^m7
9 264.75 7/6 m3 -9 935.25 12/7 M6
10 427.50 14/11 v4 = ^^d4 -10 772.50 11/7 ^5 = vvA5
11 590.25 7/5 ^d5 = vv5 -11 609.75 10/7 vA4 = ^^4
12 753.00 14/9 m6 -12 447.00 9/7 M3

The specific tuning shown is the full 11-limit POTE tuning, but of course there is a range of acceptible 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, not annihilating 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 11/9 interval, usually considered a "neutral third", is in porcupine identical to the 6/5 "minor third". This 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).

The characteristic small interval of porcupine, which is 60.75 cents in this tuning but can range from <50 to 80 cents in general, represents both 25/24 and 81/80.

Chords

Main article: Chords of porcupine

Scales

[8/5 12/7] eigenmonzos

Spectrum of porcupine tunings by eigenmonzos

Eigenmonzo Neutral Second Comments
13/12 138.573
13/11 144.605
12/11 150.637
13/10 151.405
6/5 157.821
15/13 158.710
18/13 159.154
(2\15) 160.000
8/7 161.471
14/11 161.751
7/5 162.047
(5\37) 162.162
11/8 162.171 13- and 15-odd-limit minimax
(8\59) 162.712
5/4 162.737 5-odd-limit minimax
15/14 162.897
7/6 162.986
(3\22) 163.636
9/7 163.743 7-, 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
14/13 166.037
11/9 173.704
16/13 179.736
10/9 182.404

Spectrum of porcupinefish tunings

Eigenmonzo Neutral Second Comments
12/11 150.637
6/5 157.821
(2\15) 160.000
18/13 160.307
15/13 160.860
8/7 161.471
13/12 161.531
14/11 161.751
7/5 162.047
14/13 162.100
13/10 162.149
(5\37) 162.162
11/8 162.171
16/13 162.322
13/11 162.368 13- and 15-odd-limit minimax
(8\59) 162.712
5/4 162.737
15/14 162.897
7/6 162.986
(3\22) 163.636
9/7 163.743
16/15 163.966
(7\51) 164.706
11/10 165.004
(4\29) 165.517
15/11 165.762
4/3 166.015
11/9 173.704
10/9 182.404

History

Porcupine temperament/scales were discovered by Dave Keenan, but didn't have a name until Herman Miller mentioned that his Mizarian Porcupine Overture in 15-tET 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 15, 22 was a porcupine tuning par excellence, and that was an interesting development in itself.

See also

Musical examples

Images

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