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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-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)

porcupine.png

Interval chain

Main article: Porcupine intervals
Generators
Cents
Ratios
Ups and Downs
notation
Generators
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
vP5 = ^^d5
-4
549.00
15/11~11/8
^P4 = 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
vP8 = ^^d8
-7
60.75
81/80~25/24
^P1 = 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
vP4 = ^^d4
-10
772.50
11/7
^P5 = vvA5
11
590.25
7/5
^d5 = vvP5
-11
609.75
10/7
vA4 = ^^P4
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.


Spectrum of Porcupine Tunings by Eigenmonzos

Eigenmonzo
Neutral Second

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-limit minimax
8\59
162.712
5/4
162.737 5-limit minimax
15/14
162.897
7/6
162.986
3\22
163.636
9/7
163.743 7- 9- and 11-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
[8/5 12/7] eigenmonzos: porcupinewoo15 porcupinewoo22

Spectrum of Porcupinefish Tunings

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-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

Chords of porcupine
Porcupine Notation
Porcupine modes
Porcupine Album Project

Musical examples

Images

porcupine8.jpg