Porcupine

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

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	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.

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

• Mizarian Porcupine Overture, Herman Miller, 1999. (15edo, namesake of the temperament)
• Glassic, Paul Erlich, 22edo (at least the beginning part is in porcupine).
• Night on Porcupine Mountain, Gene Ward Smith and Modest Mussorgsky, 22edo.
• being a, Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8].
• Playing Gently with Miller's Porcupine by Chris Vaisvil
• 15 Porcupines in India, Sarangi, Tambura and Sitar improvisation by Chris Vaisvil
• 15 Quills piano solo by Chris Vaisvil
• Prickly Side of Love - rock band in Porcupine Temperament with vocals by Chris Vaisvil
• Porcupine Organ Composition by Chris Vaisvil
• Among Other Things 2 by Petr Pařízek
• Porcupine Comma Pump, by Jake Freivald
• Life on Mars by Omega9
• Flying Straight Down by John Moriarty

Images