Porcupine

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 linear temperament in the porcupine family that tempers out 250/243, the porcupine comma, and whose generator is usually 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.

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, 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 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

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
Weight-skew\Order	Euclidean
Tenney	CTE ~10/9 = 164.1659¢
Tenney-Weil	CTWE ~10/9 = 164.0621¢
Equilateral	CEE ~10/9 = 163.6049¢ Eigenmonzo basis (unchanged-interval basis): 2.84375
Equilateral-Weil	CEWE ~10/9 = 163.2835¢ Eigenmonzo basis (unchanged-interval basis): 2.375
Benedetti	CBE ~10/9 = 164.3761¢ Eigenmonzo basis (unchanged-interval basis): 2.30375
Benedetti-Weil	CBWE ~10/9 = 164.3761¢ Eigenmonzo basis (unchanged-interval basis): 2.30375

2.3.5.11 Subgroup Prime-Optimized Tunings
Weight-skew\Order	Euclidean
Tenney	CTE ~11/10 = 163.8867¢
Tenney-Weil	CTWE ~11/10 = 163.9951¢
Equilateral	CEE ~11/10 = 163.1459¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 3 5 4⟩
Equilateral-Weil	CEWE ~11/10 = 162.8445¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 3 13 8⟩
Benedetti	CBE ~11/10 = 164.2393¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 605 363 60⟩
Benedetti-Weil	CBWE ~11/10 = 164.4623¢ Eigenmonzo basis (unchanged-interval basis): 2.[0 -1595 -957 90⟩

Tuning spectra

Tuning spectrum of 13-limit porcupine
Edo generator	Eigenmonzo (Unchanged-Interval)	Generator (¢)	Comments
	13/12	138.573
	13/11	144.605
1\8		150.000	Lower bound of 5-odd-limit diamond monotone
	12/11	150.637	Lower bound of 11-odd-limit diamond tradeoff
	13/10	151.405
	6/5	157.821	Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff
	15/13	158.710
	18/13	159.154
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	13- and 15-odd-limit minimax
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
	14/13	166.037
1\7	11/9	171.429	Upper bound of 5-odd-limit diamond monotone
	11/9	173.704
	16/13	179.736
	10/9	182.404	Upper bound of 9- and 11-odd-limit diamond tradeoff

Tuning spectrum of porcupinefish
Edo generator	Eigenmonzo (Unchanged-Interval)	Generator (¢)	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	5- and 7-odd-limit minimax
	15/14	162.897
	7/6	162.986
3\22		163.636
	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
	11/9	173.704
	10/9	182.404

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.