Porcupine

Porcupine is a linear temperament that equates a stack of three 6/5s to a stack of two 4/3s. 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. As a result, the generator of porcupine is a minor whole tone (10/9) which is tuned flat to around 160–165 cents such that two of them stack to a classic minor third (6/5). This is obviously in stark contrast to 12edo. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

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

In just intonation, a stack of three 6/5s is a classical diminished seventh, 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. This 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 (i.e. meantone, schismic), by tempering out 250/243, the porcupine comma. Its pergen is (P8, P4/3).

To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain:

7 is found at +6 generators (tuned to about 960-990 cents), because the 16/9 has already been flattened to merge it with (6/5)³. This makes porcupine an archy temperament.
11 is found at -4 generators (tuned to about 540-560 cents), corresponding with the acute fourth of 27/20; 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. This can also be understood as the generator 10/9 being flattened.

Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called porkypine). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy.

In the 7-limit, porcupine can be seen as as 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 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. An interval in the neutral third range is not found for a long time (until 17 generators up), and as a result that interval varies drastically depending on the tuning. 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).

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.