Porcupine

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

Porcupine is a temperament that is generated by a minor whole tone (10/9), tuned flat to around 160–165 cents, two of which represent 6/5 and three of which represent 4/3, tempering out 250/243, the porcupine comma. 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. Its pergen is (P8, P4/3). This is obviously in stark contrast to meantone temperaments, including 12edo, where the 10/9 interval is sharpened to merge with 9/8. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

One may also note that in just intonation, a stack of three 6/5's is flat of the classical minor seventh 9/5 by 25/24, and a stack of two 4/3's 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, which 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 (such as meantone and schismic).

Porcupine can be thought of as a 2.3.5.11-subgroup temperament (sometimes called porkypine) without much additional damage compared to the 5-limit; the generator here represents not only 10/9, but also 11/10 and 12/11 (equivalently, 55/54, 100/99, and 121/120 are tempered out), with the consequence that 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. 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), found at -4 generators (tuned to about 540–560 cents). This is because 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. Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup at a certain standard of accuracy.

It is also very easy to extend porcupine to prime 7, because the 16/9, found at +6 generators (tuned to about 960–990 cents), has already been flattened to merge it with (6/5)³, and therefore can be equated to 7/4. This makes porcupine 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

* In 11-limit CWE tuning, octave reduced

Besides the specific tuning shown here, 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.

Chords and harmony

Main article: Chords of porcupine

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 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.

Scales

Main article: 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	Destretched
Equilateral	CEE: ~10/9 = 163.6049 ¢	CSEE: ~10/9 = 163.2835 ¢	POEE: ~10/9 = 163.9280 ¢
Tenney	CTE: ~10/9 = 164.1659 ¢	CWE: ~10/9 = 164.0621 ¢	POTE: ~10/9 = 163.9504 ¢
Benedetti, Wilson	CBE: ~10/9 = 164.3761 ¢	CSBE: ~10/9 = 164.3761 ¢	POBE: ~10/9 = 164.1610 ¢

2.3.5.11-subgroup prime-optimized tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Equilateral	CEE: ~11/10 = 163.1459 ¢	CSEE: ~11/10 = 162.8445 ¢	POEE: ~11/10 = 164.1867 ¢
Tenney	CTE: ~11/10 = 163.8867 ¢	CWE: ~11/10 = 163.9951 ¢	POTE: ~11/10 = 164.0777 ¢
Benedetti, Wilson	CBE: ~11/10 = 164.2393 ¢	CSBE: ~11/10 = 164.4623 ¢	POBE: ~11/10 = 164.2221 ¢

11-limit prime-optimized tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Equilateral	CEE: ~11/10 = 162.4448 ¢	CSEE: ~11/10 = 162.2333 ¢	POEE: ~11/10 = 162.2522 ¢
Tenney	CTE: ~11/10 = 163.1055 ¢	CWE: ~11/10 = 162.8156 ¢	POTE: ~11/10 = 162.7474 ¢
Benedetti, Wilson	CBE: ~11/10 = 163.5299 ¢	CSBE: ~11/10 = 163.2310 ¢	POBE: ~11/10 = 163.0304 ¢

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.