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 that tempers out 250/243, the porcupine or Triyo comma, and whose generator is a minor whole tone (10/9) which is tuned flat to around 160–170 cents such that two of them stack to a classic minor third (6/5). Its pergen is (P8, P4/3). It can be thought of as a 5-limit, 7-limit, or 11-limit temperament, or a 2.3.5.11 subgroup temperament (sometimes known as porkypine). It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.

Porcupine's basic 5-limit harmonic structure can be understood by noting that tempering out 250/243 also makes (4/3)² equivalent to (6/5)³; or, in other words, two "perfect fourths" are equivalent to 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. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many of porcupine's scales.

See Porcupine family #Porcupine for technical data.

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	Ups and downs notation	#	Cents	Ratios	Ups and downs notation
0	0.0	1/1	P1	0	1200.0	2/1	P8
1	162.8	10/9, 11/10, 12/11	vM2 = ^^m2	-1	1037.2	9/5, 11/6, 20/11	^m7 = vvM7
2	325.6	6/5, 11/9	^m3 = vvM3	-2	874.4	5/3, 18/11	vM6 = ^^m6
3	488.4	4/3	P4	-3	711.6	3/2	P5
4	651.3	16/11, 22/15	v5 = ^^d5	-4	548.7	11/8, 15/11	^4 = vvA4
5	814.1	8/5	^m6 = vvM6	-5	385.9	5/4	vM3 = ^^m3
6	976.9	7/4, 16/9	m7	-6	223.1	8/7, 9/8	M2
7	1139.7	48/25, 160/81	v8 = ^^d8	-7	60.3	25/24, 81/80	^1 = vvA1
8	102.5	16/15, 21/20	^m2 = vvM2	-8	1097.5	15/8, 40/21	vM7 = ^^m7
9	265.3	7/6	m3	-9	934.7	12/7	M6
10	428.2	14/11	v4 = ^^d4	-10	771.8	11/7	^5 = vvA5
11	591.0	7/5	^d5 = vv5	-11	609.0	10/7	vA4 = ^^4
12	753.8	14/9	m6	-12	446.2	9/7	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". 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 60.3 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

• Porkypine7
• Porkypine8
• Porkypine15

Mos scales, 8/5.12/7 eigenmonzo (unchanged-interval) tuning

• Porcupinewoo15
• Porcupinewoo22

Tunings

5-limit Prime-Optimized Tunings

Weight-skew\Order	Euclidean
Tenney	CTE: ~10/9 = 164.1659¢
Weil	CWE: ~10/9 = 164.0621¢
Equilateral	CEE: ~10/9 = 163.6049¢
Skewed-equilateral	CSEE: ~10/9 = 163.2835¢
Benedetti/Wilson	CBE: ~10/9 = 164.3761¢
Skewed-Benedetti/Wilson	CSBE: ~10/9 = 164.3761¢

2.3.5.11 Subgroup Prime-Optimized Tunings

Weight-skew\Order	Euclidean
Tenney	CTE: ~11/10 = 163.8867¢
Weil	CWE: ~11/10 = 163.9951¢
Equilateral	CEE: ~11/10 = 163.1459¢
Skewed-equilateral	CSEE: ~11/10 = 162.8445¢
Benedetti/Wilson	CBE: ~11/10 = 164.2393¢
Skewed-Benedetti/Wilson	CSBE: ~11/10 = 164.4623¢

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		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.

See also

• Porcupine Notation
• Porcupine modes
• Porcupine Album Project

Music

20th century

Herman Miller

• Mizarian Porcupine Overture (1999) – in 15edo tuning, namesake of the temperament

21st century

Flora Canou

• "April Porkfest" from TOTMC Suite Vol. 1 (2023) – in 11-limit CTE tuning

CellularAutomaton

• Minnow (2024) – in 29edo tuning

Paul Erlich

• Glassic – in 22edo tuning (at least the beginning part is in porcupine.)

Jake Freivald

• Porcupine Comma Pump^{[dead link]}

Cody Hallenbeck

• Porcupine Walk (2019)
  • 15edo version · 22edo version

Lillian Hearne

• Sanctus (2015)

Andrew Heathwaite

• being a (2010) – in Porcupine[8], mode 1|6, 22edo tuning

Jollybard

• Porcupeen (2017)
• "Porcupine", from pato, with friends (2019)

Igliashon Jones

• Second Breakfast (15edo) (2018)^{[dead link]}

Löis Lancaster

• Porcupine Experience (2012) – in 22edo tuning

John Moriarty

• Flying Straight Down (2020) – in 22edo tuning

Omega9

• Life on Mars (2014)

Petr Pařízek

• Among Other Things 2

Ray Perlner

• Porcupine Lullaby (2020) – in 37edo tuning
• Porcupine[7] Modal Fugues – 7-piece playlist

Gene Ward Smith and Modest Mussorgsky

• Night on Porcupine Mountain (archived 2010) – in 22edo tuning

Chris Vaisvil

• Playing Gently with Miller's Porcupine
• 15 Porcupines in India – sarangi, tambura and sitar improvisation
• 15 Quills – piano solo
• Prickly Side of Love – rock band with vocals
• Porcupine Organ Composition

Nick Vuci

• Porcupine Prelude 1 – in 22edo tuning
• Porcupine Prelude 2 – in 22edo tuning
• Porcupine Prelude 3 – in 22edo tuning
• Porcupine Praeambulum – in 22edo tuning
• Porcupine Chorale with Prelude "Nature's Lament" – in 22edo tuning

Well-Tempered Fox

• Porcupine Major Overture (2015) – in 22edo tuning
• Waltzing in Candyland (2015) – in Porcupine[8], 15edo tuning

Diagrams