Porcupine

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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

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 0.00 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

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¢
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 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.

See also

Music

20th century

Herman Miller

21st century

Flora Canou
CellularAutomaton
Paul Erlich
  • Glassic – in 22edo (at least the beginning part is in porcupine.)
Jake Freivald
Cody Hallenbeck
Lillian Hearne
Andrew Heathwaite
  • being a (2010) – in 22edo, mode 3 1 3 3 3 3 3 3 of porcupine[8]
Jollybard
Igliashon Jones
Löis Lancaster
John Moriarty
Omega9
Petr Pařízek
Ray Perlner
Gene Ward Smith and Modest Mussorgsky
Chris Vaisvil
Nick Vuci
Well-Tempered Fox

Diagrams

Porcupine8.jpg