Porcupine
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
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¢ |
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
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 |
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
- Mizarian Porcupine Overture (1999) – in 15edo, namesake of the temperament
21st century
- "April Porkfest" from TOTMC Suite Vol. 1 (2023) – 11-limit CTE tuning
- Porcupine Walk (2019)
- Sanctus (2015)
- being a (2010) – in 22edo, mode 3 1 3 3 3 3 3 3 of porcupine[8]
- Second Breakfast (15edo) (2018) [dead link]
- Porcupine Experience (2012) – in 22edo
- Flying Straight Down (2020) – in 22edo
- Life on Mars (2014)
- Porcupine Lullaby – in 37edo
- Porcupine(7) Modal Fugues – 7-piece playlist
- Night on Porcupine Mountain (archived 2010) – in 22edo
- 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 in Porcupine Temperament with vocals
- Porcupine Organ Composition
- Porcupine Prelude 1 – in 22edo
- Porcupine Prelude 2 – in 22edo
- Porcupine Prelude 3 – in 22edo
- Porcupine Praeambulum – in 22edo
- Porcupine Chorale with Prelude "Nature's Lament" – in 22edo
- Porcupine Major Overture (2015) – in 22edo
- Waltzing in Candyland (2015) – in 15edo porcupine[8]