Porcupine
Porcupine is a linear temperament in the porcupine family that tempers out 250/243, the porcupine comma, and whose generator is somewhere around 160-165 cents. It can be thought of as a 5-limit, 7-limit, or 11-limit temperament, or a 2.3.5.11 subgroup temperament. It is one of the best temperaments in the 2.3.5.11 subgroup, with a unique combination of efficiency and accuracy.
The basic 5-limit harmonic structure of porcupine can be understood simply by noting that tempering out 250/243 makes (4/3)^2 equivalent to (6/5)^3. In perhaps more familiar musical terms, this means two "perfect fourths" equals 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, and to meantone, in which neither 4/3 nor 6/5 can be divided into any number of equal parts. The "equal tetrachord" formed by dividing 4/3 into 3 equal parts is a characteristic feature of many porcupine scales.
File:Porcupinesymmetricminor22edo.mp3
Porcupine symmetric minor scale, containing two equal tetrachords with a major wholetone between them. (Tuning in 22edo)
Interval chain[edit]
Main article: Porcupine intervals
| Generators | Cents | Ratios | Ups and Downs
notation |
Generators | 2/1 inverse | 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, not annihilating 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.
| File:OtonalPentad JI.mp3 | File:OtonalPentad 22edo.mp3 | File:OtonalPentad 29edo.mp3 |
| 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.
Spectrum of Porcupine Tunings by Eigenmonzos[edit]
| Eigenmonzo | Neutral Second | |
|---|---|---|
| 13/12 | 138.573 | |
| 13/11 | 144.605 | |
| 12/11 | 150.637 | |
| 13/10 | 151.405 | |
| 6/5 | 157.821 | |
| 15/13 | 158.710 | |
| 18/13 | 159.154 | |
| 2\15 | 160.000 | |
| 8/7 | 161.471 | |
| 14/11 | 161.751 | |
| 7/5 | 162.047 | |
| 5\37 | 162.162 | |
| 11/8 | 162.171 13- and 15-limit minimax | |
| 8\59 | 162.712 | |
| 5/4 | 162.737 5-limit minimax | |
| 15/14 | 162.897 | |
| 7/6 | 162.986 | |
| 3\22 | 163.636 | |
| 9/7 | 163.743 7- 9- and 11-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 | |
| 14/13 | 166.037 | |
| 11/9 | 173.704 | |
| 16/13 | 179.736 | |
| 10/9 | 182.404 |
[8/5 12/7] eigenmonzos: porcupinewoo15 porcupinewoo22
Spectrum of Porcupinefish Tunings[edit]
| 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-limit minimax |
| 8\59 | 162.712 |
| 5/4 | 162.737 |
| 15/14 | 162.897 |
| 7/6 | 162.986 |
| 3\22 | 163.636 |
| 9/7 | 163.743 |
| 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[edit]
Porcupine temperament/scales were discovered by Dave Keenan, but didn't have a name until Herman Miller mentioned that his Mizarian Porcupine Overture in 15-tET 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 15, 22 was a porcupine tuning par excellence, and that was an interesting development in itself.
See also[edit]
Musical examples[edit]
- "Mizarian Porcupine Overture", Herman Miller, 1999. (15edo, namesake of the temperament)
- "Glassic", Paul Erlich, 22edo (at least the beginning part is in porcupine).
- "Night on Porcupine Mountain", Gene Ward Smith and Modest Mussorgsky, 22edo.
- "being a", Andrew Heathwaite, 2010, 22edo, mode 3 1 3 3 3 3 3 3 of Porcupine[8].
- Playing Gently with Miller's Porcupine, Chris Vaisvil
- 15 Porcupines in India, Sarangi, Tambura and Sitar improvisation by Chris Vaisvil
- 15 Quills piano solo by Chris Vaisvil
- Prickly Side of Love - rock band in Porcupine Temperament with vocals by Chris Vaisvil
- Porcupine Organ Composition by Chris Vaisvil
- Among Other Things 2 by Petr Pařízek
- Porcupine Comma Pump, by Jake Freivald
- Life on Mars by Omega9
