Porcupine: Difference between revisions
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To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain, with 7 found at +6 generators (tuned to about 960-990 cents), and 11 found at -4 generators (tuned to about 540-560 cents). Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called ''porkypine''). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy. | To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain, with 7 found at +6 generators (tuned to about 960-990 cents), and 11 found at -4 generators (tuned to about 540-560 cents). Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called ''porkypine''). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy. | ||
In the 7-limit, porcupine can be seen as as an extension of [[archy]]; its Pythagorean major third is mapped to 9/7, and its fifth is tuned sharp, ranging from around 705-720 cents. | In the 7-limit, porcupine can be seen as as an 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. | ||
See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s. | See [[Porcupine family #Porcupine]] for technical data. See [[Porcupine extensions]] for a discussion on [[13-limit]] [[extension]]s. | ||
Revision as of 09:09, 20 April 2025
Porcupine is a linear temperament that equates the syntonic comma 81/80 with the 5-limit chromatic semitone 25/24. This simplifies the 5-limit to a rank-2 structure in a simple way distinct from temperaments that reduce it to an extension of pythagorean (i.e. meantone, schismic), by tempering out 250/243, the porcupine comma. Its pergen is (P8, P4/3). 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. As a result, the generator of porcupine is a minor whole tone (10/9) which is tuned flat to around 160–165 cents such that two of them stack to a classic minor third (6/5). 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.
To extend porcupine to the 7-limit and 11-limit, you may notice that both primes are found naturally in simple positions along the porcupine generator chain, with 7 found at +6 generators (tuned to about 960-990 cents), and 11 found at -4 generators (tuned to about 540-560 cents). Neither, one, or both of these additional mappings may be used (in 2.3.5.11, the temperament is sometimes called porkypine). Porcupine is one of the most efficient temperaments in the 2.3.5.11 subgroup of reasonable accuracy.
In the 7-limit, porcupine can be seen as as an 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.
See Porcupine family #Porcupine for technical data. See Porcupine extensions for a discussion on 13-limit extensions.
Interval chain
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". An interval in the neutral third range is not found for a long time (until 17 generators up), and as a result that interval varies drastically depending on the tuning. 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).
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.
Chords
Scales
- Mos scales, tuning optimized on the 2.3.5.11 subgroup
- Mos scales, 8/5.12/7 eigenmonzo (unchanged-interval) tuning
Tunings
| Euclidean | ||
|---|---|---|
| Constrained | Constrained & skewed | |
| Equilateral | CEE: ~10/9 = 163.6049 ¢ | CSEE: ~10/9 = 163.2835 ¢ |
| Tenney | CTE: ~10/9 = 164.1659 ¢ | CWE: ~10/9 = 164.0621 ¢ |
| Benedetti, Wilson |
CBE: ~10/9 = 164.3761 ¢ | CSBE: ~10/9 = 164.3761 ¢ |
| Euclidean | ||
|---|---|---|
| Constrained | Constrained & skewed | |
| Equilateral | CEE: ~11/10 = 163.1459 ¢ | CSEE: ~11/10 = 162.8445 ¢ |
| Tenney | CTE: ~11/10 = 163.8867 ¢ | CWE: ~11/10 = 163.9951 ¢ |
| Benedetti, Wilson |
CBE: ~11/10 = 164.2393 ¢ | CSBE: ~11/10 = 164.4623 ¢ |
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.
See also
Music
20th century
- Mizarian Porcupine Overture (1999) – in 15edo tuning, namesake of the temperament
21st century
- "April Porkfest" from TOTMC Suite Vol. 1 (2023) – in 11-limit CTE tuning
- Porcupine Walk (2019)
- Sanctus (2015)
- being a (2010) – in Porcupine[8], mode 1|6, 22edo tuning
- Porcupeen (2017)
- "Porcupine", from pato, with friends (2019)
- Second Breakfast (15edo) (2018)[dead link]
- Porcupine Experience (2012) – in 22edo tuning
- Flying Straight Down (2020) – in 22edo tuning
- Life on Mars (2014)
- Porcupine Lullaby (2020) – in 37edo tuning
- Porcupine[7] Modal Fugues – 7-piece playlist
- Night on Porcupine Mountain (archived 2010) – in 22edo tuning
- 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
- 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
- Porcupine Major Overture (2015) – in 22edo tuning
- Waltzing in Candyland (2015) – in Porcupine[8], 15edo tuning
Diagrams
