Porcupine/Extensions
Porcupine has various extensions to the 13-limit. Adding the 13th harmonic to porcupine is not very simple, because 13 tends to fall in between the simple intervals produced by porcupine's generator. The extensions are:
- Porcupinefowl (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
- Porcupinefish (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;
- Porkpie (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;
- Pourcup (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.
Additionally, there are alternative extensions to prime 7:
- Opossum (8d & 15) – tempering out 28/27, 40/39, 55/54, and 66/65.
- Porky (22 & 29) – tempering out 55/54, 65/64, 91/90, and 100/99;
- Coendou (29 & 36ce) – tempering out 55/54, 65/64, 100/99, and 105/104.
Porcupinefowl maps 13/8 to −2 generator steps and conflates it with 5/3 and 18/11, tempering out 40/39. This is where the generator, representing 10/9, 11/10, and 12/11, goes one step further to stand in for ~13/12. Porkpie maps 13/8 to +5 generator steps and conflates it with 8/5, tempering out 65/64. The generator now represents ~14/13. Without optimization for the 13-limit, porcupinefowl sharpens the interval class of 13 by about 30 ¢, and porkpie flattens it by about 20 ¢.
The other pair of extensions are of higher complexity, but are well rewarded with better intonation. Porcupinefish's mapping of 13 is available at −17 generator steps. This equates the sharply tuned diatonic major third of porcupine with 13/10 along with 9/7, and requires a much more precise tuning of the porcupine generator to 161.5–163.5 ¢ to tune the 13th harmonic well. Pourcup's mapping of 13 is available at +20 generator steps. They unite in 37edo, which can be recommended as a tuning for both.
Prime 17 can be found at +8 generator steps, in which case −14 generator steps represent 18/17. This conflates 16/15 with 17/16, tempering out 256/255, and 15/14 with 18/17, tempering out 85/84. It can also be found at −14 generator steps, in which case +8 generator steps represent 18/17. This conflates 17/16 with 15/14, tempering out 120/119, and 18/17 with 16/15, tempering out 136/135. Both steps tend to be tuned between around 90 and 130 ¢.
Prime 19 can be found at −13 generator steps (25/21, tempering out 400/399), or more crudely at 2 generator steps (6/5, tempering out 96/95).
Prime 23 can be found at 4 generator steps (tempering out 256/253) or −11 generator steps (tempering out 161/160). Both of these approximations are rather crude, but may be improved by varying the tuning of the generator. For a more precise (yet more complex) mapping, +26 steps is an option.
Interval chain
In the following table, odd harmonics and subharmonics 1–13 are in bold.
| # | Cents* | Approximate ratios | ||||
|---|---|---|---|---|---|---|
| 11-limit | 13-limit extensions | |||||
| Porcupine | Porcupinefish | Porkpie | Pourcup | |||
| 0 | 0.0 | 1/1 | ||||
| 1 | 162.8 | 10/9, 11/10, 12/11 | 13/12 | 14/13 | ||
| 2 | 325.6 | 6/5, 11/9 | 13/11, 16/13 | 26/21 | ||
| 3 | 488.4 | 4/3 | 13/10 | |||
| 4 | 651.3 | 16/11, 22/15 | 13/9 | |||
| 5 | 814.1 | 8/5 | 21/13 | 13/8 | ||
| 6 | 976.9 | 7/4, 16/9 | 26/15 | |||
| 7 | 1139.7 | 48/25, 64/33, 160/81 | 52/27 | 25/13 | 39/20 | |
| 8 | 102.5 | 16/15, 21/20 | 14/13, 26/25 | 27/26 | 13/12 | |
| 9 | 265.3 | 7/6 | 15/13 | 13/11 | ||
| 10 | 428.2 | 14/11 | 13/10 | |||
| 11 | 591.0 | 7/5 | 18/13 | 13/9 | ||
| 12 | 753.8 | 14/9 | 20/13 | |||
| 13 | 916.6 | 42/25 | 22/13 | 26/15 | ||
| 14 | 1079.4 | 28/15 | 24/13 | 52/27 | 13/7 | |
| 15 | 42.2 | 28/27, 49/48 | 40/39 | 26/25 | ||
| 16 | 205.0 | 28/25 | ||||
| 17 | 367.9 | 49/40, 56/45 | 16/13 | 26/21 | ||
| 18 | 530.7 | 49/36 | ||||
| 19 | 693.5 | 49/33 | ||||
| 20 | 856.3 | 49/30 | 21/13 | 13/8 | ||
| 21 | 1019.1 | 49/27 | ||||
| 22 | 1181.9 | 49/25 | 39/20 | |||
* In 11-limit CWE tuning, octave reduced
Tuning spectrum
Porcupinefowl
| Edo generator |
Eigenmonzo (Unchanged-interval) |
Generator (¢) | Comments |
|---|---|---|---|
| 13/12 | 138.573 | ||
| 13/11 | 144.605 | ||
| 1\8 | 150.000 | 8d val, lower bound of 5-odd-limit diamond monotone | |
| 11/6 | 150.637 | Lower bound of 11-odd-limit diamond tradeoff | |
| 13/10 | 151.405 | ||
| 5/3 | 157.821 | Lower bound of 5-, 7-, and 9-odd-limit diamond tradeoff | |
| 15/13 | 158.710 | ||
| 13/9 | 159.154 | ||
| 2\15 | 160.000 | Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone | |
| 7/4 | 161.471 | ||
| 7\52 | 161.538 | 52bfff val | |
| 11/7 | 161.751 | ||
| 7/5 | 162.047 | ||
| 5\37 | 162.162 | 37ff val | |
| 11/8 | 162.171 | 13- and 15-odd-limit minimax | |
| 8\59 | 162.712 | 59fff val | |
| 5/4 | 162.737 | 5- and 7-odd-limit minimax | |
| 15/14 | 162.897 | ||
| 7/6 | 162.986 | ||
| 3\22 | 163.636 | 22f val, upper bound of 7-, 9-, 11, and 13-odd-limit diamond monotone | |
| 9/7 | 163.743 | 9- and 11-odd-limit minimax | |
| 15/8 | 163.966 | ||
| 11/10 | 165.004 | ||
| 15/11 | 165.762 | ||
| 3/2 | 166.015 | Upper bound of 5- and 7-odd-limit diamond tradeoff | |
| 13/7 | 166.037 | ||
| 1\7 | 171.429 | Upper bound of 5-odd-limit diamond monotone | |
| 11/9 | 173.704 | ||
| 13/8 | 179.736 | ||
| 9/5 | 182.404 | Upper bound of 9- and 11-odd-limit diamond tradeoff |
Porcupinefish
| Edo generator |
Unchanged interval (eigenmonzo) |
Generator (¢) | Comments |
|---|---|---|---|
| 1\8 | 150.000 | 8dff val, lower bound of 5-odd-limit diamond monotone | |
| 11/6 | 150.637 | ||
| 5/3 | 157.821 | ||
| 2\15 | 160.000 | Lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone | |
| 13/9 | 160.307 | ||
| 15/13 | 160.860 | ||
| 7/4 | 161.471 | ||
| 13/12 | 161.531 | ||
| 7\52 | 161.538 | 52bf val | |
| 11/7 | 161.751 | ||
| 7/5 | 162.047 | ||
| 13/7 | 162.100 | ||
| 13/10 | 162.149 | ||
| 5\37 | 162.162 | Upper bound of 13-odd-limit diamond monotone | |
| 11/8 | 162.171 | ||
| 13/8 | 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 | Upper bound of 7-, 9-, and 11-odd-limit diamond monotone | |
| 9/7 | 163.743 | 9- and 11-odd-limit minimax | |
| 15/8 | 163.966 | ||
| 11/10 | 165.004 | ||
| 15/11 | 165.762 | ||
| 3/2 | 166.015 | ||
| 1\7 | 171.429 | 7f val, upper bound of 5-odd-limit diamond monotone | |
| 11/9 | 173.704 | ||
| 9/5 | 182.404 |