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, and 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 cents, 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 cents 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 cents.
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 |
Eigenmonzo (unchanged-interval) |
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 |