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.
Exotempering options
13 has two viable mappings within the simple intervals of porcupine. 13 can be mapped to +5 generators (tempering out 65/64, the wilsorma), or to -2 generators (tempering out 40/39, the unintendo comma, leading to the canonical tridecimal extension of porcupine). The former flattens it by about 20 cents, and the latter sharpens it by about 30.
More complex mappings
Another mapping of 13 is available at -17 generators (leading to porcupinefish). 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.
Higher primes
Prime 17 has a much more obvious mapping, as it can be found at +8 generators, which is tuned between around 80 and 120 cents. This is also the mapping of 16/15, tempering out the charisma.
Tuning spectrum
Tridecimal porcupine
| 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 | 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 |
Porcupinefish
| 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 |