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:
- Tridecimal porcupine (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
- Porkpie (15f & 22) – tempering out 55/54, 64/63, 65/63, 100/99;
- Porcupinefish (15 & 22) – tempering out 55/54, 64/63, 91/90, and 100/99;
- Porcup (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.
Tridecimal porcupine 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, tridecimal porcupine 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. Porcup'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 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.
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 | Porcup | |||
| 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
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 |