Porcupine extensions: Difference between revisions
Tridecimal porcupine is canon, so it's addressed first. - porky since this article is about extensions of 11-limit porcupine. A dedicated article for porky may be created in future. - "161.9 cents" tuning, seems arbitrary |
Implement the decanonicalization of tridecimal porcupine according to community consensus. - typo |
||
| Line 2: | Line 2: | ||
[[Porcupine]] has various [[extension]]s 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: | [[Porcupine]] has various [[extension]]s 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; | * '''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; | * '''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. | ||
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. | 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 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. | ||
| Line 31: | Line 31: | ||
! Porcupinefish | ! Porcupinefish | ||
! Porkpie | ! Porkpie | ||
! | ! Pourcup | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 220: | Line 220: | ||
== Tuning spectrum == | == Tuning spectrum == | ||
=== | === Porcupinefowl === | ||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
! Edo<br>generator | ! Edo<br>generator | ||