Porcupine extensions: Difference between revisions

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[[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''' (~~{{nowrap|~~15 & 22f}}) – tempering out 40/39, 55/54, 64/63, and 66/65;

* '''Porcupinefowl''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;

* '''Porcupinefish''' (~~{{nowrap|~~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''' (~~{{nowrap|~~15f & 22}}) – tempering out 55/54, 64/63, 65/63, 100/99;

* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, and 100/99;

* '''Pourcup''' (~~{{nowrap|~~15f & 22f}}) – tempering out 55/54, 64/63, 100/99, and 196/195.

* '''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~~{{cent}}~~, and porkpie flattens it by about 20~~{{cent}}~~.

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~~{{c}}~~ 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.

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~~{{c}}~~.

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 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.

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 ==

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-{{Breadcrumb}}
+{{Breadcrumb|Porcupine}}
 [[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''' ({{nowrap|15 & 22f}}) – tempering out 40/39, 55/54, 64/63, and 66/65;
+* '''Porcupinefowl''' (15 & 22f) – tempering out 40/39, 55/54, 64/63, and 66/65;
-* '''Porcupinefish''' ({{nowrap|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''' ({{nowrap|15f & 22}}) – tempering out 55/54, 64/63, 65/63, 100/99;
+* '''Porkpie''' (15f & 22) – tempering out 55/54, 64/63, 65/63, and 100/99;
-* '''Pourcup''' ({{nowrap|15f & 22f}}) – tempering out 55/54, 64/63, 100/99, and 196/195.
+* '''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{{cent}}, and porkpie flattens it by about 20{{cent}}.
+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{{c}} 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.
+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{{c}}.
+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 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.
+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 ==