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

* '''Porcupinefowl''' ({{nowrap|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''' ({{nowrap|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;

* '''Porkpie''' ({{nowrap|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.

* '''Pourcup''' ({{nowrap|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.

* '''[[Opossum]]''' ({{nowrap|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;

* '''Porky''' ({{nowrap|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.

* '''Coendou''' ({{nowrap|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.

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}}.

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.

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.

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{{c}}.

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|Porcupine}}
+{{Breadcrumb}}
 [[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;
+* '''Porcupinefowl''' ({{nowrap|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''' ({{nowrap|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;
+* '''Porkpie''' ({{nowrap|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.
+* '''Pourcup''' ({{nowrap|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.
+* '''[[Opossum]]''' ({{nowrap|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;
+* '''Porky''' ({{nowrap|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.
+* '''Coendou''' ({{nowrap|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.
+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}}.
-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.
+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.
-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{{c}}.
-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 ==