Porcupine extensions: Difference between revisions

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[[Porcupine]] has various [[extension]]s to the [[13-limit]].

[[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:

~~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~~.

* '''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.

~~== Exotempering options ==~~

Tridecimal porcupine maps [[13/8]] to -2 generator steps (tempering out [[40/39]]) and conflates it with [[5/3]] and [[18/11]]. Porkpie maps 13/8 to +5 generator steps (tempering out [[65/64]]) and conflates it with [[8/5]]. 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.

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 ==~~

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.

~~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 [[256/255|charisma]].

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 {{Breadcrumb|Porcupine}}
-[[Porcupine]] has various [[extension]]s to the [[13-limit]].
+[[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:
-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.
+* '''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.
-== Exotempering options ==
+Tridecimal porcupine maps [[13/8]] to -2 generator steps (tempering out [[40/39]]) and conflates it with [[5/3]] and [[18/11]]. Porkpie maps 13/8 to +5 generator steps (tempering out [[65/64]]) and conflates it with [[8/5]]. 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.
-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 ==
+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.
-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 [[256/255|charisma]].