Porcupine extensions: Difference between revisions

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* '''Porcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.

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.

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.

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[[Category:Porcupine]]

[[Category:Temperament extensions]]

~~{{Todo| expand }}~~

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 * '''Porcup''' (15f & 22f) – tempering out 55/54, 64/63, 100/99, and 196/195.
-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.
+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.
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 [[Category:Porcupine]]
 [[Category:Temperament extensions]]
-{{Todo| expand }}