Pentatonic Functional Just System: Difference between revisions

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In [[13-limit]] [[superpyth]], [[11/8]] is a sub-sub-sub-5fourth, and [[13/8]] is a sub-sub-5fifth.

In the PFJS, primes beyond 13 are classified somewhat like the FJS, based on pythagorean intervals from -5 to +6 perfect 5fourths, with priority 0, 1, -1, 2, -2, etc. Unlike FJS, however, the RoT is not the same in both directions. The RoT of a pythagorean interval of <math>i</math> cents is <math>i-68</math> through <math>i+46</math> cents. This range was chosen so that it works for the 13-limit, and it spans just over an [[2187/2048|apotome]], the large step in the pythagorean chromatic scale. The exact range was set considering a few large primes: Prime [[37/1|37]] is just barely not a 5m237, but rather a 5M237; similarly, prime [[41/1|41]] is just barely not a 5P341, but rather a 5s341.

In the PFJS, primes beyond 13 are classified somewhat like the FJS, based on pythagorean intervals from -5 to +6 perfect 5fourths ([[3/2]]'s) , with priority 0, 1, -1, 2, -2, etc. Unlike FJS, however, the RoT is not the same in both directions. The RoT of a pythagorean interval of <math>i</math> cents is <math>i-68</math> through <math>i+46</math> cents. This range was chosen so that it works for the 13-limit, and it spans just over an [[2187/2048|apotome]], the large step in the pythagorean chromatic scale. The exact range was set considering a few large primes: Prime [[37/1|37]] is just barely not a 5m237, but rather a 5M237; similarly, prime [[41/1|41]] is just barely not a 5P341, but rather a 5s341.

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Similar logic can be used for even higher primes, up to infinity. While an uneven RoT is unconventional, it is required to work well with the lower limits in the PFJS. This also solves the issue of gaps between categories in the FJS, which occurs as the RoT in the FJS, [[65/63]], is less than half of [[2187/2048]]. After all, every notation system is a compromise.

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 In [[13-limit]] [[superpyth]], [[11/8]] is a sub-sub-sub-<sub>5</sub>fourth, and [[13/8]] is a sub-sub-<sub>5</sub>fifth.
-In the PFJS, primes beyond 13 are classified somewhat like the FJS, based on pythagorean intervals from -5 to +6 perfect <sub>5</sub>fourths, with priority 0, 1, -1, 2, -2, etc. Unlike FJS, however, the RoT is not the same in both directions. The RoT of a pythagorean interval of <math>i</math> cents is <math>i-68</math> through <math>i+46</math> cents. This range was chosen so that it works for the 13-limit, and it spans just over an [[2187/2048|apotome]], the large step in the pythagorean chromatic scale. The exact range was set considering a few large primes: Prime [[37/1|37]] is just barely not a <sub>5</sub>m2<sup>37</sup>, but rather a <sub>5</sub>M2<sup>37</sup>; similarly, prime [[41/1|41]] is just barely not a <sub>5</sub>P3<sup>41</sup>, but rather a <sub>5</sub>s3<sup>41</sup>.
+In the PFJS, primes beyond 13 are classified somewhat like the FJS, based on pythagorean intervals from -5 to +6 perfect <sub>5</sub>fourths ([[3/2]]'s) , with priority 0, 1, -1, 2, -2, etc. Unlike FJS, however, the RoT is not the same in both directions. The RoT of a pythagorean interval of <math>i</math> cents is <math>i-68</math> through <math>i+46</math> cents. This range was chosen so that it works for the 13-limit, and it spans just over an [[2187/2048|apotome]], the large step in the pythagorean chromatic scale. The exact range was set considering a few large primes: Prime [[37/1|37]] is just barely not a <sub>5</sub>m2<sup>37</sup>, but rather a <sub>5</sub>M2<sup>37</sup>; similarly, prime [[41/1|41]] is just barely not a <sub>5</sub>P3<sup>41</sup>, but rather a <sub>5</sub>s3<sup>41</sup>.
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+Similar logic can be used for even higher primes, up to infinity. While an uneven RoT is unconventional, it is required to work well with the lower limits in the PFJS. This also solves the issue of gaps between categories in the FJS, which occurs as the RoT in the FJS, [[65/63]], is less than half of [[2187/2048]]. After all, every notation system is a compromise.
 {{Navbox notation}}