Overtone scale: Difference between revisions

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To construct a primodal scale, we fix a prime p to be the denominator and take intervals of the form n/p, where n ≥ p. Zhea often takes n to range over a certain "lineal segment" (Mode mp of the harmonic series where m is a positive integer) or a subset thereof.. For example, if we use p = 13 and take all n between 13 and 26 (inclusive), this would result in the scale 13:14:15:16:17:18:19:20:21:22:23:24:25:26. We may add a 3/2 to the scale root, which corresponds to adding 3p/p. (3/2 is a natural "halfway point" for harmonic scales, since if N is even, Mode N has a 3/2 as its N/2-th note.)

Primodality emphasizes unique characters of primes, as opposed to lower-complexity JI identities such as 4:5:6:7:9~~:11:13~~. Mode p and Mode 2p (called respectively the ''first'' and ''second octaves of /p'') are considered the most important for the prime's identity: for any prime p, the set of harmonics from p to 2p is unique in the sense that the sets {p/p, ..., 2p/p} and {n/n, ..., 2n/n} only intersect at {1/1, 2/1} for any positive integer n < p. Similarly, the second octaves of p and the second octave of any n < p only intersect at {1/1, 3/2, 2/1}.

Primodality emphasizes unique characters of primes, as opposed to lower-complexity JI identities such as 4:5:6:7:9 that are more common in composite modes. Mode p and Mode 2p (called respectively the ''first'' and ''second octaves of /p'') are considered the most important for the prime's identity: for any prime p, the set of harmonics from p to 2p is unique in the sense that the sets {p/p, ..., 2p/p} and {n/n, ..., 2n/n} only intersect at {1/1, 2/1} for any positive integer n < p. Similarly, the second octaves of p and the second octave of any n < p only intersect at {1/1, 3/2, 2/1}.

Approximations of [[edo]]s in certain primodalities, called ''nejis'' ("near-equal JI"), can be used to explore a prime family, while keeping the transposability, scale structures, rank-2 harmonic theory, etc. associated with that edo.

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 To construct a primodal scale, we fix a prime p to be the denominator and take intervals of the form n/p, where n ≥ p. Zhea often takes n to range over a certain "lineal segment" (Mode mp of the harmonic series where m is a positive integer) or a subset thereof.. For example, if we use p = 13 and take all n between 13 and 26 (inclusive), this would result in the scale 13:14:15:16:17:18:19:20:21:22:23:24:25:26. We may add a 3/2 to the scale root, which corresponds to adding 3p/p.  (3/2 is a natural "halfway point" for harmonic scales, since if N is even, Mode N has a 3/2 as its N/2-th note.)
-Primodality emphasizes unique characters of primes, as opposed to lower-complexity JI identities such as 4:5:6:7:9:11:13. Mode p and Mode 2p (called respectively the ''first'' and ''second octaves of /p'') are considered the most important for the prime's identity: for any prime p, the set of harmonics from p to 2p is unique in the sense that the sets {p/p, ..., 2p/p} and {n/n, ..., 2n/n} only intersect at {1/1, 2/1} for any positive integer n < p. Similarly, the second octaves of p and the second octave of any n < p only intersect at {1/1, 3/2, 2/1}.
+Primodality emphasizes unique characters of primes, as opposed to lower-complexity JI identities such as 4:5:6:7:9 that are more common in composite modes. Mode p and Mode 2p (called respectively the ''first'' and ''second octaves of /p'') are considered the most important for the prime's identity: for any prime p, the set of harmonics from p to 2p is unique in the sense that the sets {p/p, ..., 2p/p} and {n/n, ..., 2n/n} only intersect at {1/1, 2/1} for any positive integer n < p. Similarly, the second octaves of p and the second octave of any n < p only intersect at {1/1, 3/2, 2/1}.
 Approximations of [[edo]]s in certain primodalities, called ''nejis'' ("near-equal JI"), can be used to explore a prime family, while keeping the transposability, scale structures, rank-2 harmonic theory, etc. associated with that edo.