Overtone scale: Difference between revisions

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== Primodality==

[[Zhea Erose]] has considered over-p scales and chords where p is a prime, which she calls '''primodal scales'''. ''Primodality''' is an approach to JI designed to emphasize the identity of the "tonic" as the pth harmonic. Name- and notation-wise, scales and structures ~~with~~ prime p as the tonic ~~may~~ collectively be denoted simply by "/p". Zhea also uses various adjectives for specific primodalities, such as ''septimal, undecimal, tridecimal, septendecimal, novem(decimal)'' for /7, /11, /13, /17, /19, which are not to be confused with the use of these adjectives to denote prime limits.

[[Zhea Erose]] has considered over-p scales and chords where p is a prime, which she calls '''primodal scales'''. ''Primodality''' is an approach to JI designed to emphasize the identity of the "tonic" as the pth harmonic. Name- and notation-wise, scales and structures having the identity of the prime p as the tonic are collectively called a '''prime family''', and can be denoted simply by ''/p''. Zhea also uses various adjectives for specific primodalities, such as ''septimal, undecimal, tridecimal, septendecimal, novem(decimal)'' for /7, /11, /13, /17, /19, which are not to be confused with the use of these adjectives to denote prime limits.

Most importantly, primodality sees any overtone as valuable on its own, rather than relative to some fundamental. Taking a specific overtone as a tonic we can get its particular scales and colors, and even ~~specific~~ versions of "non-xenharmonic" scales, even when the corresponding fundamental is too low to be audible. In particular, primodality discards the concept of [[harmonic limit]], which Zhea considers an artificial way to look at JI harmony. Zhea argues that prime family (set of intervals over a small multiple of p) is a more natural way to categorize intervals; intervals from the same prime family (intervals with a common denominator for example, all /2, all /11 or all /13) tend to blend better together. For example, it is preferable to add 21/16 to 4:5:6:7, rather than 4/3.

Most importantly, primodality sees any overtone as valuable on its own, rather than relative to some fundamental. Taking a specific overtone as a tonic we can get its particular scales and colors and even versions of "non-xenharmonic" scales, even when the corresponding fundamental is too low to be audible. In particular, primodality discards the concept of [[harmonic limit]], which Zhea considers an artificial way to look at JI harmony. Zhea argues that prime family (set of intervals over a small multiple of p) is a more natural way to categorize intervals; intervals from the same prime family (intervals with a common denominator for example, all /2, all /11 or all /13) tend to blend better together. For example, it is preferable to add 21/16 to 4:5:6:7, rather than 4/3.

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" (a segment of the harmonic series spanning an octave starting from mp where m is a positive integer) or a subset thereof. This corresponds to taking Mode mp of the harmonic series. 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.) Zhea considers the intervals in the first octave over p of the harmonic series to be the most "natural" intervals in p-primodality.

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 == Primodality==
-[[Zhea Erose]] has considered over-p scales and chords where p is a prime, which she calls '''primodal scales'''. ''Primodality''' is an approach to JI designed to emphasize the identity of the "tonic" as the pth harmonic. Name- and notation-wise, scales and structures with prime p as the tonic may collectively be denoted simply by "/p". Zhea also uses various adjectives for specific primodalities, such as ''septimal, undecimal, tridecimal, septendecimal, novem(decimal)'' for /7, /11, /13, /17, /19, which are not to be confused with the use of these adjectives to denote prime limits.
+[[Zhea Erose]] has considered over-p scales and chords where p is a prime, which she calls '''primodal scales'''. ''Primodality''' is an approach to JI designed to emphasize the identity of the "tonic" as the pth harmonic. Name- and notation-wise, scales and structures having the identity of the prime p as the tonic are collectively called a '''prime family''', and can be denoted simply by ''/p''. Zhea also uses various adjectives for specific primodalities, such as ''septimal, undecimal, tridecimal, septendecimal, novem(decimal)'' for /7, /11, /13, /17, /19, which are not to be confused with the use of these adjectives to denote prime limits.
-Most importantly, primodality sees any overtone as valuable on its own, rather than relative to some fundamental. Taking a specific overtone as a tonic we can get its particular scales and colors, and even specific versions of "non-xenharmonic" scales, even when the corresponding fundamental is too low to be audible. In particular, primodality discards the concept of [[harmonic limit]], which Zhea considers an artificial way to look at JI harmony. Zhea argues that prime family (set of intervals over a small multiple of p) is a more natural way to categorize intervals; intervals from the same prime family (intervals with a common denominator for example, all /2, all /11 or all /13) tend to blend better together. For example, it is preferable to add 21/16 to 4:5:6:7, rather than 4/3.
+Most importantly, primodality sees any overtone as valuable on its own, rather than relative to some fundamental. Taking a specific overtone as a tonic we can get its particular scales and colors and even versions of "non-xenharmonic" scales, even when the corresponding fundamental is too low to be audible. In particular, primodality discards the concept of [[harmonic limit]], which Zhea considers an artificial way to look at JI harmony. Zhea argues that prime family (set of intervals over a small multiple of p) is a more natural way to categorize intervals; intervals from the same prime family (intervals with a common denominator for example, all /2, all /11 or all /13) tend to blend better together. For example, it is preferable to add 21/16 to 4:5:6:7, rather than 4/3.
 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" (a segment of the harmonic series spanning an octave starting from mp where m is a positive integer) or a subset thereof. This corresponds to taking Mode mp of the harmonic series. 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.) Zhea considers the intervals in the first octave over p of the harmonic series to be the most "natural" intervals in p-primodality.