Overtone scale: Difference between revisions

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Mode 30 in particular is interesting because 30 is the product of the first three primes, so it's a fairly good choice if we want a tonic that isn't a power of two. It contains modes 6 and 10 as subsets. We have the classic minor triad (from 10), the subminor triad (from 6), two major triads in 30:37:45 and 30:38:45, and the Barbados triad of 30:39:45. Chords not based on the tonic include the harmonic seventh chord (32:40:48:56). A good 13-limit subset with 16 notes in it is 30:32:33:35:36:39:40:42:44:45:48:49:50:54:55:56:60.

==~~= Over-p scales =~~==

== Primodality==

[[Zhea Erose]] has considered over-p scales where p is a prime, which she calls '''primodal scales'''. '''Primodality''' is designed to emphasize the identity of the "tonic" as the pth harmonic. 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, 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 p) is a more natural way to categorize intervals; intervals from the same prime family (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.

[[Zhea Erose]] has considered over-p scales 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. 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, 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 p) is a more natural way to categorize intervals; intervals from the same prime family (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.

@@ Line 164: / Line 164: @@
 Mode 30 in particular is interesting because 30 is the product of the first three primes, so it's a fairly good choice if we want a tonic that isn't a power of two. It contains modes 6 and 10 as subsets. We have the classic minor triad (from 10), the subminor triad (from 6), two major triads in 30:37:45 and 30:38:45, and the Barbados triad of 30:39:45. Chords not based on the tonic include the harmonic seventh chord (32:40:48:56). A good 13-limit subset with 16 notes in it is 30:32:33:35:36:39:40:42:44:45:48:49:50:54:55:56:60.
-=== Over-p scales ===
+== Primodality==
-[[Zhea Erose]] has considered over-p scales where p is a prime, which she calls '''primodal scales'''. '''Primodality''' is designed to emphasize the identity of the "tonic" as the pth harmonic. 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, 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 p) is a more natural way to categorize intervals; intervals from the same prime family (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.
+[[Zhea Erose]] has considered over-p scales 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. 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, 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 p) is a more natural way to categorize intervals; intervals from the same prime family (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.