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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]
1. circular tunings

Refers to an interval whose distance is repeated over and over again until it closes or nearly closes a cycle.

The "circle of 5ths" illustrates one example, in which reference is made to a large number of different scales or tuning systems built from a generator whose size is approximately that of a "perfect 5th", typically anywhere from about 680 to 720 cents. All meantone tunings fall into this category, as well as a number of other tunings. Pythagorean tuning is probably the oldest and longest-lasting example of this cycle.

Outside of microtonal music-theory, this particular cycle (of "5ths") ordinarily designates the cycle arising from the 12-edo tuning system, in which case the generator is a "5th" with an interval of 700 cents.

Alternatively, the 12-edo tuning can also be viewed as having a generator the size of a "4th" of 500 cents (the "circle of 4ths"), or as a semitone of 100 cents (the chromatic scale).

In all of these cases, assuming octave-equivalence, the 12th instance of the generator returns exactly to the pitch-class of the original note, thus closing the cycle to 12 pitches, an example of finity.

In April 2001, an investigation by Dave Keenan of the properties of scales built from various generators resulted in the rediscovery of the miracle family of tunings, from which Paul Erlich derived the "blackjack" scale.

For examples of how scales are built with a generator, see:

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2. periodicity-blocks

In periodicity-block theory, there are small intervals called unison-vectors, of which any linearly independent set are able to generate a kernel, which in just intonation is the lattice of adjacent parallelepipeds, each of which enloses the same number of ratios, and each of whose ratios are equivalent to those in the finite set of ratios enclosed within the central parallelepiped.

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