Graham complexity
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- This revision was by author genewardsmith and made on 2011-07-19 19:55:31 UTC.
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Original Wikitext content:
The //Graham complexity// of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [<2 0 11|, <0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6. Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads.
Original HTML content:
<html><head><title>Graham complexity</title></head><body>The <em>Graham complexity</em> of a set of pitch classes in a rank two temperament is the number of periods per octave times the difference between the maximum and minimum number of generators required to reach each pitch class. For example, consider a major triad in diaschismic temperament, with mapping [<2 0 11|, <0 1 -2|] corresponding to a generator of a '3'. That makes a 5/4 represented by [7 -2] and 3/2 by [-2 1], and 1 of course by [0 0]; so that the difference between the maximum and minimum number of generators is 1 - (-2) = 3, and so the Graham complexity of a major triad is 2*3 = 6.<br /> <br /> Given a MOS (or any generated scale) with N notes in a temperament where a given chord has a Graham complexity of C results in N-C chords in the MOS. For instance, the Graham complexity for both a major or a minor triad in meantone is 4, and hence there are 7-4 = 3 major triads in the diatonic scale, which is Meantone[7], and 3 minor triads.</body></html>