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The preceding ideas can be extended in several directions. First, one mightexamine in detail how composers have exploited the geometry of musical chords (MovieS4). Second, one could generalize the geometrical approach by considering quotientspaces that identify transpositionally and inversionally related chords (24). Third, sincecyclical rhythmic patterns can also be modeled as points on Tn/Sn, one could use thesespaces to study African and other non-Western rhythms. Fourth, one could investigate howdistances in the orbifolds relate to perceptual judgments of chord similarity. Finally,understanding the relation between harmony and counterpoint may suggest new techniquesto contemporary composers.
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