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	work description < music engine Music Theory: Nested Repetition and Musical Counterpoint Electronic dance music (EDM) is largely based on looping musical structures. Musical transformations are applied to various degrees of repetition in order to produce rhythmic syncopation and harmonic variation. Repetition and variation are often nested within larger articulations of the metrical structure. This type of music is typically composed using software applications which allow a composer/producer to manage multiple levels of musical activity by looping, cutting, pasting, and transforming musical elements. The traditional rules of musical harmony and counterpoint are also concerned with multi-leveled musical structures: related transformations of deep-level musical structure give rise to unified musical surfaces. The work of music theorists such as Heinrich Schenker, Arthur Komar, Fred Lerdahl, and Ray Jackendoff provides frameworks for analyzing nested musical relationships. EDM composition can be (unknowingly) similar in practice to an inversion of a reductional, transformation-based musical analysis, Musical elements at one level are often triggered by, or woven into, musical material at a higher metrical level. Software applications used by EDM composers enable the management of nested musical structure so that the composer is guided by ear towards musical results which would have required more conscious structural design in traditional music composition. EDM is often characterized by relatively short note patterns in duple meter. The meter is clearly delineated by the drum parts, and this leaves the other parts free to provide syncopation and variation. The regularity of meter and the prevalence of nested repetition suggest that such music might best be analyzed in terms of regularly spaced, interlocking pulses. The problem then is how to relate the patterns produced by selective, nested repetition to the principles of harmonic and contrapuntal structure. Mathematically, repeated transformation and scaling of data often leads to an attractor, a final state at which the transformations simply map the data back onto itself. Structures produced in this manner are typically self-similar, because the same transformations are in effect at each level of the hierarchical structure. Music is generally not self-similar on the surface, but the hierarchical analytical operations applied by theorists, and the hierarchical composition practices of EDM composers, suggest that the underlying space of well-formed musical possibilities might be self-similar. And so it is possible to identify basic rhythmic building blocks that can function at any level of the metrical hierarchy. Well-chosen building blocks form a compressed representation of the music that captures both generic and specific aspects of a particular musical passage. Rhythmic/harmonic analysis of "Theme from 007" Work by nineteenth century number theorists Adrien Marie Legendre, Ernst Eduard Kummer, and Edouard Lucas drew a link between carries in binary arithmetic and the patterns formed by even versus odd binomial coefficients. Carries in binary arithmetic are directly correlated to beat strength in duple meter: the number of carries that occur between the binary representation of one time-point and the next determines the beat strength of that second time-point. The patterns formed by the distribution of binomial coefficients turn out to capture the most basic counterpoint rules for musical elaboration via syncopating and/or subdividing material at successive metrical levels. These patterns may be observed as rows of odd binomial coefficients on Pascal's Triangle, often visualized as a fractal known as the Sierpinski Gasket. The connection to the Sierpinski Gasket is a manifestation of the fact that patterns can be directly evolved from each other via a cellular automaton. Rhythmic building blocks based on these patterns provide the music software environment with handles onto the hierarchical structure of music. The set of building blocks which constitute a particular piece provides a structural profile that can be compared and contrasted with other pieces. Harmonic and rhythmic operations can be carried out more efficiently and meaningfully on the building blocks than on the individual notes. Variations and hybrids of existing pieces are created by algorithms operating on the building blocks that comprise each piece.
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