The Spiral Array
This chapter presents the musical and geometric reasoning behind, and mathematical formulation of, the Spiral Array model, showing how the model successively generates representations for higher level tonal elements as a composite of each entity’s lower level components. The concept of the center of effect is defined, wherein any element in the space can generate a higher level construct, modeled, in the same space, as the centroid, a mathematical sum, of its lower level members. For example, chord representations are generated from their component pitches as the center of effect of their member pitches, and key representations are similarly derived from their defining chords. Intuitive images illustrate the construction of the model, beginning from the “rolling up” of the Harmonic Network, and through the stages of defining higher level entities in the interior of the model. Theorems describe mathematical properties of the different levels of representations. The chapter concludes with a summary of the definitions and a visual depiction of the resulting array of spirals: nested helices comprising the outermost pitch class spiral, the major/minor chord double helix inside the pitch class spiral, and the major/minor key double helix inside the major/minor chord double helix.
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