Structure of The Spiral Array
The model covering basic pitch, major chords, minor chords, major keys and minor keys comprises five concentric helixes. Starting with a formulation of the pitch spiral, inner spirals are generated by a convex combination of points on outer spirals. For example, the pitches C, E, and G are represented as points by the cartesian coordinates C(x,y,z), E(x,y,z) and G(x,y,z). The convex combination formed by the points CEG is a triangle, and represents the "center of effect" of the three pitches. This convex combination represents the triad, or chord, CEG (the C major chord) in the spiral array model. The geometric center (or other point chosen by a weighting of the constituent points, as seen in the equations below) of the C major chord (formed by CEG) can be called the "center" of the C major chord, and assigned a point CM(x,y,z). Similarly, keys may be constructed by the centers of effect of their I, IV, and V chords.
- The outer helix represents pitches classes. Neighboring pitch classes are a music interval of a perfect fifth, and spatially a quarter rotation, apart. The order of the pitch classes can be determined by the circle of fifths. For example, C would be followed by G, which would be followed D, etc. As a result of this structure, and one of the important properties leading to its selection, vertical neighbors are a music interval of a major third apart. Thus, a pitch class's nearest neighbors and itself form perfect fifth and major third intervals.
- By taking every consecutive triad along the helix, and projecting their centers of effect, a second helix is formed inside the pitch helix, representing the major chords.
- Similarly, by taking the proper minor triads and projecting their centers of effect, a third helix is formed, representing the minor chords.
- The major key helix is formed by projections of the I, IV, and V chords from points on the major chord
- The minor key helix is formed by similar projects of minor chords.
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