Spiral array model
In ""music theory", the ""spiral array model"" is a type of pitch space. It aims to represent human perceptions of pitches, chords and keys as a mathematical model comprised of 5 concentric helixes (an "array of spirals"). It was invented by Prof. Elaine Chew, and first published as her MIT doctoral dissertation. Further research by Chew and others have produced modifications of the spiral array model, and, particularly, applied it to various music theory and practical problems such as key finding and pitch spelling.
The spiral array model can be viewed as an extension of the tonnetz, which maps pitches into a two-dimensional lattice structure. Unlike the tonnetz, the spiral array models higher order structures such as chords and keys in the same space as the low level structure: pitches. This allows the spiral array model to produce geometric interpretations of relationships between low and high level structures. For example, you can measure the "distance" between a particular pitch and a particular key. Like the tonnetz, when applied to equal temperament, the spiral array model folds into a torus as octaves overlap.
Structure of the Spiral Array
The structure comprises 5 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 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, a repeating sequence of 12 semitones, or pitch classes, representing the chromatic scale. Each semitone is placed a quarter rotation away from its predecessor. The order of the semitones is determined by the circle of fifths, so the nearest neighbor along the spirals are related by a major fifth. 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, the nearest neighbor above a semitone is a major third higher. Thus the nearest neighbors of a pitch are its perfect fifth and major third, ahead and above it, respectively.
- 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.
See also
References
Chuan, C.-H., Chew, E. (2005). Applying the Spiral Array Key-finding Algorithm to Polyphonic Audio. In Proceedings of the 9th INFORMS Computing Society Conference (invited sessions on Music, Computation and AI), Annapolis, MD, Jan 5-7, 2005. Chew, Elaine (2002). The Spiral Array: An Algorithm for Determining Key Boundaries Chew, Elaine (2000). Towards a Mathematical Model of Tonality. Ph.D. dissertation. Operations Research Center, MIT. Cambridge, MA.