In music theory, the spiral array model is an extended type of pitch space. It represents human perceptions of pitch, chord and key in the same geometric space, as a mathematical model involving concentric helixes (an "array of spirals"). It was proposed in 2000 by Prof. Elaine Chew in her MIT doctoral thesis "Toward a Mathematical Model of Tonality". Further research by Chew and others have produced modifications of the spiral array model, and, applied it to various problems in music theory and practice, 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. Just like 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 geometric distance between a particular pitch and a particular key (both represented as points). Like the tonnetz, when applied to equal temperament, the spiral array model folds into a torus as octaves overlap.
Read more about Spiral Array Model: Structure of The Spiral Array, Equations
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... The pitch spiral P, is represented in parametric form by Where k is an integer representing a semitone, r is the radius of the spiral, and h is the "rise" of the ... changing the relative values of these weights, the spiral array model effects how "close" the resulting chord is to the three constituent pitches ...
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