Diatonic set theory

Application of musical set theory to the diatonic scale

Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. The name is something of a misnomer as the concepts involved usually apply much more generally, to any periodically repeating scale.

Music theorists working in diatonic set theory include Eytan Agmon, Gerald J. Balzano, Norman Carey, David Clampitt, John Clough, Jay Rahn, and mathematician Jack Douthett. A number of key concepts were first formulated by David Rothenberg (the Rothenberg propriety), who published in the journal Mathematical Systems Theory, and Erv Wilson, working entirely outside of the academic world.

## See also[edit]

* Bisector
* Diatonic and chromatic
* Generic and specific intervals

## Further reading[edit]

* Balzano, Gerald, "The Pitch Set as a Level of Description for Studying Musical Pitch Perception", Music, Mind and Brain, the Neurophysiology of Music, Manfred Clynes, ed., Plenum Press, 1982.
* Carey, Norman and Clampitt, David (1996), "Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues", Perspectives of New Music 34, no. 2: 62–87.
* Grady, Kraig, (2007), "An Introduction to the Moments of Symmetry", Wilson Archives, anaphoria.com
* Johnson, Timothy (2003), Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals, Key College Publishing. ISBN 1-930190-80-8.

### Precursors[edit]

* Rahn, Jay (1977), "Some Recurrent Features of Scales", In Theory Only 2, nos. 11–12: 43–52.
* Rothenberg, David, (1977), "A Model for Pattern Perception with Musical Applications", Mathematical Systems Theory, part I: 11, 199–234 doi:10.1007/BF01768477; part II: 353–372 doi:10.1007/BF01768486; part III: (1978) 12, 73–101 doi:10.1007/BF01776567.
* Wilson, Erv (1975), "Handwritten letter to John Chalmers pertaining to 'Moments of Symmetry'/'Tanabe Cycle', 26 April 1975, 27 pages, anaphoria.com

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Musical set theory

* All-interval tetrachord
* All-trichord hexachord
* Complement
* Forte number
* Identity
* Interval class
* Interval vector
* Multiplication
* Permutation
* Pitch class
* Pitch interval
* Pitch-interval class
* Set
* List
* Similarity relation
* Transformation
* Z-relation

Diatonic
set theory|

* Bisector
* Cardinality equals variety
* Common tone (Deep scale property)
* Diatonic scale
* Generated collection
* Generic and specific intervals (Myhill's property)
* Maximal evenness
* Rothenberg propriety
* Structure implies multiplicity

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