In diatonic set theory maximal evenness is the quality of a collection or scale which for every generic interval there are is either one or two consecutive (adjacent) specific intervals, in other words a scale which is "'spread out as much as possible.'" This property was first described by music theorist John Clough and mathematician Jack Douthett in "Maximally Even Sets" (1991). (Johnson 2003, p.27, 150) Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights of set theory to properties of the diatonic collection such as maximal evenness, Myhills property, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. ... In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. ... In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle (interval class), in other words the number of half steps between notes. ... Music theory is a set of systems for analyzing, classifying, and composing music and the elements of music. ... A mathematician is a person whose area of study and research is mathematics. ...

Myhill's property, that there are two specific intervals for every generic intervals, is maximally even, and both properties are true of the diatonic collection, for instance adjacent notes are separated only by half-tones and whole-tones (1 and 2). The whole-tone scale is also maximally even, for instance adjacent notes are separated only by whole-tones. In diatonic set theory Myhills property is the quality of scales or collections with exactly two specific intervals for every generic interval, and thus also have the properties of maximal evenness, cardinality equals variety, structure implies multiplicity, and be a well formed generated collection. ... The musical interval of a half step, semitone, or minor second is the relationship between the leading tone and the first note (the root or tonic) in a major scale. ... The musical interval of a major second — also called a whole-tone — is the relationship between the first note (the root or tonic) and the second note in a major scale (and also a minor scale). ...

Second-order maximal evenness is maximal evenness of subcollections with regards to a larger collection which is maximally even. Diatonic triads and seventh chords possess second-order maximal evenness, being maximally even in regards to the maximally even diatonic scale but are not maximally even with regards to the chromatic scale. (ibid, p.115) This nested quality resembles Fred Lerdahl's "reductional format" for pitch space from the bottom up: Fred Lerdahl, Fritz Reiner Professor of Musical Composition at Columbia University, is a composer and musicologist, most well known for his work on music theory regarding pitch space and cognitive restraints on compositional systems or musical grammars. ... Reduction or reducing has several meanings: In mathematics, reduction is the process of manipulating a series of equations or matrices into a desired simpler format. ... In music pitch space is pitch relations, ie nearness or farness, represented through geometric models, most often multidimensional, how near or far pitches are from each other. ...

(Lerdahl, 1992)

The musical theory of maximal evenness has been used as part of the Ising model especially in physics to model electron behaviour (Johnson 2003, p.144). Music theory is a set of systems for analyzing, classifying, and composing music and the elements of music. ... The Ising model, named after the physicist Ernst Ising, is a model in statistical mechanics. ... Wikibooks Wikiversity has more about this subject: School of Physics sci. ... Properties The electron (sometimes called negatron; commonly represented as e−) is a subatomic particle. ...

Further reading

• Clough, John and Douthett, Jack (1991). "Maximally Even Sets", Journal of Music Theory 35: 93-173.

Source

• Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1930190808.