Maximal evenness
Encyclopedia
In diatonic set theory
maximal evenness is the quality of a collection or scale which for every generic interval
there are either one or two consecutive (adjacent) specific interval
s, 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)
Myhill's property
, that there are two specific intervals for every generic interval, 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.
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 regard to the maximally even diatonic scale but are not maximally even with regard to the chromatic scale. (ibid, p.115) This nested quality resembles Fred Lerdahl
's "reductional format" for pitch space
from the bottom up:
The musical theory of maximal evenness has been used as part of the Ising model
in physics
to model electron
behavior. (Johnson 2003, p.144).
In a dynamical approach, spinning concentric circles and iterated maximally even sets have been constructed as an alternate approach to Riemann theory. This approach leads to some interesting connections between diatonic and chromatic theory. (Douthett, 2008)
Diatonic set theory
Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights 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...
maximal evenness is the quality of a collection or scale which for every generic interval
Generic interval
In diatonic set theory a generic interval is the number of scale steps between notes of a collection or scale. The largest generic interval is one less than the number of scale members...
there are either one or two consecutive (adjacent) specific interval
Specific interval
In diatonic set theory a specific interval is the shortest possible clockwise distance between pitch classes on the chromatic circle , in other words the number of half steps between notes. The largest specific interval is one less than the number of "chromatic" pitches. In twelve tone equal...
s, in other words a scale which is "spread out as much as possible." This property was first described by music theorist
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
John Clough and mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Jack Douthett in "Maximally Even Sets" (1991). (Johnson 2003, p.27, 150)
Myhill's property
Myhill's property
In diatonic set theory Myhill's property is the quality of musical 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...
, that there are two specific intervals for every generic interval, 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.
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 regard to the maximally even diatonic scale but are not maximally even with regard to the chromatic scale. (ibid, p.115) This nested quality resembles Fred Lerdahl
Fred Lerdahl
Alfred Whitford Lerdahl is the Fritz Reiner Professor of Musical Composition at Columbia University, and a composer and music theorist best known for his work on pitch space and cognitive constraints on compositional systems or "musical grammar[s]." He has written many orchestral and chamber...
's "reductional format" for pitch space
Pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...
from the bottom up:
| C | E | G | C | |||||||||
| C | D | E | F | G | A | B | C | |||||
| C | D♭ | D | E♭ | E | F | F♯ | G | A♭ | A | B♭ | B | C |
-
- (Lerdahl, 1992)
The musical theory of maximal evenness has been used as part of the Ising model
Ising model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
to model electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
behavior. (Johnson 2003, p.144).
In a dynamical approach, spinning concentric circles and iterated maximally even sets have been constructed as an alternate approach to Riemann theory. This approach leads to some interesting connections between diatonic and chromatic theory. (Douthett, 2008)
Further reading
- Clough, John and Douthett, Jack (1991). "Maximally Even Sets", Journal of Music Theory 35: 93-173.
- Douthett, Jack and Krantz, Richard (2007). "Maximally Even Sets and Configurations: Common Threads in Mathematics, Physics, and Music", Journal of Combinatorial Optimization 14: 385-410.
- Douthett, Jack and Krantz, Richard (2008). "Dinner Tables and Concentric Circles: A Harmony in Mathematics, Music, and Physics", College Mathematics Journal 39: 203-211.
Sources
- Johnson, Timothy (2003). Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals. Key College Publishing. ISBN 1-930190-80-8.
- Lerdahl, Fred (1992). "Cognitive Constraints on Compositional Systems", Contemporary Music Review 6 (2), pp. 97-121.
- Douthett, Jack (2008). "Filter Point-Symmetry and Dynamical Voice-Leading", Music Theory and Mathematics: Chords, Collections, and Transformations (Eastman Studies in Music):72-107. Ed. J. Douthett, M. Hyde, and C. Smith. University of Rochester Press. ISBN 1-58046-266-9.