Diatonic set theory is a subdivision or application of musical set theory which applies the techniques and insights

Musical analysis

Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

 of discrete mathematics

Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...

 to properties of the diatonic collection such as maximal evenness

Maximal evenness

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 specific intervals, in other words a scale which is "spread out as much as possible." This property was first described by music theorist John...

, 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...

, well formedness, the deep scale property

Deep scale property

In diatonic set theory, the deep scale property is the quality of pitch class collections or scales containing each interval class a unique number of times. Examples include the diatonic scale...

, cardinality equals variety

Cardinality equals variety

The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the same distance in pitch class space...

, and structure implies multiplicity

Structure implies multiplicity

In diatonic set theory structure implies multiplicity is a quality of a collection or scale. This is that for the interval series formed by the shortest distance around a diatonic circle of fifths between member of a series indicates the number of unique interval patterns formed by diatonic...

. 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

David Rothenberg

David Rothenberg is a professor of philosophy and music at the New Jersey Institute of Technology, with a special interest in animal sounds as music...

, who published in the journal Mathematical Systems Theory, and Erv Wilson

Erv Wilson

Ervin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...

, working entirely outside of the academic world.

See also

• Bisector
  Bisector (music)
  In diatonic set theory, a bisector divides the octave approximately in half and may be used in place of a generator to derive collections for which structure implies multiplicity is not true such as the ascending melodic minor, harmonic minor, and octatonic scales. Well formed generated collections...
  
  
• 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...
  
  
• 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...
  
  
• Diatonic and chromatic
  Diatonic and chromatic
  Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to intervals, chords, notes, musical styles, and kinds of harmony...
  
  
• Rothenberg propriety
  Rothenberg propriety
  In music, Rothenberg propriety denotes an important concept in the general theory of scales which was introduced by David Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context by Gerald Balzano, who termed it coherence...
  
  

Further reading

• Johnson, Timothy (2003), Foundations of Diatonic Theory: A Mathematically Based Approach to Music Fundamentals, Key College Publishing. ISBN 1-930190-80-8.
• 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

Precursors

• Wilson, Erv
  Erv Wilson
  Ervin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...
  
  (1975), Letter to Chalmers pertaining to MOMENTS OF SYMMETRY/TANABE CYCLE
• Rahn, Jay (1977), "Some Recurrent Features of Scales", In Theory Only 2, no. 11-12: 43-52.
• Rothenberg, David, (1978), A Model for Pattern Perception with Musical Applications parts I, II, and III , Mathematical Systems Theory, 11, 199-234, 353-372, 12, 73-101.