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 (including major, natural minor, and the modes). (Johnson 2003, p.41) 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 music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... In music, a scale is an unordered collection of notes or pitches, as opposed to a series of intervals, which is a musical mode. ... In music, specifically, musical set theory an interval class, or unordered pitch-class interval, is an interval measured by the distance between its two pitch classes ordered so they are as close as possible. ... In music theory, a diatonic scale is a scale whose notes are built on the natural staff positions of lines and spaces, with no accidentals, with or without a key signature. ... In music theory, the major scale is one of the diatonic scales. ... A minor scale in musical theory can be viewed as the sixth mode of the major scale. ... In music, a mode is an ordered series of musical intervals, which, along with the key or tonic define the pitches. ...

The common tone theorem describes that scales possessing the deep scale property share a different number of common tones for every different transposition of the scale, suggesting an explanation for the use and usefulness of the diatonic collection. (ibid, p.42) In music transposition is moving a note or collection of notes (or pitches) up or down in pitch by a constant interval. ...

Further reading

• Winograd, Terry. "An Analysis of the Properties of 'Deep Scales' in a T-Tone System", unpublished.
• Gamer, Carlton (1967). "Deep Scales and Difference Sets in Equal-Tempered Systems", American Society of University Composers: Proceedings of the Second Annual Conference: 113-22 and "Some Combinational Resources of Equal-Tempered Systems", Journal of Music Theory 11: 32-59.
• Browne, Richmond (1981). "Tonal Implications of the Diatonic Set" In Theory Only 5, no. 6-7: 6-10.

Source

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