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)

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)

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 "An Analysis of the Properties of 'Deep Scales' in a T-Tone System", unpublished.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.