Enharmonic systems: A theory of key signatures, enharmonic equivalence and diatonicism

Abstract

Key signatures and enharmonic equivalence are taken as points of departure for a study of the diatonic–chromatic relationship. Key signatures are modelled by signature vectors, i.e. seven-dimensional vectors with integer coordinates, each coordinate indicating the number of sharps (+) or flats (−) assigned to one of the seven letter names. A definition of standard signature vectors (those corresponding to the usual key signature of some major key) is readily formulated. These constructions do not depend on any convention regarding enharmonic equivalence of note names, but enharmonic equivalence (EE) conditions are intimately connected with key signatures, and in fact may also be formalized in terms of signature vectors, called in this context EE vectors. The canonical EE vector gives rise to a familiar twelve-note enharmonic system, but other systems are possible. Connections with maximal evenness and other properties familiar in diatonic set theory are investigated. The usual staff notation, including key signatures, may be realized within any enharmonic system, and various transformations (diatonic and chromatic transposition, and signature transformations that alter the key signature) may be applied to music thus notated. The interaction between the EE vector defining a system and the signature vector defining a seven-note subset thereof is subtle and sometimes unexpected. Several non-canonical enharmonic systems are explored, including those used in the Twelve Microtonal Etudes of Easley Blackwood.

Keywords:

• Key signatures
• Enharmonic equivalence
• Diatonic scales

Acknowledgements

The author thanks David Clampitt, Thomas Noll, Dmitri Tymoczko and an anonymous referee for helpful comments and suggestions.

Notes

^*Modifiers beyond±2, though extremely rare in musical scores, are not unknown 2. Triple sharps occur in Valentin Alkan's Etude, Op. 39, No. 10 (1857), m. 291, and Max Reger's Sonata for Clarinet and Piano, Op. 49, No. 2 (1900), fourth movement, m. 91. Triple flats occur in the Piano Sonata No. 1 (1914) of Nikolai Roslavets, mm. 152–153.

^*The terminology used in Theorem 4.1 is standard in the diatonic set theory literature. The maximal evenness condition in part (a), first defined by Clough and Douthett [1], means, loosely speaking, that the notes of are distributed as uniformly as possible around the circle of Figure 2. Myhill's Property, mentioned in part (b), means that every diatonic interval comes in precisely two chromatic sizes; this is a version of the principle ‘cardinality equals variety’ 13. The interval equation in part (c) of the theorem states that is generated by the ‘perfect fifth’ interval. A seven-note generated scale is well-formed, as defined by Carey and Clampitt 14 15, if the generated ordering is related to the scalar ordering by some automorphism of ℤ ₇; as noted in Section 1, this is always the case for the fifths in .

^*Key signatures with voice crossings were first studied by Tymoczko 7.

^*Blackwood discusses 22-note tuning at some length in his book [8, pp. 302–304], but does not actually mention the particular arrangement that appears in the Etude.