Getting involved with mathematical music theory

Abstract

This essay advocates the integration of mathematical reasoning into the teaching of music theory. It offers a didactical pathway through a series of attractive results from well-known algebraic approaches to the study of diatonicity. In particular, strategies are discussed that let the students re-enact processes of investigation and discovery. The content centers around the study of intervals and chords within the diatonic scale. This involves the distinction between generic and specific descriptions, combinatorics of voice leading and chord inversion, as well as the explanatory power of the cycle of fifths. In order to reach its goals, the essay strives towards the introduction of unorthodox media, such as the Tonkreisel, the Rotating Square, or the Diatonic Cord (rope).

Keywords:

• diatonic theory
• diatonic seventh chords
• parsimonious voice leading
• Cardinality Equals Variety
• structure implies multiplicity
• circle of fifths
• well-formed scales
• Tonkreisel
• MIDI SolFa Mode-Go-Round

Acknowledgements

I cordially thank Thomas Fiore, Jason Yust, and the anonymous reviewers for numerous helpful suggestions and a careful reading of the manuscript.

Notes

¹In the literature, such as Regener (1973), Clough and Myerson (1985), Carey and Clampitt (1989), Agmon (1989), Clough and Douthett (1991), Agmon (1991), Carey and Clampitt (1996), Douthett (2008), Clampitt (2008), these concepts have been explicitly and implicitly developed at different levels of generality. In the present essay I resist the temptation to recapitulate the underlying history of ideas.

²The term “octave” carries both interval meanings, namely the generic span of seven scale steps and the specific pitch height size of log _b(2:1), which is commonly normalized to the value log ₂(2:1)=1. In scale theory both meanings are subject to generalizations and one may find terms such as “interval of periodicity” or “pseudo-octave.”

³The term was coined by the Jazz pianist and teacher Barry Harris.

⁴A thorough understanding of the different kinds of violations of the Cardinality-Equals-Variety property requires additional engagement. The interested reader is referred to Clampitt (2008).

⁵The concept appears, for example, in Peirce (1931–5, Vol. 3, Section 468 and Vol. 4, Sections 434 and 564).

⁶One should not conflate diagrams merely with graphical figures, but at this point I prefer to omit such sophistications.

⁷Here the term “interval cycle” is simply used in distinction from the term “interval chain.” In American set theory “interval cycle” refers more specifically to a cycle of chromatically-equal intervals, like a diminished seventh or whole tone scale. In the present paper, the equality of the (either generic or specific) intervals is not presupposed.

⁸See https://sites.google.com/site/solfamodegoround/.

⁹For example, “add2-chords” are prominent in the choral works of Morton Lauridsen.

¹⁰I lay my iPad on a bar stool or some similar piece of furniture, such that I can go around while playing. Alternatively, the app offers the possibility to let the inner playground rotate.

¹¹Theoretical explanations and tutorials may be accessed at https://sites.google.com/site/solfamodegoround/home/.