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A Ma thema tical Model Of Tonal Function Robert T. Kelley Lander University <http://www. robertkelleyphd.c om/tonalfunction.h tm> Abstract I propose a mat hematical mode l tha t formalizes the der iv ation of rec en t tonal harmonic theories that posit potential harmonic functions for each scale degree, such as Daniel Harrison’s Harmonic Function in Chromatic Music and Ian Quinn’s “Harmonic F unction without Primary T riads”. Mathematical groups that model tonal scale-degree functions will help to clarify the use of these func- tions to aid in compositio n and analysis. The mathematica l represe ntation of pitches is based on the ordered pair notation introduced by Alexander Brinkman (Spectrum 1986). F ollowing an intuitive analytic al discussion of the mathematical groups and the algebraic functions that relate them, I give examples of the use- ful distinctions among the tonal scale-degree functions that are claried by this theory. I shall then use the distinctions among scale-degree functions supported by the mathematical model to reinforce and rene Ian Quinn’s functional desig- nations as well as to contribute to the current systems of part-writing techniques based on scale-degree functions. Some of you may have been among the theorists who crammed into the crowded room in Boston last November where Ian Quinn was presenting his intriguing approach to theory pedagogy. While I was sitting on the oor in the front of that room, I found that he and I had been thinking along much the same lines with regard to the most economical and protable wa ys of teaching harmon y to undergraduates. Ian, ho we ver, was wa y ahead of me. His harmonic model, which emphasizes scale-degree function while simultaneously downplaying the role of primary triads as generators of those functions, is supported by his systematic use 1

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