The Linear Diophantine Equation in Music Theory
Year: 2001 Authors: Jack Douthett; Richard Krantz
Core claim
Unidirectional P-cycles and related maximally even sets can be constructed using the equation cl_g - dg = ±1.
Topics
music theory, microtonal scales, circle of fifths, P-cycles
Domains
linear Diophantine equations, coprime integers, set classes, music composition, microtonal design, auditory structure
Methods
theoretical construction, music-theoretic analysis, scale generation
Media
microtonal musical scales, Balzano’s 20-fold system, Bohlen-Pierce 13-note scale
Paper text
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BRIDGES Mathematical Connections in Art, Music, and Science
The Linear Diophantine Equation in Music Theory
Jack Douthett Department of Arts and Science, TVI Community College Albuquerque, New Mexico 87106 email: jdouthett@tvi.cc.nm.us
Richard Krantz Department of Physics, Metropolitan State College of Denver, Denver, Colorado 80217-3362 email: krantzr@mscd.edu
In Western music, modulation usually occurs between closely-related keys, keys adjacent to each other on the circle of fifths. According to the New Harvard Dictionary of Music¹ “Because of the way in which sharps and flats are added to key signatures along the circle (of fifths), the number of pitches in common between the starting key and each successive key outward in either direction decreases by one.” In recent works in music theory, microtonal versions of the circle of fifths have become known as unidirectional P-cycles. A P-cycle is a cycle of three or more sets from the same set-class such that there is a map between adjacent sets that leaves all but one of its notes fixed. The note that moves is changed by only a half-step. A P-cycle is said to be unidirectional if all the sets in the set-class are included in a single cycle. Clough and Myerson² were the first to construct an algorithm that generates the complete family of sets that support unidirectional P-cycles, and Clampitt³ has shown that such sets are maximally even sets, as defined by Clough and Douthett⁴, whose chromatic cardinality, c, and the diatonic cardinality, d, are co-prime. Such sets are also generated sets in the sense that it is possible to find a generating interval (in fact there are two) that will generate each set in the unidirectional P-cycle. This may be formalized by a linear Diophantine equation: where g and are the chromatic and diatonic lengths of the generators. In this presentation, we will demonstrate how the linear Diophantine equation may be used to construct microtonal musical scales (i.e. Balzano’s 20-fold system⁵ and the Bohlen-Pierce 13-note scale⁶) having the modulation properties associated with the familiar circle of fifths.
¹ “Modulation,” in The New Harvard Dictionary of Music, ed. D. M. Randell (Belknap, Cambridge, MA, 1986). p. 503. ² J. Clough and G. Myerson, “Musical Scales and the Generalized Circle of Fifths,” Am. Math. Monthly 93(9), 695 - 701, (1986). J. Clough and G. Myerson, “Variety and Multiplicity in Diatonic Systems,” J. Music Theory 29, 249-270 (1985). ³ D. Clampitt, “Pair-Wise, Well-Formed Scales, Structured and Transformational Properties,” Ph. D. dissertation (SUNY at Buffalo, Buffalo, NY, April 1997). ⁴ J. Clough and J. Douthett, “Maximally Even Sets,” J. Music Theory 35, 93 – 173 (1991). ⁵ G. Balzano, “The Group-Theoretic Description of 20–Fold Pitch Systems,” Comput. Music J. 4. 66 – 84 (1980). ⁶ M. V. Mathews, J. R. Pierce, A. Reeves, and L. A. Roberts, “Theoretical and Experimental Explorations of the Bohlen-Pierce Scale,” J. Acoust. Soc. Am. 84, 1214 – 1222 (1984).