Scales, Sets, and Interval Cycles: A Taxonomy

Abstract

Recent studies in theory of by Agmon, Balzano, Carey and Clampitt, Clough and Douthett, Clough and Myerson, and Gamer have in common central role of interval cycle. Based on scale features defined in these studies, and an additional feature called distributional evenness defined here, a taxonomy is proposed for pitch-class sets (pcsets) correspond to interval cycles or to certain conjunctions thereof. Pairwise implicative relationships among features are explored. Of twenty sets of features that are consistent with these relationships, thirteen are found to be instantiated by actual pcsets and seven others are shown to be incapable of instantiation. Most instantiated feature-sets correspond to infinite classes of pcsets which are shown to be enumerable; one such feature-set is found to be uniquely realized (up to transposition) in usual diatonic pcset. Earlier versions of this paper, by Clough and Engebretsen, were presented at conferences of Society for Music Theory, New York, November 1995, and European Society for Cognitive Sciences of Music, Uppsala, Sweden, June 1997. WORKS CITED Agmon, Eytan. 1989. A mathematical model of diatonic system. Journal of Music Theory 33: 1-25. . 1991. Linear transformations between cyclically generated chords. Musikometrika 3: 15-41. . 1996. Coherent tone-systems: a study in theory of diatonicism. Journal of Music Theory 40: 39-59. Balzano, Gerald J. 1980. The group-theoretic description of 12fold and microtonal pitch systems. Computer Music Journal 4: 66-84. Browne, Richmond. 1981. Tonal implications of diatonic set. In Theory Only 5/6-7: 3-21. Carey, Norman, and David Clampitt. N.d. Two theorems concerning rational approximations: number theory and music theory. Typescript. -. 1989. Aspects of well-formed scales. Music Theory Spectrum 11: 187-206. Clampitt, David. 1997. Pairwise well-formed scales: structural and transformational properties. Ph.D. diss., State University of New York at Buffalo. Clough, John. 1994. Diatonic interval cycles and hierarchical structure. Perspectives of New Music 32/1: 228-53. Clough, John, and Gerald Myerson. 1985. Variety and multiplicity in diatonic systems. Journal of Music Theory 29: 249-70. Clough, John, and Gerald Myerson. 1986. Musical and generalized circle of fifths. American Mathematical Monthly 93: 695-701. Clough, John, and Jack Douthett. 1991. Maximally even sets. Journal of Music Theory 35: 93-173. Cohn, Richard. 1991. Properties and generability of transpositionally invariant sets. Journal of Music Theory 35: 1-32. This content downloaded from 207.46.13.33 on Wed, 12 Oct 2016 04:55:22 UTC All use subject to http://about.jstor.org/terms 104 Music Theory Spectrum . 1996. Maximally smooth cycles, hexatonic systems, and analysis of late-romantic triadic progressions. Music Analysis 15: 9-40. Gamer, Carlton. 1967. Some combinational resources of equaltempered systems. Journal of Music Theory 11: 32-59. Hanson, Howard. 1960. The harmonic materials of modern music: resources of tempered scale. New York: Appleton-Century-Crofts. Lewin, David. 1982. A formal theory of generalized tonal functions. Journal of Music Theory 26: 23-60. . 1996. Cohn functions. Journal of Music Theory 40: 181-216. Messiaen, Olivier. 1944. Technique de mon langage musicale. Paris: A. LeDuc. Morris, Robert D. 1987. Composition with pitch-classes. New Haven: Yale University Press. Rahn, Jay. 1996. Turning analysis around. Black Music Research Journal 16: 71-90. Soderberg, Stephen. 1995. Z-related sets as dual inversions. Journal of Music Theory 39: 77-100. Starr, Daniel, and Robert Morris. 1977-78. A general theory of combinatoriality and aggregate. Perspectives of New Music 16/1: 3-35, and 16/2: 50-84. Vieru, Anatol. 1995. Diatonicity and chromaticity. Muzica 21: 64-68. Wilson, Erv. 1975. Letter to John Chalmers re the basic structure of moments of symmetry. Winograd, Terry. N.d. An analysis of properties of deep scales in a T-tone system. Typescript. Wooldridge, Marc. 1992. Rhythmic implications of diatonic theory: a study of Scott Joplin's ragtime piano works. Ph.D. diss., State University of New York at Buffalo. This content downloaded from 207.46.13.33 on Wed, 12 Oct 2016 04:55:22 UTC All use subject to http://about.jstor.org/terms