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Fowler, Michael D. (2019): Mapping k-combinations and Dih(4) in John Cage's Variations I as utilities for determinate and indeterminate realization strategies. In: Journal of Mathematics and Music, Vol. 13, No. 2: pp. 171-191
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Abstract

This article analyses the 1958 graphic score, Variations I, by John Cage (1912-1992). I firstly trace the resistance that the work has established towards traditional analysis, given its meta-score qualities, and the "distance metric problem," which arises from the necessity to generate musical parameter data from the measurement of perpendiculars between points and lines printed on six transparent sheets. I propose that an extension to Cage's instructions allows the symmetry of the transparencies to be described as the dihedral group of order 8 (). I also report on the size of the k-combinations (with and without repetition) of the transparencies and account for the sound densities and sound classes of the work. This analysis then allows for the development of two opposing realization frameworks that are determinate and indeterminate in nature.