1. Iannis Xenakis, Formalized Music, rev. ed. (Stuyvesant, N.Y.: Pendragon Press, 1992).  Page number references to this book are given in the text henceforth.

2. Throughout this article, we concern ourselves with pitches, as opposed to pitch classes, unless otherwise indicated.

3. Two minor errors in PLANE 2 of the diagram as given by Xenakis have been corrected in Figure 1. First, the region corresponding to ABC in the leftmost figure on the ff line has been shaded, where it was blank in the original. Second, a missing overline has been added to the character C in the rightmost figures on each line (both ppp and ff). Also, I have deduced that the broken line between A B C and (A B + A B)C should have its arrowhead pointing at the latter rather than the former.

4. Xenakis observes that the form of Equation 2 is more computationally efficient than the disjunctive normal form of Equation 1 in the sense that it involves fewer total union, intersection, and complementation operations. The difference is ten operations versus fourteen, assuming that intermediate results are available for multiple re-uses once computed. In particular, observe that once A B + A B has been computed, it can be used to compute G by means of a single complementation, as opposed to computing G from scratch using A, B, and C. This fact has little practical implication for the composition, however, since not all of the different operations involved are sequentially illustrated. PLANES 1 and 2 each contain fewer than the respective fourteen and ten diagrams, with the G C diagram, furthermore, being repeated.  Xenakis indicates that seventeen operations, rather than fourteen, are required to compute the right-hand side of Equation 1 (Formalized Music, 173).  This would be the case if the complements A, B, and C could not be multiply re-used in computations. Such re-use of intermediate results is, of course, necessary to achieve the computational efficiency associated with the alternative form specified by Equation 2.

5. B�lint A. Varga, Conversations with Iannis Xenakis (London: Faber & Faber, 1996).

6. Iannis Xenakis, Herma (London: Boosey & Hawkes, 1967).

7. James Tenney, META / HODOS: A Phenomenology of 20th-century Musical Materials and An Approach to the Study of Form; and META Meta / Hodos, 2nd ed. (Oakland: Frog Peak Music, 1992).

8. Ibid., 87.

9. Ibid., 94.

10. Ibid., 113.

11. Varga, Conversations, 85.

12. Albert S. Bregman, Auditory Scene Analysis: The Perceptual Organization of Sound (Cambridge, Mass.: MIT Press, 1990).

13. One might be tempted to explain the apparent inconsistencies by invoking the notion of "fuzzy sets," first introduced into the scholarly literature in 1965 by engineer and mathematician Lotfi A. Zadeh as a generalization of the classical set concept (Lotfi A. Zadeh, "Fuzzy Sets," Information and Control 8 (1965): 338-353). Is it possible that Xenakis anticipated this concept in 1961?

A "fuzzy set" is one whose elements can possess degrees of confidence of membership between zero (in which case the element certainly is not in the set) and one (in which case the element certainly is in the set). In a classical (or "crisp") set, on the other hand, the degree of membership of any element is either precisely one or precisely zero.

If the degree of membership of an element x in a given fuzzy set A is, for example, 0.8, then the degree of membership of x in A is 1 - 0.8 = 0.2. Thus, one would expect to observe x in a randomly drawn sample set whether it was drawn from A or A (albeit with greater frequency in a sample set drawn from A).

Nonetheless, I maintain that a fuzzy set model is not appropriate for Herma, for the following reasons. First, nowhere in his detailed theoretical discussion of Herma in Formalized Music does the composer introduce any concept relatable to fuzzy sets. Indeed, his description of complementation clearly assumes classical sets: "If class A has been symbolized or played to [the listener] and he is made to hear all the sounds of R except those of A, he will deduce that the complement of A with respect to R has been chosen" (p. 171).

Furthermore, it should be noted that it is impossible in principle for a listener or analyst to determine with certainty the precise degree of membership of any given pitch in a fuzzy pitch set from a finite random sampling of the set in question. That is, if it were given that fuzzy sets were employed in Herma, it would not be possible to definitively characterize them either upon hearing the work or upon inspection of the score. The composer unequivocally advances a model of the work as a set-theoretic argument cast in pitch. The introduction of fuzzy sets would not only represent an unmotivated complication of the exposition, but would prevent the terms in that exposition from being determinable with perfect confidence.

14. Iannis Xenakis, Herma, Yuji Takahashi, piano. Denon 33CO-1052 [1972]. Compact disc.

15. Nouritza Matossian, Xenakis (New York: Taplinger, 1986), 147.

16. Brian J. C. Moore, An Introduction to the Psychology of Hearing, 4th Ed. (San Diego: Academic Press, 1997), 246.

17. Matossian, Xenakis, 151.

18. Iannis Xenakis, "La crise de la musique serielle," Die Gravesaner Bl�tter 1 (1955): 2-4. Cf. Matossian, Xenakis, 85-86.

19. Varga, Conversations, 96.

20. I would like to thank Profs. David Lidov and James Tenney for their encouragement and guidance during the writing of this article. As well, I would like to thank two anonymous reviewers for their comments and MTO Editor Eric Isaacson both for his helpful suggestions and for performing some computer simulations which instigated those reported in Paragraph 4.1.

End of footnotes