2. Our use of the word "topological" here and elsewhere should not be confused with its usual meaning in mathematics, but the metaphorical connection between the two meanings should not be dismissed. The intuitive (and somewhat naive) sense often associated with "topology" is "rubber sheet geometry." Here we wish to transfer that intuition to musical systems as "rubber band geometry." For the reader who wishes a clearer distinction, perhaps a formal definition of "musical topology" might read, "the study of the properties of musical structures that remain invariant under certain transformations." This now shifts the emphasis to "certain transformations," which is precisely the object of the present study. At any rate we will periodically remind the reader of our special sense of the term topological by the use of quotation marks or, more simply, by the abbreviation TT.
3. #intj(S,T) is identical to David Lewin's interval function IFUNC which he first posited in "Re: Intervallic Relations Between Two Collections of Notes," Journal of Music Theory 3.2 (November 1959): 298-301.
4. While we are not attempting to "interpret" these basic relationships, to understand the distinctions we are making here and throughout, it may be helpful to review the last section ("Philosophical musings") in David Lewin's classic article "Forte's Interval Vector, My Interval Function, and Regener's Common-Note Function," Journal of Music Theory 21:2 (fall 1977) 227ff.
5. The idea of characterizing a pc set by listing its contiguous intervals when the set is in "close" position has a history that stretches back at least 80 years. The better known references to this practice are Richard Chrisman, "Describing Structural Aspects of Pitch-Sets Using Successive-Interval Arrays," Journal of Music Theory 21:1 (spring 1977) and Eric Regener, "On Allen Forte's Theory of Chords," Perspectives of New Music 13:1 (1974). Also, Robert Morris uses "interval succession" in Composition With Pitch-Classes: A Theory of Compositional Design (New Haven: Yale University Press, 1987). In The Harmonic Materials of Twentieth-Century Music (New York: Appleton- Century-Crofts, 1960) Howard Hanson employs "intervallic order" throughout to accompany his version of ic vectors. But the most fascinating use (pointed out to the author by Richard Cohn in private correspondence) can be found in Ernst Lecher Bacon's "Our Musical Idiom" first published in The Monist 27:1 (October 1917). Not only does Bacon employ an intervallic string notation, he uses it to define a chord's normal order and gives what appears to be a complete list of set classes mod 12.
6. David Lewin, Generalized Musical Intervals and Transformations (New Haven: Yale University Press, 1987): 175ff.
7. John Clough and Gerald Myerson, "Variety and Multiplicity in Diatonic Systems," Journal of Music Theory 29.2 (fall 1985).
8. Clough and Myerson (op cit.) refer to this set of specific strings as "species."
9. In Clough and Myerson (op cit.) this equality appears as "cardinality equals variety."
10. The reader is invited to work a few examples to help clarify this concept. Suggestion 1: let q = <1123>, q' = <3211>, r = r' = <1211213>. Suggestion 2: let q = q' = <1222>, r = <1112132>, r' = <2312111>. Suggestion 3: let q = 1123, q' = <3211>, r = <1112132>, r' = <2312111>. Note especially the effect produced by "symmetric" strings as preparation for the comments in [2.20].
11. To give some indication of what might lie ahead in a general study of these patterns, consider a seemingly innocent change in this example's generic string from <1133> to q = <1313>. The generic string now has two degrees of transpositional symmetry and two degrees of inversional symmetry. If a = <1616> and b = some circular permutation of <2534>, the pattern of specific strings is:
12. We assume the reader has some knowledge of "interval vectors" and related concepts which, over the past 30 years have made appearances in a wide variety of forms and contexts. To trace the history of this concept would take us too far afield in the present study. But the most important connections, especially relating "multiplicity" and "common tones" are brought out by David Lewin in "Forte's Interval Vector, My Interval Function, and Regener's Common-Note Function."
13. There is a convergence of ideas here that is worth noting. First, John Clough and Jack Douthett ("Maximally Even Sets," Journal of Music Theory 35.1-2: 118) define the multiplicity function DFUNC(X,k,I) as "the numbers of intervals of [chromatic length] k and [diatonic length] I in the set X." If the interval string of X is r, then DFUNC is clearly identical to our "substring counting function" since DFUNC(X,k,I) = #(r;I,k) in all cases. But for the present study we find it useful to allow all values of #(r;I,k) including zeroes so as to arrange these values in matrix form. Second, Robert Morris (op cit., p.40) defines a cyclic interval succession function CINT which is related to our "substring listing function" r;x,y. An example should make this relationship clear. If X = {0,3,5,9} in C12, say, for m=2 CINTm(X) = (5,6,-5,6) = (a,b,c,d). X's interval string is <3243>, so CINTm(X) gives us the y-values and x = m for r;x,y:
14. At first this may seem like a circuitous path to the obvious, but we will find increasingly that the matrix form for displaying interval multiplicities reveals more than the (summary) vector form. In Part II, int-matrices will be indispensible in understanding efficient linear transformation saturation.
15. Just as the interval string is a circle, the corresponding interval multiplicity matrix is a torus. So when the "left edge" of the matrix is reached, the reading is continued at the "right edge."
16. The set of y's corresponding to non-zero MINT entries for a given x is Clough and Myerson's "spectrum" of x (op cit.).
17. Jay Rahn, "Coordination of Interval Sizes in Seven-Tone Collections," Journal of Music Theory 35:1-2 (spring-fall 1991).
18. Freedom from ambiguity is what Eytan Agmon terms "coherence." See for example, "Coherent Tone Systems: a Study in the Theory of Diatonicism," Journal of Music Theory 40.1 (spring 1996).
19. Allen Forte, The Structure of Atonal Music (New Haven: Yale University Press, 1973).
20. A concise statement of this principle can be found in John Rahn, Basic Atonal Theory (New York: Schirmer Books, 1980): 107ff.
21. Keep in mind that we are working with ic-vectors at this point. q = <2131> is a string in C7, an odd-sized space, so V(q) = [213] accurately reflects the common tone pattern; but if q was from an even-sized space, the last component of V(q) must be doubled to read it as a common-tone pattern.
22. There is some resonance here with Richard Cohn's concept of "transpositional combination" whereby pc sets are constructed by combining two or more transpositionally related subsets ("Transpositional Combination in Twentieth Century Music," Ph.D. Dissertation, University of Rochester, 1987). In Part II it will be noted that a "Riemann cover" is in effect one possible generalization of Cohn's idea to microtonal spaces.
23. G. Mazzola has developed the idea of "partial" covers, what he terms "minimal cadence sets," an interesting special case of the concept of covering a scale. His work is described in Daniel Muzzulini, "Musical Modulation by Symmetries," Journal of Music Theory 39.2 (fall 1995).
End of Footnotes