1. John Clough and Jack Douthett, "Maximally Even Sets," Journal of Music Theory 35.1 & 2 (1991) : p.96. 

2. There is some redundancy in this notation since any necklace class has any number of ME tokens; for example, <(7;12)> represents the same necklace class as, say, <(7;9)> and <(7;16)>. While it would be more cumbersome and less intuitive, this notation could be improved by partitioning all ME strings with M(m,x,k) defined as all strings of the form <m;xm%k> where m,x,k are integers with m >= 1; x = 1,2,...; k <= [[m/2]]; xm%y >= m; and % stands for "plus or minus." N(m,x,k) would then represent all necklaces which can be derived from any member of M(m,x,k) by substitution. Or, using the "generating" case and the above notation, we might simply use N(m,k) defined as <(m;m+k)>. Thus <7;12>, <7;9>, <7;16>, as well as <1171117> and so on, all belong to the necklace class N(7,2) = <(7;9)> = <aabaaab>. 

3. This particular WARPSET presents a situation that occasionally arises when resultant strings are symmetric. Note that a' is a circular permutation of a, i.e., a and a' represent the same shape, so for any p in C13, (p,a') = (p-4,a) (mod 13) (see [1.13] in Part I). If we "canonize" a = <445> as the "correct" string form and require that all a-based structures be put in that "root position" form, the CHORDSET given in [5.6] must be abandoned for {0a,1a,11a,4b,5b,6b,7a,4a,5a,11b,12b). This in turn makes scale elements 4, 5, and 11 into roots which can be associated with both a and b strings, whereas scale elements 2, 8, and 9 can't be associated as roots with any chords. 

4. Clough and Douthett, op. cit., pp.170-71.

5. "System" is used here in the sense of the ordered triple of [2.25] in Part I. "Diatonic system" may be confused with that term as used by C&D (---) and Agmon (---), and for that reason we will mostly use the term "Riemann Diatonic System" introduced below. 

6. Diatonic tetrachords (e.g., the dominant 7th-chord) are not included here since they would unduly complicate the present study. But the reader may want to keep these important structures in mind as we procede since they present an interesting dilemma. In the usual diatonic system, clearly one could view a 7th-chord as either a triad with an added third *or* the complement of a triad with respect to the diatonic scale (at the end of the day they produce the same structure). However, as the reader may note after reading about the next RD system, these two possible construction principles, when lifted into RD(2), produce different results. In RD(2) the basic structure is a 5-note chord. If we use the additive principle, the "hyper7th-chord" will have six notes. But if we use the complement principle, the "hyper7th-chord" will have eight notes. 

7. The reader is invited to generate and compare the AVLINTs for the three other possible triadic white note systems which can be associated with the usual diatonic string using g = <115>, <133> (necklaces) or <124> (asymmetric). It is then an interesting exercise to search for non-diatonic scale strings for each of these generic strings which might be profitably employed as a basis for compositional praxis (both inside and outside C12).

8. Clearly the converse, a preference for maximal variety coupled with a secondary desire to discover as much symmetry as possible, is most attainable by choosing COVER2 -- or by choosing neither or both. In the larger sense, if we were to adopt a preference for variety throughout, we could have stopped with Part I since, when we find ourselves in any (structured) musical universe, what we can say about any string can be said about symmetric strings as well, and cover covariance is simply an objective fact associated with whatever system we happen to find ourselves in at the moment. But in beginning Part II by focusing on necklaces and then on maximal evenness and (hyper)diatony we already (perhaps covertly) stated a preference for symetric structures which a fortiori guides our present and subsequent choices.

9. David Lewin, "A Formalized Theory of Generalized Tonal Functions," Jounal of Music Theory 26.1 (1982). 

10. Lewin, op. cit., p.26.

11. Lewin, op. cit., p.37-39.

12. Lewin, op. cit., p.48.

End of Footnotes