In [traditional harmony], the fundamental material associated with a tonal center is a set of pitches associated with the tonal center; in a particular horizontal order, we call them diatonic scales. More generally, consider also, modes and ragas with ascending and descending aspects.

On the traditional western scales, triads (vertical structures) are constructed on each pitch of the scale, where, of course, the scale ordering is a significant structuring of the fundamental pitch set. [Cyclic permutations of the ordering generate well tempered modes.] The triads are to some extent, treated like sets both in terms of melodic line subelements and as chords, by virtue of the use of horizontal permutations of the pitches and chordal inversions. The triads obviously "cover the scale" but do not partition it; in fact every triad shares two pitches with two other triads, and shares one pitch with two other triads, generating a topology on the scale set. For example, in C major

             III I  VI                    I

              B
              G  G                        G - B D    V
              E  E  E                     E
                 C  C         IV    F A - C
                    A

The less a pitch set has in common with another, it seems, the more strongly the two are harmonically related.

With the above considerations, it would appear that the standard way of thinking of chordal structures and their horizontal ordering as being the substance of harmony, is in reality too rigid an academic construct; that one ought to consider harmony more generally from the viewpoint of the set theoretical language that is implied by the traditional chordal analysis. Vertical harmony can be, and frequently is implied by horizontal chord presentation, i.e., a sequence of pitches of the chord.

Similarly then, as a logical extension to the dodecaphonic set, replace the chromatic scale with the chromatic set, and consider the chromatic harmony implied by a class of subsets. The possibilities increase enormously:

        1) Cardinality of the fundamental pitch set increases
           from 7 to 12.

        2) There is no longer any reason to construct triads
           on a chromatic scale (or a dodecaphonic row), so that
           restriction is removed.

        3) Thinking of a dodecaphonic row, an obvious choice
           of choosing a possible class of subsets is by overlapping
           segments of the row, each of which can then be treated
           autonomously, in varying degrees of vertical and horizontal
           presentation, and similarly with respect to each other.
           Though this seems like almost too much freedom, there is
           the  [Wolpe 1959]  concept of chroma to consider.
           This article should be allowed to speak for itself and
           not be interpreted.

         4) The transcendence from dodecaphonic row to dodecaphonic
            subsets or subsegments, as Wolpe has shown breaks down
            the concept of octave equivalence inherent in
            the language of strict pitch serialism, thus, in practice,
            increasing the cardinality of the fundamental pitch set
            from 12 further to 88 or more.  For example in the set

                                  Bb A B

            composed of intervals of minor and major seconds, the
            translations of any of the pitches by octave(s) is now a
            meaningful and significant transformation of the set, which
            can be considered new pitch sets distinct from, but clearly
            related to its parent.

          5) Pitch sets within a composition need not remain static,
             but become dynamic objects, subject to transformations
             by octave translations of its pitches, pitch accretion
             and deletion, set theoretic transformations of union,
             intersection, difference, symmetric difference,
             complementation.  As with any other sequence of musical
             transformations, the structural clarity or lack thereof
             is up to the purposes and artistry of the composer.

The concept of tonality or more generally of a sound center, while explicitly banned in serial music, is once more obtainable in music built on pitch sets. The center can be a pitch or a set, as one previously considered the tonic note or the I chord of a diatonic pitch sequence. The choice of which may change during the course of a piece, and is a matter of the composer's own voice.

There are no historically established cadential formulas by which a composer may say with his music and to the listener, "Here, this is the current center."

In the absence of cadential formulas, much less a gestural tradition of any kind, the most primitive and powerful principle of establishing a center is by repetition and return, simple devices that are audible to the attentive ear.

The method of composing by pitch sets is quite unlike Schoenberg's method in that it does not make the composer's task easier, but rather considerably more free and more difficult since more choices have to be made. There is therefore also a greater possibility for higher and lower art, depending on the composer's choices, and the artistry with which they are made.

The notion of transcending serialism through pitch sets is not mine. It is implied in the serialistic works of [Anton von Webern], and brought consciously into being by [Stefan Wolpe]. It was taught to me by both Wolpe and Raoul Pleskow. The idea that groups of transformations underlie the logical structure of all music, and that they are particularly appropriate to shaping musical structures from the vast possibilities contained in the pitch set language I believe to be original, though Xenakis was certainly an influence.

Wolpe has left us a rich and pliable medium for the art of musical composition that has not yet been completely explored. The future is still out there.