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But what procedural possibilities for composition are revealed by this system? This paper only begins to explore these. There are certainly implications for polyphony: tones for sound events could be taken from a given contonation while its primary tone could be continuously or intermittently sounded below (or even merely assumed). When the melody reached a certain tone common to another contonation, the primary tone beneath could be changed to that of the new contonation, thereby effecting a "modulation" to the new contonation. The events could then continue, with their tones now being taken from the new contonation. The resulting texture could be analogous to early organum (of, say, Léonin) wherein the tenor comprises a lower, extremely long-breathed chant melody beneath comparatively florid melodic material in the duplum. The fundamental tones of each contonation in series would be analogous to the tenor, while the events above would be analogous to the duplum. The print version of this paper contains excerpts from The Madman's Diary: A Monodrama for Tenor, CD-ROM, Strings and Percussion by Peter Lucas Hulen, which provide models for this type of texture and modulatory procedure.

A system of contrapuntal procedures could also be derived from the pitch relationships within each contonation; for example, single intervallic relationships could be categorized as "consonant" or "dissonant." The tones of the E contonation in all their possible simple intervallic relationships are shown in Figure 7. They are given without their cents deviation from equal temperament, but with the value of each interval in cents. These intervals have been arranged according to an arbitrary system of "perfect consonance," "imperfect consonance," "dissonance," and "empty consonance" (discussed below). Their order within each group is insignificant. The white noteheads would be the first note in the interval (an "existing" note e.g. in a cantus firmus) and the black notes would indicate the second note in the interval (a "proposed note e.g. of counterpoint); thus, all of the possible simple intervals and their simple inversions appear twice, depending on which note is considered primary. The only compound intervals appear where, for example, a (simple) second is considered dissonant whereas its compound form as a ninth, or its inversion as a seventh, is considered consonant. In these cases the ninth above is given with the seventh below in a "consonant" category, while the second is given in the "dissonant" category, and the two are connected with an asterisk or cross. This would be the case for the F# against the G# and vice versa.

Figure 7

Figure 7

The criteria for categorization according to "consonance" or "dissonance" are as follows: Octaves (1200 cents), unisons (0 cents), just fifths (702 cents) and just fourths (498 cents) are discarded as "empty" consonances. Given the now historically-established meaning of thirds and sixths, certain of these intervals may now be considered "perfect consonances," in contrast to the octaves and fifths of 16th century practice. These would include all of the just major thirds (386 cents) and minor sixths (814 cents), just minor thirds (316 cents) and major sixths (884 cents), as well as a minor third/major sixth ten cents from equal temperament (290/910 cents) and a major third/minor sixth thirteen cents from just intonation (373/827 cents). (All intervals given include both their simple and compound forms unless otherwise indicated.) The imperfect consonances would include all other thirds and sixths from 247-455 and 745-953 cents, respectively, (technically, the extremes of these are large major seconds, small fourths, large fifths and small minor sevenths), as well as seconds of a 9:8 ratio (204 cents) and larger and their inversions (sevenths), the compound form of the 182-cent second (a ninth) and its inversion (a seventh), and the "just" tritone (first in the harmonic series) of 583/617 cents which, removed from the context of dominant function in tonality, sounds very consonant. The dissonances include the simple form of the 182-cent second, all seconds simple and compound smaller than 182 cents with their inversions, and all tritones other than the "just."

This categorization provides a workable balance of perfect and imperfect consonances and dissonances, which are divided not merely according to their ratios and/or audible, harmonic consonance, but also according to relative ideas of "perfection" and more recently-accepted ranges of consonance and dissonance, both of which have been historically and socially mediated. The result is a range of relative consonance and dissonance accessible to the conventionally-minded listener on the one hand, yet possessing a considerable degree of inherent dissonance and a fresh range of expression with regard to musical convention on the other. Intervals in these categories may then be applied in a manner analogous to counterpoint, according to, perhaps, a scheme of rhythmic relationships within and between melodic or quasi-melodic events. In the print version of this paper there are excerpts from The Madman's Diary which are constructed according to such a simple contrapuntal procedure.