Chords could also be constructed, for example, by superimposing consonances, either "perfect" or "imperfect." A chord with all perfect consonances could be termed "perfect," while a chord with a combination of perfect and imperfect consonances could be termed "imperfect." For the superimposition of "perfect" and "imperfect" consonances as listed in Figure 7, without regard to inversion or the designation of a primary tone (a "tonic"), there are fifteen possible triadic combinations per contonation, eight possible tetrads and one possible pentad. The perfect/imperfect chord combinations for the E contonation are shown in Figure 8. They are organized into groups according to the presence of tones beginning with the primary tone (E) and ascending. All chords containing E are given first; all those remaining which contain F# are next; those left with G# follow. All possible combinations contain at least one of these three tones. This is merely the order in which they are systematically identified and is not intended to suggest the primacy of the tone by whose presence they are accounted for. Their order within each group is also insignificant. Since relationships within each contonation are the same, this chart could simply be transposed to the levels of other primary tones to show the chordal possibilities in other contonations. (N.B.: 1)  Specific deviations from equal temperament and/or interval sizes in cents are omitted here; 2)  The allowance of the 182-cent second as a consonance only if it is inverted to a 7th or compounded to a ninth--in this case involving the F# and G# of this contonation--can be observed in the reference given to the chords with an asterisk.) This accounts for all the possibilities in terms of mere combination: yet to be considered are interrelated matters of the possible primacy of a given tone in each triad, inversions of these combinations, and the implications of and for any "voice leading" procedure. The print version of this paper contains an excerpt and analysis from The Madman's Diary which consists of a simple succession of the simultaneities listed in Figure 8.

Figure 8

*The F# and G# must share a seventh or ninth relationship and not be inverted to a second.

Figure 8

Since the web of relationships connecting the tones in this overall system is derived from the pitch ratios of the harmonic series, an apt designation for the system as a whole, for the eight-tone scale which it features, and for music composed according to its procedural implications, might be ratiotonic. The overall implications for this ratiotonic system have only begun to be explored. Yet to be proposed are other means for managing "dissonance" within the system, (for example, a scheme for the application and progression of simultaneities), the means to integrate conventional acoustic instruments, the construction of new acoustic instruments which vibrate and resonate according to the prescriptions of the system, and the computer-based creation of digital synthesis applications which meet the specific pitch/frequency requirements of the system. Surely the most important step in addressing these and other considerations would be to compose ratiotonic music with the materials and technology at hand. Its theoretical implications could thus be refined in the crucible of practice, as the music created is subjected to the aesthetic sensibilities of its participants.

CONCLUSION

Ratiotonic temperament provides a potential system of order for the material element of tone in art music. It does not follow the historically established paradigm of attempting to manipulate a spiral of just fifths into a circle of adjusted fifths, a procedural model that has finally resulted in the homogenization of intervals into a few artificially equivalent types and classes, nor does it even begin with the superimposition of fifths that resulted in just intonation in the first place, neither does it construct some alternative equal division of the octave to create a new, arbitrarily defined scale; rather, it is constructed according to intervallic relationships to the nearest cent--or to actual frequency ratios--found within the harmonic series. This allows for the construction of a highly interconnected group of tone series which reflect directly, rather than approximately for the sake of theoretical convenience, the acoustical properties of all pitched sound. In this respect the "detuned" sound of various successive and simultaneous intervals as heard could be considered in a way natural, and those "tuned" according to twelve-tone equal temperament artificial, even though only the latter would sound "right" to most listeners unaccustomed to alternate tunings.

This system of tone order, like any other, reveals procedural possibilities for composition. These include implications for the combination of melodic or quasi-melodic events, the patterning of modulations between series of tones at differing pitch levels, and the control of relative dissonance. There are surely other implications which have yet to be addressed. Like any procedure for creating art, a considerable amount of practice must be undertaken on the crafting and construction of these materials in order to develop methods which more fully realize the artistic potential of the system.