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DESCRIPTION

This paper describes a system of tone order having the same internal pitch relations that exist between partials in the harmonic series. The system which has been developed has eight primary tones (analogous to the twelve "tonic" pitches as the are conventionally understood) which have the same ratio relations to one another as the fourth octave of harmonics over a fundamental, and within which, the pitches of each of their tone families (analogous to "diatonic sets") have the same relations to their primary tone  as harmonics to a fundamental up through the fourth octave. These families of tones are called contonations, where tones are to a contonation as stars are to a constellation. The relative complexity of this system, in terms of tuning, necessitates the use of electronic instruments capable of fine detuning to specific frequency values in hertz or pitch values in cents.

Figure 1

Figure 1

Figure 1 shows the true intervallic relationships of the harmonic series over a fundamental of E1, or 41.25 hertz. These are expressed as conventionally-notated pitches, which assume twleve-tone equal temperament, with the addition of a figure which shows their deviation from equal temperament as measured in cents, against the E1 as a standard of zero cents. The E1 has been chosen because it is the lowest conventional tone playable on the contrabass. Fretless string instruments may be considered somewhat more adaptable to an alternate system of tuning than instruments producing more technically arbitrary divisions of their vibrating medium, though less technically demanding music allowing for pitch matching with an electronic instrument might still have to be written for them. Note that the E1 of equal temperament based on A4 at 440 hertz is 41.203 hertz. The E1 given here is unadjusted for equal temperament, as though it were tuned from A 440 using only the just fifths produced by the harmonics of its own strings, and is therefore 41.25 hertz. Now note that the fourth octave of this set contains eight tones.

Figure 2

Figure 2

In Figure 2 these eight tones have been transposed down three octaves to serve as the primary tones (or "tonics") for each of the eight contonations (or "diatonic sets"). The tones in each of these contonations have the same relations to their primary tone as harmonics to their fundamental.

Figure 3

Figure 3

The tones in the contonation over E1 are exactly the same as the harmonic series shown in Figure 1. Figure 3 shows the contonation over F#1 (plus four cents), the second primary tone in the collection. Note that the relationships between the notes in the F# contonation are the same as those in E, with the values of deviation from equal temperament all increased by the same coefficient of plus four cents. In the contonation over G#1 (minus fourteen cents, see Fig. 4) all the tones are transposed up a major third from from the E contonation and adjusted to their coefficient of minus fourteen cents, and so forth. Also note in Figure 3 that in the fifth octave only the tones representing even-numbered harmonics are present--the tones in the fourth octave are repeated in the octaves above without any additional tones between them that would correspond to theoretically sounding harmonics. This would be necessary to keep the system manageable, and to keep the already considerably "detuned" tones in the system aurally accessible to some listeners.

Figure 4

Figure 4

Another area where aural simplicity would be desirable is the system of common-tone transpositional relationships among the tones of all eight contonations. Figure 4 shows the contonation over G#. The arrows pointing away from many  of the notes point to letter names indicating the primary tone(s) of the contonation(s) containing exactly the same given tone that is present in the G# contonation; thus, most of the tones in each of the eight contonations are common to at least one other contonation, and all eight contonations have at least one tone in common with every other contonation, allowing for a full range of common-tone transpositional relationships. A "Table of Contonations" in the Appendix below includes a range of tones for all eight contonations and their transpositional cross-references.