As we have seen one can distribute freely pitches around a given symmetry and therewith construct modalities for possible transpositional schemes. With this basic principle we can of course free ourselves from our twelve-note division of an octave and bring the theory into full usage with other perfect divisions of an octave that precipitate other prime numerical symmetries. Each of these symmetries must be with the sum of its members a prime number.

For example the tritone consisting of two members or the augmented triad which consists of three members. A microtonic division of the octave resulting in prime numerical symmetrical constructs could be for example, the number 15. With 15 notes equally dividing an octave we have the augmented triad as well as the symmetrical division of the number 5.

We can as well utilize this symmetry as a nucleus and develop modes from the other 10 remaining pitches. Then one can use the 15 steps of this microtonal chromatical scale for a given transpositional scheme. We can even go back to even smaller divisions of an octave where perhaps prime numerical symmetries occur. An example could be 8, 9 or 10 perfect divisions of an octave.

At this point one can propose, within the spectrum of human perception, a choice of free pitches not necessarily being a part of a symmetrical 15 or a10 note division of an octave. We could have a well-tempered symmetrical tritone , an augmented triad or perhaps a geometrically prime numerical division of an octave as 5, 7, 11, 13, etc. Then distribute whatever tones we wish and pattern the transpositions upon these unsymmetrical tones and the chosen prime numerical symmetry.

© Copyright by Paul Amrod