On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation

Abstract

For well-formed generalized Pythagorean scales, it is explained how to fill in a bidimensional table, referred to as a scale keyboard, to represent the scale tones, arranged bidimensionally as iterates and cardinals, together with the elementary intervals between them. In the keyboard, generalized diatonic and chromatic intervals are easily identified. Two factor decompositions of the scale tones, which are particular cases of duality, make evident several properties of the sequence of intervals composing the octave, such as the number of repeated adjacent intervals and the composition of the generic step-intervals. The keyboard is associated with two matrix forms. When they are mutually transposed, the keyboard is reversible, as in the 12-tone Pythagorean scale. In this case, the relationship between the two main factor decompositions is given by an involutory matrix.

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