What is it about?
erred to as 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 on 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 mutual transpose, 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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Why is it important?
The distribution of the scale tones are analyzed in the current work by using the approach for generalized Pythagorean scales described in Cubarsi (2020) developed in the specific level of the frequency domain.
Perspectives
The approach provides tools to estimate how close the tones of a cyclic scale are from those of an equally tempered scale, which will be studied in a future work.
Rafael Cubarsi
Universitat Politecnica de Catalunya
Read the Original
This page is a summary of: On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation, Journal of Mathematics and Music, April 2023, Taylor & Francis,
DOI: 10.1080/17459737.2023.2194301.
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