Distributional Analysis of n-Dimensional Feature Space for 7-Note Scales in 22-TET

摘要

Many scale features have been defined in an effort to account for the ubiquity of the diatonic scale in tonal music. In 12-TET, their relative influences have been difficult to disentangle. In 22-TET however, the features are spread differently across different scales. We sought here to to establish a set of 7-note scales in 22-TET that represent the major clusters within the whole population of scales. We first calculate numerous features of every 7-note scale in 22-TET that may relate to their perception in harmonic tonality. This feature space is then reduced by the step-by-step removal of features which may be most completely expressed as linear combinations of the others. A k-medoids cluster analysis leads finally to the selection of 11 exemplar scales, including approximations of four different tunings of the diatonic scale in just intonation.