Interval Scale as Group Generators

The purpose of this paper is to propose “interval scale,” a new concept defined as a set of ordered pitch-class intervals. Unlike an ordinary scale, this concept restricts usable intervals and doesn’t restrict pitch classes directly. This provides possibilities that interval scales can be used in atonal music that uses all pitch classes and can be used to express some differences in a similar way to ordinary scales, depending on the selection of the elements. In this paper, we first present two existing musical pieces that can be interpreted as being based on interval scales, and see the possible effectiveness of this concept to express some senses of tonality. Next, we show that an interval scale is a generating set of a mathematical group and prove the necessary and sufficient condition for an interval scale to generate all pitch classes as a condition of atonality. Furthermore, the relationship between tone row and interval scale is examined, and the necessary and sufficient condition for an interval scale to be preserved by several tonerow transformations is proved. These results will provide a basic understanding and some criteria of selecting interval scales for composers who create music based on this concept.