Coherence and sameness in well-formed and pairwise well-formed scales

Abstract

A common theme running through many of the scale studies in recent years is a concern for the distribution of intervals and pitch classes. The question of good distribution becomes increasingly complex with the increase in parameters. Complexity increases when the cardinality, N, increases, and when the number of step sizes increases relative to cardinality. Complexity is also shown to be dependent upon the relative sizes of the step intervals. Two measures of scalar complexity are the properties known as difference and coherence. Difference rates a scale according to the number of distinct specific intervals it contains, whereas coherence concerns conflicts between generic and specific intervallic measures. There are two types of conflicts, ‘ambiguity’ and ‘contradiction’. This paper demonstrates that well-formed scales have, as a class, the highest coherence rank—fewest numbers of ambiguities or contradictions—for scales of a given cardinality. They are, then, in this sense, ‘minimally complex’. The paper concludes with a conjecture about pairwise well-formed scales, namely that these types are more complex than well-formed ones, but less so than all others.

Keywords:

• Scale theory
• Well-formed scales
• Distribution modulo 1
• Coherence
• Fibonacci sequence