Introduction to Scale Theory over Words in Two Dimensions

Recently, an interaction between the mathematical discipline of combinatorics on words and musical scale theory has led to various interesting results. So far, the focus was mainly on scales generated by a single interval. The paper proposes an extension of word scale theory to tone systems of higher dimensions, i.e. generated by more than one interval. It is shown that the number of specific varieties for any non-zero generic interval in n-dimensional comma-demarcated generated tone systems is between 2 and 2. Therefore, generating patterns in two-dimensional systems are words over a four-letter alphabet. A concept of quasi pairwise well-formed words is introduced as a weakening of Clampitt’s pairwise well-formedness. The main result of the paper is that a four-letter word is a generating pattern in a comma-demarcated two-dimensional system if and only if it is quasi pairwise well-formed.