Generalized diatonic scales

Abstract

The article studies the problem of generalizing the concept of ‘diatonic scale’ for a given ambient chromatic of N tones: ‘Which subset A⊂ℤ N shall be considered as a generalized diatonic scale?’ Each generic type of well-formed scale has exactly two specific manifestations in chromatic universes, which are large ME*-scales, i.e. which are maximally even non-degenerate well-formed scales, whose cardinality exceeds half of the chromatic cardinality. A qualitative distinction between these two large ME*-scales of the same type can be comfortably made on the basis of the shuffled Stern–Brocot tree, which is introduced in Section 2. The shuffled Stern–Brocot tree represents the same abstract binary tree as the traditional Stern–Brocot tree, but has a different planar arrangement. Candidates and final choices for generalized diatonic scales are studied in Section 3. Candidates are those large ME*-scales A⊂ℤ N which are tightly generated by a prime residue class m mod N. According to this property there is no non-degenerate m-generated well-formed scale W properly included between the translation T 1(A c ) of the complement of A and A itself. This property is equivalent to the fact that the associated ratio m/N is a chromatic number, i.e. that its penultimate predecessor on the Stern–Brocot tree is a convergent for m/N, which again is equivalent to the fact that the ratio m/N corresponds to a right branching on the shuffled Stern–Brocot tree. Tight generatedness of a large ME*-scale is also equivalent to the fact that the small-scale step of chromatic size 1 is at the same time also the rarer step, while the large step of size 2 is at the same time also the more frequent step. A generalized diatonic scale A minimizes the cardinality difference between |A| and |A c | among all tightly generated large ME*-scales in ℤ N . For N divisible by 4, this definition reproduces exactly the family of hyperdiatonic scales, as studied by Agmon (Journal of Music Theory, 33(1), 1–25, 1989) and Clough and Douthett (Journal of Music Theory 35, 93–173, 1991). For N odd we exclude the trivial case of 2-generated scales and obtain a rich inventory of generalized diatonic scales, such as the 7-generated 11-tone scale in ℤ19. An interesting point for N odd is also the generalization of the tritone as a minimal limited transposition subset. We show that Ivan Wyschnegradsky has already done pioneering work on this subject (1916). His 11-generated 13-tone scale in the quarter-tone chromatic ℤ24 is a hyperdiatonic scale in the sense of Agmon (1989) and Clough and Douthett (1991) as above, and his argumentation in favour of this scale anticipates central points of the discussion in this article.