Modes, the Height-Width Duality, and Handschin’s Tone Character

Table of Contents* 1. Introduction* 1.1 Scale Theory and Word Theory* 1.2 Glarean Revisited* 1.3 Height-Width Duality and the Generic/Specific Dichotomy* 2. Scale Step Patterns and Scale Foldings* 2.1 A Portrait of the Authentic Ionian Mode* 2.2 Generic Height and Width* 2.3 Plain Adjoints of Well-formed Words* 3. Thoughts on Jacques Handschin's Tone Character* 3.1 First Objection by Dahlhaus* 3.2 Second Objection by Dahlhaus* 4. Further Distinctions Among the Modes, via Algebraic Combinatorics on Words* 4.1 Divider Incidence* 4.2 Double-Neighbor Polarity* 4.3 Central Words as Factors in Standard and Anti-Standard Words* 4.4 Sensitive Intervals* 5. Conclusions* 5.1 Two Degrees of Freedom Bound in a Duality* 5.2 Elements of Harmonic Tonality in a Modal Perspective* 5.3 Music-theoretical Interpretation of Mathematical Facts* Acknowledgments* References1. IntroductionWhere actual musical practice is concerned, the relevant historical fact is that people have evidently internalized the diatonic pitch set-carried it around in their heads as a means of organizing, receiving, and reproducing meaningful sound patterns-as far back as what is as of now the very beginning of recorded musical history, some three and a half millennia ago. [1] Thus says Richard Taruskin, in the first chapter ("The Curtain Goes Up") of The Oxford History of Western Music, concerning what he calls "our most fundamental musical possession" (Taruskin 2005, 30, 29). Scale theory studies have addressed the robust nature of the usual diatonic in terms of properties such as non-degenerate well-formedness, Myhill's Property, Cardinality Equals Variety for lines, and self-similarity, to mention only some equivalent characterizations (Clough and Myerson 1985, Carey and Clampitt 1989, Carey and Clampitt 1996a). Other concepts may be productively studied in relation to the aforementioned equivalences, such as maximal evenness and coherence (Clough and Douthett 1991, Agmon 1989, Agmon 1996, Carey 2002, Carey 2007). These properties, however, are attributable to the general diatonic scale or set (and to analogues of the diatonic, notably the usual pentatonic) under octave equivalence; they do not easily distinguish among the modal varieties. Moreover, the studies above have not generally engaged with tonality, surely the seat of some of the deepest questions in music theory. [2] We argue that mathematical word theory provides a way to extend musical scale theory, to begin to approach issues of modality and tonality. We hasten to acknowledge that, given the abstraction of the word-theoretical level of description, we can hope to do no more than to open a new perspective on these perennial topics. The present article is a contribution to a mathematical theory of music that aims to complement philological methods of accessing knowledge about the diatonic modes and related music-theoretical concepts. The methodology can be characterized as an "experimental philology" that is concerned to study the interdependence of concepts irrespective of their emergence in historical discourse. Those music-theoretical ideas with historical philological anchors that participate in a multiplicity of mathematical interdependencies are those that deserve to be the focal point of our attention. For example, Myhill's Property for the diatonic is the property that every non-zero generic interval comes in two specific varieties, e.g., generic diatonic thirds are either major or minor. This is known to every music theorist, as is the fact that the diatonic set is non-degenerate well-formed, i.e., is generated by an interval of a given size, all instances of which span the same number of step intervals. (2) That these properties are equivalent, however, is a different type of knowledge from the mere conjunction of the two facts, and therefore deserves additional music-theoretical interest. …