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

Abstract

The theory of well-formed modes is a modal refinement of the theory of well-formed scales. The mathematical approach is based on various results from the subdiscipline of algebraic combinatorics on words. Section 1 provides anchors and motivations for this investigation both in music theory and in mathematics and traces some earlier cross-connections. An overview of the theory is presented in terms of a dichotomy between generic and specific levels of description. Section 2 presents a first group of basic theoretical results. Height-width duality mediates between scale step patterns and fifth-fourth folding patterns. Both are encoded through divided words, on two-letter alphabets, such as aaba | aab and yx | yxyxy . The letters a and b denote ascending whole and half steps, and the letters x and y denote ascending perfect fifths and descending perfect fourths, respectively. These words are well-formed words; i.e., in the language of word theory, they are conjugate to Christoffel words and inherit a duality that is akin to Christoffel duality . Qualitative differences between the modal varieties of the same underlying scale can be detected and formalized through word-theoretical arguments. For example, a property we refer to as divider incidence characterizes modes corresponding to standard words. Positive standard words generalize the ascending authentic Ionian mode. Sturmian morphisms provide a transformational meta-language for the study of well-formed modes. Section 3 revisits Jacques Handschin’s concept of tone character and defends it on the basis of the mathematical results against two criticisms that had been raised by Carl Dahlhaus. Section 4 explores distinctions among the modes based upon considerations of word theory, especially divider incidence and concomitant properties that support major-minor tonality. A concluding section connects these arguments with other lines of investigation.