Scale Theory, Serial Theory and Voice Leading

Abstract

Article S cale T heory , S erial T heory and V oice L eading dmitri tymoczko dmitri tymoczko S cale T heory , S erial T heory and V oice L eading Musical norms evolve over time. In the eleventh century parallel perfect fifths were tolerated, and perfect fourths were considered more consonant than thirds. In the eighteenth century parallel perfect fifths were not tolerated, and thirds were considered more consonant than perfect fourths. And in the twentieth century all manner of traditional prohibitions collapsed as composers revelled in the sense that everything was permitted. Despite such changes, however, certain musical principles remain relatively constant across styles. One of the most important of these dictates that harmonies should, in general, be connected by efficient voice leading. That is, notes should be distributed among individual musical voices so that no voice moves very far as harmonies change. Ex. 1 supplies a few representative passages. In each case, the music exploits the shortest possible path between successive chords, in a sense to be defined below. 1 Efficient voice leading is not simply a matter of performers' physical comfort, although that is certainly a factor. It also enables listeners to segregate the auditory stimulus into a series of independent musical lines, which is a prerequisite for understanding polyphony. 2 Efficient voice leading is so ubiquitous that we tend to take it for granted. But upon reflection it is rather remarkable that composers manage so consistently to find the shortest route from chord to chord. There are 57,366,997,447 distinct voice leadings between two hexachords. 3 There are 288 bijective voice leadings from a half-diminished seventh chord to the twelve dominant seventh chords. 4 Yet almost any composer, theorist or undergraduate student of Ex. 1 Efficient voice leading in several musical styles: (a) Ad organum faciendum (eleventh century); (b) Landini, Sy dolce non sono; (c) J.S. Bach, Das wohltemperierte Klavier I, Fugue No. 3 in CC major, BWV 869 (final cadence, upper four voices only); (d) Wagner, the 'Tarnhelm' motive (upper three voices only); and (e) a common jazz ii-V-I voicing (upper four voices only)dmitri tymoczko harmony can quickly find the most efficient voice leading between two hexachords, or a maximally efficient bijective voice leading from a particular half-diminished seventh chord to any of the twelve dominant seventh chords. The puzzle lies in the mismatch between the large number of possibilities and our apparent ease in sorting through them.We can complicate our puzzle by noting that students receive little explicit conceptual instruction in identifying efficient voice leadings. Music teachers tend to provide examples augmented with relatively uninformative exhortations: we enjoin our students to obey the 'law of the shortest way' -that is, to minimise the overall voice-leading distance between successive chords 5 -without telling them precisely what this means or how to do it. And students comply, easily and for the most part w...

Set-Class Similarity, Voice Leading, and the Fourier Transform

“…(These are the absolute values of the numbers above the arrows in the voice leading.) In principle, there are many different measures of voice-leading size but no compelling reason to choose one over another (Tymoczko 2006;hall and Tymoczko 2007). In this article, however, it is convenient to use the Euclidean metric, according to which the size of a collection of real numbers x 1 , x 2 , .…”

“…(These voice leadings have been described as the n "interscalar transpositions" between the chords; that they are the only possibilities follows from the fact that voice crossings always increase the Euclidean size of a voice leading [Tymoczko 2006]. ) For example, to identify the minimal voice leading between {0, 3, 9} and {3, 10, 11}, we need to consider the three voice leadings (0, 3, 9) 3, 5, 2 (3, 10, 11), (0, 3, 9) 2, 4, 6 (10, 11, 3), (0, 3, 9) 1, 0, 1 (11, 3, 10).…”

A Model of Triadic Post-Tonality for a Neoconservative Postmodern String Quartet by Sky Macklay

“…Finally, they say of their analysis that it "demonstrates that Bryars' harmonies are not to be dismissed simply as non-functional sonorities". Instead, "The parsimonious voice leading apparent in the music is a systematic, not arbitrary, consequence of the transformational system Bryars employs" (Roeder and Although Tymoczko (2008a) defines triadic transformations (e.g., neo-Riemannian transformations) as a strictly harmonic phenomenon that "does not specify any particular mapping between its notes" (p. 10), Richard Cohn (1998) explains that these transformations invert a triad, "mapping major and minor triads to each other", and that "because the inversional axis is defined in relation to the triad's component pitch classes, rather than as a fixed point in pitch-class space, this class of transformations is now referred by the term "contextual inversion" (p. 170). The present study asserts that Macklay's use of this class of triadic transformation is often equivalent to a contextually fluid voice mapping that can be described as a dual process of inversion.…”

“…As in the kinematic graphs of earlier analyses, the arrows connect terms to their transforms (i.e., triads signified by ϒ or gamma, the third letter of the Greek alphabet), therefore representing triadic transformations (usually involving voice leading by unordered pc intervals 1 and 2), and epitomizing the imaginary representative point that "jumps" from term to term, representing every possible progression. Like Tymoczko's (2008a) systems, the terms of the Macklay kinematic represent particular chords (e.g., "C +"), and not just chord types (e.g., "major triad") (pp. 1-49).…”

Automatic Transcription of Polyphonic Vocal Music

“…Most symbolic voice separation approaches are based on voice leading rules, which have been investigated and described from a cognitive perspective in a few different works [26][27][28]. Among these rules, three main principles emerge: (1) large melodic intervals between consecutive notes in a single voice should be avoided; (2) two voices should not, in general, cross in pitch; and (3) the stream of notes within a single voice should be relatively continuous, without long gaps of silence, ensuring temporal continuity.…”