'From where do these chords come?': Theoretical Traditions In The Enlightenment Boston University, 23 October 2015

Accepted manuscrip

Ganymed's heavenly descent

Schubert's song “Ganymed” has attracted a great deal of interest from analysts due to its progressive tonal plan, often seen as a challenge to Schenkerian theories of tonal structure, and evocative text. This article draws upon a spatial theory of tonal meaning which helps both to resolve the epistemological impasse faced by reductive theories of tonal structure, and to better access Schubert’s interpretation of Goethe’s text through spatial metaphors that derive from the harmony of the song. It also highlights an allusion to Beethoven's Op. 53 “Waldstein” Piano Sonata in the song that has previously gone unremarked, and identifies this as part of a network of references to Beethoven’s sonata that act both as homage to and critique of Beethoven's middle-period style. These serve both as a window into the song, and into Schubert’s aesthetic stance vis-à-vis his most pre-eminent musical forebear. The theory of tonal space draws upon previous publications, but is re-explained in music-theoretical terms relating to diatonicity and triadicity here. It realizes latent directional metaphors in the diatonic sharp-flat and triadic dominant-subdominant dimensions, which are of hermeneutic value for tonal music. Such a theory helps us interpret Schubert’s tonal plan, explain his choices of keys, and better understand his reading of Goethe's text and aesthetic priorities in setting it to music.Accepted manuscrip

A space for inflections: following up on JMM's special issue on mathematical theories of voice leading

Journal of Mathematics and Music's recent special issue 7(2) reveals substantial common ground between mathematical theories of harmony advanced by Tymoczko, Hook, Plotkin, and Douthett. This paper develops a theory of scalar inflection as a kind of voice-leading distance using quantization in voice-leading geometries, which combines the best features of different approaches represented in the special issue: it is grounded in the concrete sense of voice-leading distance promoted by Tymoczko, invokes scalar contexts in a similar way as filtered point-symmetry, and abstracts the circle of fifths like Hook's signature transformations. The paper expands upon Tymoczko's ‘generalized signature transform’ showing the deep significance of generalized circles of fifths to voice-leading properties of all collections. Analysis of Schubert's Notturno for Piano Trio and ‘Nacht und Träume’ demonstrate the musical significance of inflection as a kind of voice leading, and the value of a robust geometrical understanding of it.Accepted manuscrip

Generalized Tonnetze and Zeitnetze, and the topology of music concepts

The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.Accepted manuscrip

Testing Schenkerian theory: an experiment on the perception of key distances

The lack of attention given to Schenkerian theory by empirical research in music is striking when compared to its status in music theory as a standard account of tonality. In this paper I advocate a different way of thinking of Schenkerian theory that can lead to empirically testable claims, and report on an experiment that shows how hypotheses derived from Schenker’s theories explain features of listener’s perception of key relationships. To be relevant to empirical research, Schenker’s theory must be treated as a collection of interrelated but independent theoretical claims rather than a comprehensive analytical method. These discrete theoretical claims can then lead to hypotheses that we can test through empirical methods. This makes it possible for Schenkerian theory improve our scientific understanding of how listeners understand tonal music. At the same time, it opens the possibility of challenging the usefulness of certain aspects of the theory. This paper exemplifies the empirical project with an experiment on the perception of key distance. The results show that two features of Schenkerian theory predict how listeners rate stimuli in terms of key distance. The first is the Schenkerian principle of “composing out” a harmony, and the second is the theory of “voice-leading prolongations.” In a regression analysis, both of these principles significantly improve upon a model of distance ratings based on change of scalar collection alone.Accepted manuscrip

Harmonic qualities in Debussy's "Les sons et les parfums tournent dans l'air du soir"

This analysis of the fourth piece from Debussy's Préludes Book I illustrates typical harmonic techniques of Debussy as manipulations of harmonic qualities. We quantify harmonic qualities via the magnitudes and squared-magnitudes of the coefficients of the discrete Fourier transform (DFT) of pitch class sets, following Ian Quinn. The principal activity of the piece occurs in the fourth and fifth coefficients, the octatonic and diatonic qualities, respectively. The development of harmonic ideas can therefore be mapped out in a two-dimensional octatonic/diatonic phase space. Whole-tone material, representative of the sixth coefficient of the DFT, also plays an important role. I discuss Debussy's motivic work, how features of tonality – diatonicity and harmonic function – relate to his musical language, and the significance of perfectly balanced set classes, which are a special case of nil DFT coefficients.Accepted manuscrip

Restoring the structural status of keys through DFT phase space

One of the reasons for the widely felt influence of Schenker’s theory is his idea of long-range voice-leading structure. However, an implicit premise, that voice leading is necessarily a relationship between chords, leads Schenker to a reductive method that undermines the structural status of keys. This leads to analytical mistakes as demonstrated by Schenker’s analysis of Brahms’s Second Cello Sonata. Using a spatial concept of harmony based on DFT phase space, this paper shows that Schenker’s implicit premise is in fact incorrect: it is possible to model long-range voice-leading relationships between objects other than chords. The concept of voice leading derived from DFT phases is explained by means of triadic orbits. Triadic orbits are then applied in an analysis of Beethoven’s Heiliger Dankgesang, giving a way to understand the ostensibly “Lydian” tonality and the tonal relationship between the chorale sections and “Neue Kraft” sections

Geometric generalizations of the Tonnetz and their relation to Fourier phase space

Some recent work on generalized Tonnetze has examined the topologies resulting from Richard Cohn’s common-tone based formulation, while Tymoczko has reformulated the Tonnetz as a network of voice-leading relationships and investigated the resulting geometries. This paper adopts the original common-tone based formulation and takes a geometrical approach, showing that Tonnetze can always be realized in toroidal spaces,and that the resulting spaces always correspond to one of the possible Fourier phase spaces. We can therefore use the DFT to optimize the given Tonnetz to the space (or vice-versa). I interpret two-dimensional Tonnetze as triangulations of the 2-torus into regions associated with the representatives of a single trichord type. The natural generalization to three dimensions is therefore a triangulation of the 3-torus. This means that a three-dimensional Tonnetze is, in the general case, a network of three tetrachord-types related by shared trichordal subsets. Other Tonnetze that have been proposed with bounded or otherwise non-toroidal topologies, including Tymoczko’s voice-leading Tonnetze, can be under-stood as the embedding of the toroidal Tonnetze in other spaces, or as foldings of toroidal Tonnetze with duplicated interval types.Accepted manuscrip

Probing questions about keys: tonal distributions through the DFT

Pitch-class distributions are central to much of the computational and psychological research on musical keys. This paper looks at pitch-class distributions through the DFT on pitch-class sets, drawing upon recent theory that has exploited this technique. Corpus-derived distributions consistently exhibit a prominence of three DFT components, 5, 3, and 2, so that we might simplify tonal relationships by viewing them within two- or three-dimensional phase space utilizing just these components. More generally, this simplification, or filtering, of distributional information may be an essential feature of tonal hearing. The DFTs of probe-tone distributions reveal a subdominant bias imposed by the temporal aspect of the behavioral paradigm (as compared to corpus data). The phases of 5, 3, and 2 also exhibit a special linear dependency in tonal music giving rise to the idea of a tonal index.Accepted manuscrip

Analysis of analysis: importance of different musical parameters for Schenkerian analysis

While criteria for Schenkerian analysis have been much discussed, such discussions have generally not been informed by data. Kirlin [Kirlin, Phillip B., 2014 “A Probabilistic Model of Hierarchical Music Analysis.” Ph.D. thesis, University of Massachusetts Amherst] has begun to fill this vacuum with a corpus of textbook Schenkerian analyses encoded using data structures suggested byYust [Yust, Jason, 2006 “Formal Models of Prolongation.” Ph.D. thesis, University of Washington] and a machine learning algorithm based on this dataset that can produce analyses with a reasonable degree of accuracy. In this work, we examine what musical features (scale degree, harmony, metrical weight) are most significant in the performance of Kirlin's algorithm.Accepted manuscrip