Probing Questions About Keys: Tonal Distributions Through the DFT

Abstract

Abstract. 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, f5, f3, and f2, 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 f5, f3, and f2 also exhibit a special linear dependency in tonal music giving rise to the idea of a tonal index.

A Clustering-Based Approach to Automatic Harmonic Analysis: An Exploratory Study of Harmony and Form in Mozart’s Piano Sonatas

“…We began by finding the pair of DFT coefficients that most frequently appear as one of the three largest for each cluster. We predicted that â 5 and â 3 , the coefficients that dominate pitch-class profiles for keys (according to Aarden, 2003;Cuddy and Badertscher, 1987;Krumhansl and Cuddy, 2010;Sapp, 2011;Yust, 2017bYust, , 2019 would also be principal qualities for most of the cluster centroids. For the remaining clusters which did not have these two coefficients among the top three, we found another pair that accounted for most of these.…”

“…Other important coefficients, â 3 and â 4 , indicate when a vector is concentrated around some division of the pitch-class circle by three or four respectively, and so are useful for identifying triads and seventh chords. Previous research (Yust, 2017b(Yust, , 2019Bernardes et al, 2016) has shown that two dimensions of the DFT, â 3 and â 5 , are effective in estimating the key of passages of tonal music and sorting harmonic functions. Because typical pitch-class profiles of major and minor keys have most of their energy in â 3 and â 5 , a two-dimensional space on the phases of these, denoted φ 3 and φ 5 , can serve as a map of key relatedness (Krumhansl, 1990;Yust, 2017b).…”

“…We also show that applying the discrete Fourier transform (DFT) on pitch-class vectors produces an effective map of keys and harmonic functions. This method has previously been used to characterize musical keys (Cuddy and Badertscher, 1987;Krumhansl, 1990;Yust, 2017b), as a feature description for musical audio (Harte et al, 2006;Ramiréz et al, 2020), and as a tool for computational analysis of musical works and large corpora (Yust, 2019;Novarro-Cáceres et al, 2020;Harding, 2020Harding, , 2021Chiu, 2021;Viaccoz et al, 2022;Harasim et al, 2022). In our Experiment 1 we use the DFT to classify clusters into three basic types (triadic, tetradic, and scalar).…”

Dimensions of Atonality: A Response and Extension of von Hippel and Huron (2020)

“…This method of analyzing probe-tone data was also applied by Cuddy and Badertscher (1987). Much has been written recently to explain this procedure (e.g., Amiot, 2016;Quinn, 2006;Yust, 2016Yust, , 2017Yust, , 2019 which need not be fully recounted here. To give brief summary, the pitchclass vector turns a pitch-class set or distribution into a periodic signal, like (1,0,0,0,1,0,0,1,0,0,0,0) for the C major triad, and converts it into six periodic components, Fourier coefficients f1-f6.…”

“…We can plot a pitch-class vector in this space using the DFT, and the nearest major or minor key in the space will be the best-fitting key for that pitch-class set or distribution. In Yust (2017), I show the conditions under which this procedure breaks down, which correspond to conditions of tonal ambiguity that we can classify using other elements of the DFT. Not surprisingly, we find all of these methods of achieving tonal ambiguity being used in the twelve-tone row data set under investigation.…”