Macroharmonic Progressions through the Discrete Fourier Transform

Abstract

This article examines macroharmony through the lens of the discrete Fourier transform (DFT) using computational analysis. It first introduces the DFT, giving an interpretive framework to understand the theory of chord quality first introduced by Ian Quinn (2007) before extending the theory to macroharmonies. Subsequently, the paper discusses different approaches—including different weighting and windowing procedures—to retrieving pitch data for computational analysis. An analysis of macroharmony in Domine Jesu from Maurice Duruflé’s Requiem, Op. 9 follows. I show that the DFT reflects intuition, reveals form-functional macroharmonies in the movement, and provides us with a perspective to find novel hearings.

Phasic Dissonance, Timbral Contrast: Analysing 20th‐Century Guitar Harmonies with Discrete Fourier Transform

“…For example, f 3 is most closely associated with set classes [048] and [014589]. 5 The former represents 'augmented' or 'triadic' quality (see Chiu 2021); the latter indicates Fig. 1 The six different Fourier components one can create through cyclical arrangement of the chromatic aggregate (Yust 2016) 'hexatonicity' (these designations will be explained in more detail below, using a C major triad as an analytical case study).…”

“…For example, f 3 is most closely associated with set classes [048] and [014589] 5. The former represents ‘augmented’ or ‘triadic’ quality (see Chiu 2021); the latter indicates ‘hexatonicity’ (these designations will be explained in more detail below, using a C major triad as an analytical case study). Following Ian Quinn (2007), we refer to these qualities as ‘prototypes’; a full list thereof is included in Table 1 6…”

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

“…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).…”