Insiders’ Choice: Studying Pitch Class Sets Through Their Discrete Fourier Transformations

Abstract

This contribution responds to a growing interest in the application of Discrete Fourier Transform (DFT) to the study of pitch class sets and pitch class profiles. Theoretical fundaments, references to previous work and explorations of various directions of study have been eloquently assembled by Emmanuel Amiot. Recent pioneering work in the application to music analysis and the reinterpretation of theoretical knowledge has been accomplished by Jason Yust. The intention of this paper is to show ways to make Yust’s strategies and methods more easily accessible and reproducible for a broader readership, especially students. This includes the introduction of concepts as well as interactive experiments with the help of computation and visualization tools.

The theoretical starting point is the interpretation of pitch class sets in terms of their characteristic functions, i.e. as pitch class profiles with values 0 and 1. Apart from the magnitudes of the respective partials, the study of their phases is particularly illuminating. The paper shows how the contents of this approach can be made accessible in a four steps proceedure.

1 The Diamond Jubilee: Lewin’s Remarkable 1959 Article

The publication of Lewin’s pioneering article [4] dates back precisely 60 years now. Although the condensed article presents an absolutely convincing application of Discrete Fourier Transform to a concrete music-theoretical problem it took the music-theoretical community nevertheless almost half a century in order to show an interest in this project. In fact, it was after Lewin’s second go [5], that a small group of younger visionary theorists realized, that the special case of vanishing Fourier coefficients might just be the tip of an mathe-musical iceberg. Before immersing ourselves into the very basics, it is useful to recapitulate Lewin’s original question.

If we have two chords (pitch class sets) X (of m tones) and Y (of n tones) we may form $m \times n$ tone pairs (x, y) with $x \in X$ and $y \in Y$. Each pair defines an interval $y-x \in \mathbb {Z}_{12}$. And the $m \times n$ interval-instances can be counted in terms of the interval function $IF(X,Y): \mathbb {Z}_{12} \rightarrow \mathbb {N}$, where each index $IF(X,Y)(i) = \sharp \{(x, y) \in X \times Y \, | \, y - x = i\}$ is the multiplicity of i among the $m \times n$ interval-instances from X to Y. Figure 1 shows two such pairs, namely $X_1 = \{2, 5, 7, 11\}$ and $Y = \{0, 4, 7\}$ with the interval function $IF(X_1, Y) = [1, 1, 2, 0, 0, 3, 0, 1, 1, 1, 1, 1]$ and $X_2 = \{1, 5, 7, 11\}$ and $Y = \{0, 4, 7\}$ with the interval function $IF(X_2, Y) = [1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2]$. It turns out that the chord $Y = \{0, 4, 7\}$ is uniquely determined by $X_1$ and $IF(X_1, Y)$, because there is no other chord $\tilde{Y}$, such that $IF(X_1, Y) = IF(X_1, \tilde{Y})$. This is different in the case of $X_2$ and $IF(X_2, Y)$, where there are seven other 3-chords, sharing the interval function from $X_2$ with Y: $IF((X_2, Y) = IF((X_2, \tilde{Y}_k)$ for $\tilde{Y}_1 = \{0, 1, 4\}, \tilde{Y}_2 = \{0, 1, 10\}$, $\tilde{Y}_3 = \{0, 7, 10\}, \tilde{Y}_4 = \{0, 4, 6\}$, $\tilde{Y}_5 = \{1, 6, 10\}$, $\tilde{Y}_6 = \{4, 6, 7\}$, and $\tilde{Y}_7 = \{6, 7, 10\}$.

Fig. 1.

Left side: the chord $Y = \{0, 4, 7\}$ is uniquely determined by the chord $X_1 = \{2, 5, 7, 11\}$ and the interval function $IF(X_1, Y) = [1, 1, 2, 0, 0, 3, 0, 1, 1, 1, 1, 1]$. Right side: the chord $Y = \{0, 4, 7\}$ is not uniquely determined by the chord $X_2 = \{1, 5, 7, 11\}$ and the interval function $IF(X_2, Y) = [1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2]$.

Lewin’s elegant explanation for this different behavior takes advantage of the famous convolution theorem of Fourier theory (e.g. see [3] Theorem 1.10 on p.7). The discrete Fourier transform $\widehat{f}: \mathbb {Z}_n \rightarrow \mathbb {C}$ of a complex-valued function $f: \mathbb {Z}_n \rightarrow \mathbb {C}$ is defined by virtue of $\widehat{f}(k) = \sum _{j = 0}^{n-1} f(j) exp(-2 \pi i k j / n)$. The interval function IF(X, Y) can be written as a convolution of the characteristic functions $\chi _X, \chi _Y: \mathbb {Z}_{12} \rightarrow \{0, 1\} \subset \mathbb {C}$, namely $IF(X, Y)(k) = \sum _{j = 0}^{n_1} \chi _X(j ) \cdot \chi _Y(k-j).$ The vanishing of one of the Fourier-coefficients $\widehat{IF(X, Y)}(k)$ of the interval function than implies the vanishing of the Fourier-Coefficient at that same index k of—at least—one of the two characteristic functions $\chi _{X}$ or $\chi _{Y}$, and thereby provides freedom for the value of that coefficient for the other characteristic function. This is how the concrete ambiguities on the right side of Fig. 1 emerge, because $\widehat{\chi _{\{1, 5, 7, 11\}}}(k) = 0$ for all odd indices k, while $\widehat{\chi _{\{0, 4, 7\}}}(k) \ne 0$, throughout.

Eventually Lewin’s ideas initiate a gradual reclamation of the Fourier approach by music theorists. Several established music-theoretical concepts are being successively translated into the Fourier domain, and their interaction is being studied under new perspectives. This starts with the interpretation of the absolut values of the Fourier coefficients. A special case of the above discovery is the equation $IF(X, X)(k) = |\widehat{\chi _X}(k)|^2$ for $k \in \mathbb {Z}_{12}$, which Amiot ([3], p. 15) calls Lewin’s Lemma. According to this formula two chords are Z-related (or belong to the same set class) if and only if the Fourier coefficients of their characteristic functions have the same magnitudes.

Avoiding the mathematical formalism Quinn [8] meets the challenge to interpret the vanishing of Fourier coefficients within a broader investigation of chord quality for a music-theoretical readership. The vanishing of a coefficient—a a Fourier balance—is often accompanied with high or maximal values of others. Amiot [1] undertakes systematic investigations into the link between the concept of maximally even sets of d tones and the maximality of the absolute value $|\widehat{\chi _X}(d)|$ among all d-chords X.

Recent exciting work is dedicated to the interpretation of the phases of Fourier coefficients. Traditional spaces of chords or tonal regions, such as Douthett and Steinbach’s chicken wire torus or Weber’s regional space maybe regained within tori of phases. This implies the possibility for robust analytical methods, because the Fourier coefficients of chords and scales of different cardinalities can be located in the same phase spaces and the interpretation of the notes of a score may freely switch between strict and fuzzy encodings through characteristic functions and pitch class profiles, respectively. These new developments are due to Amiot [2, 3] and Yust [9,10,11,12,13,14]. Yust’s analytical work entails new ideas about a generalized concept of tonality.

One may easily anticipate that this process will continue. Sooner rather than later it is desirable to let students of theory or composition participate in these new developments. So the question arises how one may suitably integrate carefully selected content into a theory class. The present paper offers a few proposals in this direction.

2 Step One: Partial Chords and Fourier-Prototypes

The present pedagogical approach complements that of the Fourier Scratching—an earlier attempt [6, 7] to convey the nature of the Fourier transform through musical interaction in a predefined rhythmical playground. The Fourier Scratching deliberately covers the entire space of complex-valued functions $f: \mathbb {Z}_{n} \rightarrow \mathbb {C}$ for low dimensions n in a musically meaningful way, and the project intends to pique the players’ curiosity for the complex numbers and the DFT. The Fourier Scratching addresses a general public and uses the musical rhythms as medium for the exploration of mathematical circumstances.

This is different here, where the main goal is to access the musical meaning of the Fourier coefficients of characteristic functions of pitch class sets. The applications mentioned in Sect. 1 use real-valued functions only, and therefore another approach lends itself for accessing the required knowledge, especially for the study of Yust’s articles. Although the discrete Fourier Transform of functions $f: \mathbb {Z}_{12} \rightarrow \mathbb {C}$ is mathematically easier to handle than the case of continuous periodic functions $F: \mathbb {R} /\mathbb {Z} \rightarrow \mathbb {C}$, it appears that composition students usually bring a basic understanding of trigonometric functions and additive synthesis as a “dowry” from their knowledge in sound and signal processing, while they have seldom come across the complex exponential function.

It is therefore helpful to inspect continuous cosine functions over an octave periodic pitch domain $\mathbb {R} /12 \mathbb {Z}$ and drag them along with their restrictions to $\mathbb {Z} /12 \mathbb {Z}$. It is crucial though that the familiar association – say—of a periodic time-dependent movement of an air particle, a loudspeaker membrane, a Cello string etc. must be given up. It is only allowed to serve as a metaphor. There is no time parameter involved in this application.

Fig. 2.

The 3rd, 4th and 5th partial chords (with suitable phase shifts) corresponding to the periods $\frac{12}{3}, \frac{12}{4}$, and $\frac{12}{5}$, respectively. Their values are represented (1) as dots on the vertical lines of the $\mathbb {Z}_{12}$-grid and (2) as restrictions of (shifted) continuous cosine functions. The tones with positive values are framed. They form the associated prototypes: a hexatonic scale, an octatonic scale and Guido’s hexachord respectively. The phases are chosen to orient these scales at the anchor tone $C = 0$.

The analogous transfer is indeed useful, as the partial chords—the DFT analogues to the partials of additive synthesis—are not musical chords in the usual sense. They can be visualized as discretized cosine functions and as such they have positive and negative values. It is important to bring the phase parameter into play and to understand that the range of the phase has to be adopted to the individual period of each partial chord, i.e. the period $\frac{12}{k}$ of the k-th partial. In visualizations the phase can be suitably shown in terms of a leftward shift of a cosine function. In Fig. 2 it is indicated through a vertical line; in Fig. 4 it is shown as a colored rectangle (in both cases measured from the right border of the graph in leftward direction). Figure 2 shows three partial chords, namely for $k = 3, 4$ and 5.

While the partial chords are still somewhat alien with respect to common musical intuition one obtains a link to prominent musical structures by inspecting the Fourier prototypes: the carrier sets of their positive values.

Fig. 3.

Characteristic function $\chi _X = (0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1)$ of the Prometheus chord $X = \{1, 3, 6, 7, 9, 11\}$. The (only auxiliary) continuous rendering arises from the superposition of the continuous partials.

Fig. 4.

Partial decomposition of the characteristic function of the Prometheus chord $X = \{1, 3, 6, 7, 9, 11\}$, see Fig. 3. The schematized keyboard layout within each partial-figure indicates the location of the 12 chromatic notes along the octave period. The gray/light rectangle on the right side below the keyboard indicates the size of the period, while the light part on the very right within it represents the phase shift.

3 Step Two: Partial Decomposition

After having achieved some familiarity with the partial chords one may extend the analogous transfer to the inspection of additive synthesis. Here it is useful to have tools for interactive experiments.¹ Knowing the “right” parameters one may then re-synthesize the characteristic functions of given chords X in terms of superpositions of the appropriate 12 partial chords. Figure 4 shows the partials of the characteristic function $\chi _X = (0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1)$ (see Fig. 3) of the Prometheus chord $X = \{1, 3, 6, 7, 9, 11\}$.

4 Step Three: Inspecting Fourier Coefficients

Careful exploration of the partial decompositions of various chords shows that the different partials typically have different impacts to the superposition. In Fig. 4 one can clearly observe a high impact of the sixths partial, whose prototype is the whole-tone scale.

At this stage one may go one step further: Disregarding the meanwhile familiar partials and extracting only their impacts in terms of magnitudes and phases brings the Fourier coefficients as such to the foreground. For a complete Fourier-portrait of a chord X it is sufficient to inspect Partials $\widehat{\chi _X}(1)$ to $\widehat{\chi _X}(6)$, because of the symmetry of the DFT of a real valued function $f: \mathbb {Z}_{12} \rightarrow \mathbb {R}$. For characteristic functions $f = \chi _X$ the zeroth coefficient $\widehat{\chi _X}(0) = \sharp X$ simply measures the cardinality of the chord X and doesn’t need further attention. For the main six coefficients the following musically motivated names have been proposed: $\widehat{f}(1)$: chromaticity, $\widehat{f}(2)$: dyadicity, $\widehat{f}(3)$: triadicity, $\widehat{f}(4)$: octatonicity, $\widehat{f}(5)$: diatonicity, $\widehat{f}(6)$: wholetone-property.

Fig. 5.

CDF-tool for the display of the squared magnitudes and the phases (in a clock format) of the main six Fourier coefficients of a characteristic function. The figure allows the comparison of the Fourier coefficients of two chords with the same interval content (Z-related pitch class sets) $X = \{0, 1, 4, 5, 7\}.$ and $Y = \{0, 1, 2, 5, 8\}.$ One observes that the squared magnitudes coincide, while the phases show a quite different behavior.

For the reconstruction of Yust’s analyses it is recommended to have a tool at disposal for the calculation of squared magnitudes and phases in the range between 0 and 12 (see Fig. 5).

5 Step Four: Analyzing Progressions in Two-Phase Plots

It is a common popular procedure in serval approaches to harmony to interpret chord progressions as trajectories in certain harmonic configuration spaces. Discrete Fourier analysis supports such an approach quite naturally. The consideration of two Fourier-phases at once takes place in a torus $\mathbb {R} /2 \pi \mathbb {Z} \times \mathbb {R} /2 \pi \mathbb {Z}$. Yust normalizes these spaces to a period of 12 in both dimensions.

Fig. 6.

The left figure shows a phase plot of the coefficients 4 (octatonicity) and 5 (diatonicity) of the trajectory formed by the first eight chords of the C-major prelude of the well-tempered piano (part I) by Johann Sebastian Bach (score above). The right figure shows a phase plot of the same coefficients for the first eight chords of Alexander Scriabin’s prelude Op. 74 No. 2 (score below). In both windows the width and height of the rectangle around the location of chord no. 8 indicate the magnitudes of the 4th and the 5th Fourier-coefficients of this chord.

The two sample analyses of chord progressions in Bach and Scriabin in Fig. 6 share an interesting characteristic: In both trajectories there is a more static and a more versatile coefficient. Interestingly, the Fourier coefficients change roles. Mobile octatonicity under static diatonicity is typical for traditional harmonic tonality (with seventh chords) and is exemplified by the progression in Bach. The Scriabin prelude exemplifies a form of mobile diatonicity under static octatonicity. It is an interesting proposal by Jason Yust to interpret a different activity allocation among the coefficients as a different type of tonality.