Pitch-Space Lattices: Tonnetze and other Transpositional Networks

Year: 2011 Authors: José Oliveira Martins

Core claim

Incremental changes to the generic Tonnetz yield coherent T-nets that unify several pitch-space models and support analysis of tonal and atonal music.

Topics

Tonnetz, transpositional networks, pitch-space lattices, interval cycles

Domains

mod-12 group theory, lattice geometry, interval cycles, music theory, composition analysis, visualization

Methods

classification framework, lattice modeling, comparative analysis

Media

pitch-class lattices, 12-tone chords, diminished-seventh chords

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2011: Mathematics, Music, Art, Architecture, Culture

Pitch-Space Lattices: Tonnetze and other Transpositional Networks

José Oliveira Martins Eastman School of Music University of Rochester 26 Gibbs St. Rochester, NY, 14604, USA e-mail: jmartins@esm.rochester.edu

Abstract

The paper proposes a three-fold classification for pitch-space lattices based on a music-theoretic construct known as the Tonnetz. It argues that applying incremental changes on some of the constructive features of the generic Tonnetz (Cohn 1997) results in a set of coherent and analytically versatile transpositional networks (T-nets), which brings under a focused perspective diverse pitch structures such as Tonnetze, affinity spaces, Alban Berg’s “master array” of interval-cycles, and other types of networks. The paper also explores the music-modeling potential of progressive and dynamic T-nets by attending to characteristic compositional deployments in the music of Witold Lutosławski.

The Tonnetz

Figure 1a is a music-theoretic construct—known as the Tonnetz—that coordinates pitch classes (pcs) distancing perfect fifths (7 semitones) and major thirds (4 semitones) along the horizontal and vertical dimensions respectively. This construct adopts a closed mod-12 group-theoretic perspective (it is a geometrical torus), and has become a central framework for the modeling of (major and minor) triadic relations in the so-called neo-Riemannian literature that has developed since the mid-1990s. ${}^1$ Richard Cohn [1] has generalized the structure of the Tonnetz in order to accommodate other trichordal relations and other modular cardinalities. Figure 1b presents Cohn’s generic Tonnetz, a two-dimensional lattice that assigns an arbitrary pc-reference 0 and maps the surrounding pcs according to the pc-distance from the origin 0 under the lattice grid structured by the two axes $x$ and $y$ . ${}^2$

I would like to thank Akiyuki Anzai, Ruben Moreno Bote, C. Douglas Haessig, and Dmitri Tymoczko for their valuable input.

1. The mod12- version of the Tonnetz reformulates its nineteenth-century (theoretically infinite) precursor based on just intonation. The literature on neo-Riemannian theory is significantly extensive. For a historical overview and a variety of approaches to neo-Riemannian theory and analysis see the volume 42.2 of the Journal of Music Theory [2][5]. Particularly germane to this paper is [1].
2. The figure recaptures Cohn’s Figure 6, p. 10 [1].

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Fig. 1. (a) Neo-Riemaniann Tonnetz. (b) Richard Cohn’s (1997) generic Tonnetz.

Transpositional Networks (T-nets)

The generic transpositional network. The constructive features of the generic Tonnetz constitute the starting point for this paper. Figure 2 introduces a generic layout for a two-dimensional lattice, or transpositional network (T-net), where a given $pc(i,j)$ (pitch class $pc$ in position $(i,j)$ ) forms transpositional relations with adjacent pcs in the lattice. These relations are captured by the associated directed intervals $dx(i,j)$ and $dy(i,j)$ along the $x-$ and $y$ -axes respectively, such that $dx(i,j)=pc(i+1,j)-pc(i,j)$ and $dy(i,j)=pc(i,j+1)-pc(i,j)$ . Furthermore, the generalization developed here requires we consider the degree of variation of directed intervals along and across the $x-$ and $y$ -axes. The notation $\Delta dx|x$ and $\Delta dy|y$ refers to the change of transpositional values $dx$ along the $x$ -axis, and $dy$ along the $y$ -axis respectively; whereas $\Delta dx|y$ and $\Delta dy|x$ refer to the change of transpositional values of $dx$ across the $y$ -axis, and $dy$ across $x$ -axis respectively. By defining and varying certain constraints on the four values of $\Delta d$ , the paper proposes a three-fold framework of T-nets (here classified as homogeneous, progressive, and dynamic), which captures important features of diverse music-theoretic pitch models such as Tonnetze, affinity spaces, Alban Berg’s “master array” of interval-cycles, and other pitch constructs, and has considerable music-analytical potential for both the tonal and atonal repertoires.

Pitch-Space Lattices: Tonnetze and Other Transpositional Networks

Fig. 2. Generic layout of a transpositional network (T-net).

Homogeneous T-nets. Cohn’s generic Tonnetz can now be examined in light of the generic framework for T-nets introduced in Figure 2. Given any $x$ and $y$ values in a Tonnetz, a straight (horizontal or vertical) path constitutes an interval cycle (i-cycle), and any two parallel paths project the same i-cycle. In other words, there’s no change of $dx$ (or $dy$ ) in a straight path, and there’s no change of $dx$ (or $dy$ ) in parallel paths. I refer to the Tonnetz as the homogeneous T-net in order to reflect the lack of variation in the transpositional values across the lattice. In a homogeneous T-net: $\Delta dx\mid x=\Delta dy\mid y=\Delta dx\mid y=\Delta dy\mid x=0$ . Figure 3 represents a homogeneous T-net that captures important trichordal relations for set-class (013), which is analytically relevant for a passage by J.S. Bach discussed by David Lewin. In this T-net, $dx=11$ and $dy=2$ , and the four forms of $\Delta d=0$ .

Fig. 3. A homogeneous $T$ -net, where $dx=11$ and $dy=2$ , and $\Delta d=0$ .

Progressive T-nets. The constraints that structure transpositional relations for the homogeneous T-net are now expanded to allow for a variation of transpositional values across both the $x$ - and $y$ -axes. The result is

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a lattice; here referred to as progressive $T$ -net. Figure 4a presents a familiar example of such construct, the so-called Alban Berg’s “master array” of i-cycles, which coordinates all i-cycles in both vertical and horizontal dimensions. Figure 4b recasts Berg’s array as a T-net, emphasizing transpositional relations. Unlike in homogenous T-nets, however, in progressive T-nets parallel paths of transpositional values vary throughout the network; in the case of the Berg’s array, the transpositional values increase by 1 from left to right and from bottom to top in the network; in other words: $\Delta dx\mid y=\Delta dy\mid x=1$ . In general, the constraints for a progressive T-net are $\Delta dx\mid y=\Delta dy\mid x\ne 0$ and $\Delta dx\mid x=\Delta dy\mid y=0$ .

Fig. 4. (a) Alban Berg’s “master array” of interval cycles. (b) Berg’s array represented as a progressive $T$ -net.

Figure 5 shows a progressive T-net, whose transpositional levels in parallel paths are incremented by 9, i.e., $\Delta dx\mid y=\Delta dy\mid x=9$ (while $\Delta dx\mid x=\Delta dy\mid y=0$ ). This progressive T-net models the closing section of Witold Lutosławski’s song “Rycerze” (Ilakowicz Songs 1956-57). The passage (mm. 181-199, reduced in Figure 6a) presents a series of “vertical” 12-tone chords, each being formed by the stacking of three distinct “fully-diminished-seventh chords.” The transpositional framework of Figure 5 is “filled in” by the stacking of diminished-seventh chords in Figure 6b. Each stacking of three diminished-seventh chords is also a segment of what I refer to elsewhere as an affinity space.[16] Affinity spaces are (often non-octave) periodic constructs in which an interval pattern or modular unit (a fully-diminished-seventh chord in the case of Figure 6b) is transposed twelve times in different parts of the space. Lutosławski’s closing passage is highlighted in the figure: while the 12-tone chord progression circles around all available $dy$ -levels < 5, 2, 11, 8, (5)> returning to its starting point at $dy=5$ , the stacking of the diminished-seventh chords does not cover all the theoretically available $dx$ -levels < 1, 10, 7, 4> , as the stacking does not use $dx=4$ ; having done so, however, would create a repetition of the “lower” diminished-seventh chord and

Pitch-Space Lattices: Tonnetze and Other Transpositional Networks

thus a tetrachordal redundancy in the 12-tone chord. ${}^{11}$ The transpositional levels for the stacking and succession of fully diminished-sevenths in Lutosławski’s passage also ensure that no common tones are retained in adjacent tetrachords both vertically (along the $y$ -axis) and horizontally (along the $x$ -axis). ${}^{12}$

Fig. 5. A progressive $T$ -net: $\Delta dx\mid y=\Delta dy\mid x=9$ , $\Delta dx\mid x=\Delta dy\mid y=0$ .

(a)

(b) Fig. 6. (a) Lutoslawski’s 12-note chord succession (stacking of “fully- diminished-seventh chords”) in the closing section of Rycerze (mm. 181-199). (b) Rycerze’s closing section modeled by the mapping of affinity spaces within the progressive $T$ -net.

Dynamic T-nets. This section expands further the versatility of T-nets by modeling a constant contraction or expansion of the network in both straight and parallel paths. Such networks are here referred to as dynamic $T$ -nets. Whereas in progressive T-nets the transpositional values of $dx$ do not vary along the $x$ -axis, and the values of $dy$ do not vary along the $y$ -axis, such is not the case in dynamic T-nets where the transpositional values of $dx$ and $dy$ do vary along their respective axes. The constraints on the rate of change for directed intervals in a dynamic T-net are thus $\Delta dx\mid x=\Delta dy\mid y\ne 0$ and $\Delta dx\mid y=\Delta dy\mid x\ne 0$ .

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Figure 7 presents a dynamic T-net, which models the longer central section of Lutosławski’s Postludium I (Three Postludes, 1959). The transpositional changes for this dynamic T-net are $\Delta dx\mid x=\Delta dy\mid y=\Delta dx\mid y=\Delta dy\mid x=11$ . Lutosławski’s passage (mm. 41-60) reduced in Figure 8 achieves a gradual contraction of pc-space by juxtaposing a series of affinity-space segments that tend towards a semitonal cluster, while retaining a constant interval of transpositio $t=1$ (mod 12) (except hexachord 8). Figure 9 maps the succession of affinity-space segments of the section into the dynamic T-net of Figure 7. Given the contracting aspect of the dynamic T-net from left to right and bottom to top, the affinity spaces are no longer mapped along the $y$ -axes as in previously discussed T-nets, but are rather mapped in zig-zag lines along the southeast-northwest direction.

Fig. 7. Dynamic T-net: $\Delta dx\mid x=\Delta dy\mid y=\Delta dx\mid y=\Delta dy\mid x=11$ .

Fig. 8. Lutosławski’s Postludium I, mm. 41-60 (Three Postludes, 1959); pitch reduction of gradually contracting affinity-space segments.

Pitch-Space Lattices: Tonnetze and Other Transpositional Networks

Fig. 9. Mapping of affinity-space segments into the dynamic $T$ -net.

The paper proposes a framework that coordinates several models of pitch space whose constructive features rely on the concept of interval cycles and transpositional relations. This general model brings under a focused perspective diverse pitch structures such as tonnetze, affinity spaces, Alban Berg’s “master array” of interval-cycles, and several types of transpositional networks, here referred to as homogeneous, progressive and dynamic T-nets. While the paper engages in incremental modifications to the constructive features of the Tonnetz, additional steps could be taken in the exploration of further deformations. For instance, Figure 10 presents a dynamic T-net, ${}^{14}$ which further deforms the changes of transpositional values while retaining a consistent structure: $\Delta dx\mid x=1,\Delta dy\mid y=10,\Delta x\mid y=\Delta y\mid x=3$ .

Fig. 10. Fully dynamic $T$ -net.

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References

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