Generalized Tonnetze and Zeitnetze, and the topology of music concepts

Abstract

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.

Keywords:

• Tonnetz
• topology
• geometry
• Fourier transform
• voice leading

2010 Mathematics Subject Classifications:

• 37F20
• 11F23

Acknowledgements

Thanks to Dmitri Tymoczko, whose comments on an earlier version were instrumental in clarifying and improving the content and presentation of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 London (2002) suggests using this term for a different kind of network, but his definition has not entered into common usage.

2 An invertibility condition similar to the transposability condition might also be added to this definition, so that the Tonnetz graph must include all representatives of each T${}_n$I-type set class. All the Tonnetze and Zeitnetze appearing as examples below would satisfy such a condition.

3 This relationship, however, is not one-to-one. The point representing {C,G} in Ph${}_5$ also represents {F,D}. The relationship only becomes one-to-one if we consider all possible phase spaces $0\le k\le u/2$ and the magnitudes of each of these Fourier coefficients.

4 Throughout, “voice-leading spaces” will refer to what Callender, Quinn, and Tymoczko (2008) specifically call “OP-space.” That is, it is voice-leading space folded to recognize octave and permutational equivalences.

5 A common-tone voice leading between ic1 dyads that is equally efficient to the one on ic5 dyads is in fact possible (moving one note by two semitones at each stage) but it involves a voice crossing. Stated with respect to the dual space, we can say that edges can only rotate through a vertical axis (corresponding to the wedge voice leading) not the horizontal axis (corresponding to transposition). Rotations of the latter type correspond to a voice crossing.

6 The exception is a choice such that $k_1$, $k_2$, and u are not mutually prime; see Yust (2015b). Ideally we should also apply some constraints on the choice of $k_1$ and $k_2$ relating to the compactness of regions, but determining exactly how this should be done would be somewhat involved and therefore beyond our present scope.

7 Our drawings of the regions, with straight lines as boundaries, however, are not completely accurate and should be seen as an idealization. In fact, if the interval defining a given boundary has a larger magnitude in one dimension than the other, this imparts a sinusoidal curve to the boundary. In practical situations, the amount of curvature is typically slight and therefore can be safely disregarded.

8 Dmitri Tymoczko (personal communication, October 2018) has challenged this point with the argument that diatonicity is not an intuitive description of $f_5$ in other universes (say, for quarter-tones, u = 24). The obvious response to this is that $f_5$ has only been equated with diatonicity in contexts where 12-tET is taken for granted – in other words, “diatonicity” or Quinn's (2006) “diatonicness” has only ever referred to $f_{5/12}$, not $f_{5/u}$ for all u. Nonetheless, Tymoczko's challenge raises an interesting point, which is that certain concepts related to harmonic qualities should generalize away from a particular discretization of the octave. This is true of diatonicity: we have an intersubjectively robust intuitive sense of diatonicity in, say, 31-tone equal temperament. The appropriate generalization to capture this for u>12 is $|f_{7/u}||f_{12/u}|/|f_0{|}^2$. That is, diatonicity is determined, generally, by the combined size of two components, $f_7$ and $f_{12}$ (normalized by the total power, $f_0^2$). The special status of these particular components ultimately comes, presumably, from the fact that they give the best approximations to the acoustic perfect fifth, ${\log }_2\left(3/2\right)$, so an alternate, more basic, definition might be given by the correlation with the spectrum of the perfect fifth with $f_0$ (and values of k exceeding some threshold) excluded.

9 For more on this distinction, see Yust (2015b, 2018b).

10 Spelling itself is by definition a notational phenomenon, and as such it is accurately described by a system of voice leading on seven-note collections, or generalized key signatures (Tymoczko 2005). Although Ph${}_5$ mimics the central axis of this voice-leading space when restricted to relatively even seven-note collections (Yust 2016), the two are derived from a very different set of assumptions and should not be conflated (Tymoczko and Yust 2019). However, it is routine in music to treat enharmonic notation as an imperfect tool for expressing an underlying musical reality, often relating to key or harmonic function (though by no means limited to that). To the extent that one believes key perception relates to pitch-class distributions – a theory that remains debatable, but is supported at this point by a large amount and variety of empirical evidence (Krumhansl and Cuddy 2010) – tonal composers' use of spelling often reflects an underlying reality expressed by Ph${}_5$ relationships.

11 The relationship of intervals to Fourier components (i.e. the index of a phase space, Ph${}_k$) has been explored by Quinn (2006) and Amiot (2007). Both show that a one-to-one relationship does not hold for interval content per se, but does for intervals as generators.

12 See also Yust (2019).

13 There are three barycentric coordinates given by projections onto the lines from each vertex of the triangle to the midpoint of the opposite edge, and their values are the proportion of the distance from the opposite edge. The three coordinates are constrained to sum to a constant, so only two of the three are independent.

14 Compare to the trichord space of Callender, Quinn, and Tymoczko (2008) and Tymoczko (2011), which has the same topology. This space is derived differently, from voice-leading considerations, but is similar to the spaces discussed here in that it can be understood as a quotient of a direct product of three Ph${}_1$ spaces.

15 Theoretically this reasoning could be inverted so that the chord resolves to G♯ minor with A and C resolving down by semitone. This is less compelling because of the general tonal tendency to associate descent in Ph${}_5$ (decending fifths, ascending leading tones) with resolution.

16 Note that the total is 24 (rather than $4^3=64$) because certain intervals are excluded, namely those that are long in two dimensions at once. We already noted that the diminished seventh interval has this property (it is long in both Ph${}_1$ and Ph${}_5$). Similarly the long Ph${}_3$ interval (the minor third that does not span triadic positions) is never the augmented second (indeed, it would hard to interpret the tonal meaning such an interval), and also is never allowed to be the major sixth interval.

17 However, this is not the only possible method of folding. See the section on spherical Tonnetze below.

18 Toussaint's definition is different than Osborn's, effectively only including the maximally even patterns (which, it turns out, can be generated though a version of the Euclidean algorithm). Osborn's definition is somewhat preferable since a new term is not needed for maximally even patterns.

19 Although they may be reduced or mapped onto one another. For instance, in the 3-torus version of the seventh-chord Tonnetz, there are two distinct planes with (025) Tonnetze and two with (037) Tonnetz. These are mapped onto one another in the folded version. Each (036) Tonnetz has eight triangles as a plane in the 3-torus, but only four in the fattened 2-torus.

20 Reasoning from χ can thus show that these are the only two examples of two-dimensional spherical Tonnetze. We could also observe that the existence of a spherical Tonnetz satisfying the transposability condition implies the existence of a Platonic solid with u vertices and whose symmetry group has a cyclic subgroup of order u. By process of elimination we are left then with only these two, tetrahedral and octahedral. For example, there is no cubic Tonnetz because the order-8 subgroups of the symmetry group of a cube are not cyclic.

21 This can be done by specifying that the balanced voice leading plane occurs within the region, which is then defined as a convex hull around that point. Note that the balanced voice leading is the same plane for three major-third related triads, so these regions all cross one another in the space. For instance, the C major region and E major region cross in a line that corresponds to balanced voice leadings that hold E constant, C major and A♭ major cross in a line of balanced voice leadings holding C constant, and all three planes intersect in a point that corresponds to balanced voice leadings between trichords of sum class 11 ($=0+4+7=11+4+8=10+5+8$ mod 12). The illustration in Tymoczko 2012 redraws the Tonnetz to eliminate these visually confusing intersections, which is helpful for visualizing its structure, but somewhat obscures the underlying geometry.

22 A three-dimensional toroidal Tonnetz was also proposed by Gollin (1998), but, lacking the minor seventh chords, it is impossible to think of Gollin's Tonnetz as actually filling any space, including a three-dimensional toroidal one. Tymoczko (2012) points out that Gollin did not really explicitly realize the geometry of his Tonnetz, and therefore he might have been thinking of something like a voice-leading Tonnetz, in which case the tritone duplications in his illustration would be superfluous. If, on the other hand, he had something like Douthett's Tonnetz in mind, as he suggests in referring to the geometry as toroidal, the tritone duplications are necessary, and we could give a retroactive theoretical justification for them along the lines suggested in Yust (2018a), using phase spaces. Nevertheless, Tymocko's critique illustrates the importance of establishing clear theoretical foundations when proposing something like a geometry of chord relations.