The Tonnetz –  a tone network

Leonhard Euler was the first to describe the relationships among pitches by using a network called the Tonnetz  [1]. Although this structure has been largely generalised, see for instance [2,3], the original idea was to create a diagram mirroring the acoustical proximity of pitch classes (pitches modulo octave) of the chromatic scale. See Figure 1.

 

euler_tonn

Figure 1: The Tonnetz. As you can see every vertex is labelled with the name of a pitch class, i.e. without distinguishing a C belonging to the second octave of the piano (usually denoted by C2) from its higher counterpart C4.

Two consecutive notes on the horizontal axis of the diagram, equipped with the orientations of the arrows showed in Figure 1, form a perfect fifth interval (PV). On the vertical axis,  consecutive notes are a major third apart (MIII), from top to bottom. Observe that a change of the orientation of the axes would simply invert the intervals: A perfect fifth’s inversion is a perfect fourth, while the inversion of a major third is a minor sixth.

 

The Spiral Array

The Tonnetz has inspired important modern musical models. For instance, the spiral array [4] can be described as a spiralisation  of the Tonnetz. It is defined as a 3-dimensional helix where the position of the ith pitch class has cylindrical coordinates

    \[ p(i) = (\sin(i\pi/2) , \cos(i\pi/2) , ih), \]

where h\in\mathbb{R} is constant, and i\in\mathbb{Z}.

Hence, subsequent pitches on the helix are arranged to form perfect fifth intervals. Moreover, the periodicity of the trigonometric functions implies that

    \[ \pi_{x,y} (p(i)) = \pi_{x,y}(p(i+4)), \]

where \pi_{x,y}:\mathbb{R}^3\to\mathbb{R}^2 is the canonical projection. Thus, two of these points differ only in their last coordinate, and represent a major third interval. See Figure 2 for a representation of the spiral array, and an example of the two configurations of pitch classes described above.
spiral

Figure 2: The spiral array. Two consecutive pitch classes lying on the helix are a perfect fifth apart (green arrow). The blue arrow connects two pitch classes a major third far from each other.

Chords on the spiral

A chord on the spiral array can be represented as a polygon. See, e. g. Figure 3.  The great advantage of this geometric viewpoint is that once a geometric entity has been coherently chosen to represent a musical object, it is possible to take advantage of the powerful mathematical tools and theory defined on such spaces. Afterwards, geometric properties can be decoded  into musical ones.

spiral_ton

Figure 3: Chords on the spiral array. The C major triad is depicted as a red triangle, G and F major are rendered in green and blue, respectively.

Consider a chord \mathbf{C} realised by superposing n pitch classes \{p_1, \dots, p_n\}, and hence represented as a polygon\mathcal{C}, with as many vertices, on the spiral array. The idea is to represent \mathbf{C} as a single point in \mathbb{R}^3. The most natural candidate is the  centre of mass of \mathcal{C}:

    \[ b_{\mathcal{C} } = \left(\frac{\sum_{i=1}^n  x_i}{n},\frac{\sum_{i=1}^n  y_i}{n},\frac{\sum_{i=1}^n  z_i}{n} \right), \]

where p_i = (x_i,y_i,z_i) for every i\in \{1,\dots, n\}.

In the equation above every pitch class composing the chord \mathbf{C} (here we identify pitches with their associated points on the spiral array) contributes equally to define the position of the centroid b_{\mathcal{C}. However, every pitch contributes to the perception of a chord in a different way, and with a different weight. For example, a major and a minor triads can be distinguished by comparing their thirds, while the fifth is not contributing to the musical definition of these chords (the same argument holds, symmetrically, for an augmented and a major triad). Thus, it is possible to weight the contribution of each pitch class p_i\in\mathbb{C}, by considering the sum \sum_{i=1}^n\lambda_i p_i, such that \sum_i\lambda_i = 1. We mentioned above that a chord composed by 3 pitch classes can be represented as a triangle T on the spiral array.

 

t

Figure 4: Barycentric coordinates.

For the sake of intuition consider let T = [(1,0,0),(0,1,0),(0,0,1)]\in\mathbb{R}^3. The centre of mass corresponds to the point (1/3,1/3,1/3), where every pitch contributes equally to the representation of the chord associated to the triangle. See Figure 4. By considering points lying on the edges of T, the triad is reduced to an interval (the superposition of two notes), whilst in the interior of the triangle lie infinite points , with non-zero coordinates, corresponding to as many representation of the triad in our geometric space. Let f:\mathcal{C}\to\mathbb{R}^3 be the function that associates a chord with its representative point in \mathbb{R}^3.

 

Tonalities on the spiral

A tonality can be defined as the set of chords that arises from the harmonisation of a particular scale. Here, we will only consider major tonalities, in order to reduce the analysis to major scales. For example, consider the major scale of C:

    \[ \{C,D,E,F,G,A,B\}, \]

and superpose to each of its pitch classes two thirds, according to the scale itself:

    \[ \begin{matrix} \mbox{I} & \mbox{C major:} & C & E & G \\ \mbox{II} &\mbox{D minor:} &D & F & A \\ \mbox{III} &\mbox{E minor:} &E & G & B \\ \mbox{IV} &\mbox{F major:} &F & A & C \\ \mbox{V} &\mbox{G major:} &G & B & D \\ \mbox{VI} &\mbox{A minor:} &A & C & E \\ \mbox{VII} &\mbox{B diminished:} &B & D & F \end{matrix} \]

The strategy, presented in [4] to represent a tonality on the helix consists in considering the I, IV and V degree of the harmonisation, as it is shown in the table above. In geometrical terms, one considers the three triangles depicted in Figure 3. As we did for chords, it is thus possible to represent a whole tonality with a single point, by considering the point given by f(I,IV, V) for each tonality, where I,IV,V are the triangles associated the degrees of a tonality.

 

spiral2

Figure 5: Major tonality representation on the spiral array.

By representing in such way all the available major tonalities, we get a new set of points in Ton= \{t_1,\dots,t_{12}\}\subset\mathbb{R}^3. By taking advantage of the metric nature of \mathbb{R}^3, we can associate a tonality to an arbitrary chord, by considering the distance between f(\mathcal{C}) and the points in Ton.

 

References:

[1]  Euler, Leonhard. Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae. ex typographia Academiae scientiarum, 1739.

[2] Douthett, Jack, and Peter Steinbach. “Parsimonious graphs: A study in parsimony, contextual transformations, and modes of limited transposition.”Journal of Music Theory (1998): 241-263.

[3] Tymoczko, Dmitri. “The Generalized Tonnetz.” Journal of Music Theory 56.1 (2012): 1-52.

[4] Chew, Elaine. “The spiral array: An algorithm for determining key boundaries.”Music and artificial intelligence. Springer Berlin Heidelberg, 2002. 18-31.