Spiral array model

In music theory, the spiral array model is an extended type of pitch space. It represents human perceptions of pitch, chord and key in the same geometric space, as a mathematical model involving concentric helices (an "array of spirals"). It was proposed in 2000 by Prof. Elaine Chew in her MIT doctoral thesis Toward a Mathematical Model of Tonality. Further research by Chew and others have produced modifications of the spiral array model, and, applied it to various problems in music theory and practice, such as key finding (symbolic and audio ), pitch spelling, tonal segmentation, and similarity assessment .

The spiral array model can be viewed as a generalized tonnetz, which maps pitches into a two-dimensional lattice structure. The spiral array wraps up the two-dimensional tonnetz into a three-dimensional lattice, and models higher order structures such as chords and keys in the interior of the lattice space. This allows the spiral array model to produce geometric interpretations of relationships between low- and high-level structures. For example, it is possible to model and measure geometrically the distance between a particular pitch and a particular key, both represented as points in the spiral array space. To preserve pitch spelling, because musically A# ≠ Bb in their function and usage, the spiral array does not assume enharmonic equivalence, i.e. it does not fold into a torus.

Structure of the spiral array

The model as proposed covers basic pitches, major chords, minor chords, major keys and minor keys, represented on five concentric helices. Starting with a formulation of the pitch helix, inner helices are generated as convex combinations of points on outer ones. For example, the pitches C, E, and G are represented as points by the cartesian coordinates C(x,y,z), E(x,y,z) and G(x,y,z). The convex combination formed by the points CEG is a point inside the triangle outlined by the three pitches, and represents their "center of effect" (ce). This interior point, C_M(x,y,z), represents the C major chord in the spiral array model. Similarly, keys may be constructed by the centers of effect of their I, IV, and V chords.

1. The outer helix represents pitches classes. Neighboring pitch classes are a music interval of a perfect fifth, and spatially a quarter rotation, apart. The order of the pitch classes can be determined by the circle of fifths. For example, C would be followed by G, which would be followed D, etc. As a result of this structure, and one of the important properties leading to its selection, vertical neighbors are a music interval of a major third apart. Thus, a pitch class's nearest neighbors and itself form perfect fifth and major third intervals.
2. By taking every consecutive triad along the helix, and projecting their centers of effect, a second helix is formed inside the pitch helix, representing the major chords.
3. Similarly, by taking the proper minor triads and projecting their centers of effect, a third helix is formed, representing the minor chords.
4. The major key helix is formed by projections of the I, IV, and V chords from points on the major chord
5. The minor key helix is formed by similar projects of minor chords.

Equations

The pitch spiral P, is represented in parametric form by:

$P(k)={\begin{bmatrix}x_{k}\\y_{k}\\z_{k}\\\end{bmatrix}}={\begin{bmatrix}rsin(k\cdot \pi /2)\\rcos(k\cdot \pi /2)\\kh\end{bmatrix}}$

where k is an integer representing a semitone, r is the radius of the spiral, and h is the "rise" of the spiral.

The major chord C_M is represented by:

$C_{M}(k)=w_{1}\cdot P(k)+w_{2}\cdot P(k+1)+w_{3}\cdot P(k+4)$

where $w_{1}\geq w_{2}\geq w_{3}>0$ and $\sum _{i=1}^{3}w_{i}=1.$

The weights "w" effect how close the center of effect are to the fundamental, major third, and perfect fifth of the chord. By changing the relative values of these weights, the spiral array model effects how "close" the resulting chord is to the three constituent pitches. Generally in western music, the fundamental is given the greatest weight in identifying the chord (w1), followed by the fifth (w2), followed by the third (w3).

The minor chord C_m is represented by:

$C_{m}(k)=u_{1}\cdot P(k)+u_{2}\cdot P(k+1)+u_{3}\cdot P(k-3)$

where $u_{1}\geq u_{2}\geq u_{3}>0$ and $\sum _{i=1}^{3}u_{i}=1.$

The weights "u" function similarly to the major chord.

The major key T_M is represented by:

$T_{M}(k)=\omega _{1}\cdot P(k)+\omega _{2}\cdot P(k+1)+\omega _{3}\cdot P(k-1)$

where $\omega _{1}\geq \omega _{2}\geq \omega _{3}>0$ and $\sum _{i=1}^{3}\omega _{i}=1.$

Similar to the weights controlling how close constituent pitches are to the center of effect of the chord they produce, the weights $\omega$ control the relative effect of the I, IV, and V chord in determining how close they are to the resultant key.

The minor key T_m is represented by:

$T_{m}(k)=\nu _{1}\cdot C_{M}(k)+\nu _{2}\cdot (\alpha \cdot C_{M}(k+1)+(1-\alpha )\cdot C_{m}(k+1))+\nu _{3}\cdot (\beta *C_{m}(k-1)+(1-\beta )\cdot C_{M}(k-1)).$

where $\nu _{1}\geq \nu _{2}\geq \nu _{3}>0$ and $\nu _{1}+\nu _{2}+\nu _{3}=1$ and $0\geq \alpha \geq 1$ and $\beta \geq 1.$

Resources

• Chew, Elaine (2014). Mathematical and Computational Modeling of Tonality: Theory and Applications. Springer.
• Chew, Elaine (2000). Towards a Mathematical Model of Tonality (Ph.D.). Massachusetts Institute of Technology.
• François, Alexandre (2012). "MuSA_RT".