Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

- 411 -

of a twelve-elemented set of tonalities $Fenh$ , a three-elemented set ${T,D, S}$ of tonal functions and a two elemented set ${M aj,min}$ of modes. The harmonic tensor $ht : R × R --> R$ is the sum of three user-defined tensors within these factors:

hton : Fenh× Fenh --> R hfun : {T,D, S}× {T,D, S}--> R hmod : {M aj,min} ×{M aj,min} --> R

The tonality tensor $h ton$ is supposed to be homogeneous with respect to the translation within $F enh$ , i.e. it satisfies $h (x,y) = h (x+ z,y +z) ton ton$ for all $x,y,z (- F enh$ . The full harmonic tensor $ht$ is defined by the formula:

h((t1,f1,m1) ~ (t2,f2,m2) := ht(0 ~ (t2- t1))+ hf(f1 ~ f2)+ hm(m1 ~ m2).

The user has to specify $25 = 12+ 9+ 4$ values in the theory settings.

Figure 1:

Snapshot of the user interface for controlling the harmonic tensor $ht$ . It consists of three parts: tonality distance matrix (left), function distance matrix (center) and mode distance matrix (right)

The corresponding panels (c.f. Figure 1) are called tonality distance matrix, function distance matrix and mode distance matrix. The harmonic tensor $ht$ is a pseudometric or a metric if and only if all the three tensors $hton$ , $hfun$ and $hmod$ are.

3.2 Riemann Logic in the >Classic< HarmoRubette

We first describe a slightly simplified definition in direct application of the tone profile method (c.f. Subsection 2.2) and recall Mazzola’s original account afterwards. We are concerned with the space $TON ES = Hoct$ of pitch classes and the corresponding tone profiles $CHORDS = P ro(Hoct)$ . The definition of a map $locusProfile : R --> P ro(Hoct)$ is given from a user defined map:

FunctionScale : {0}× {T,D, S}× {M aj,min} --> R >=0,

and a homogenity assumption that the 6 profiles are shaped in the same way in all 12 tonalities. In other words, the user specifies the (not yet normalized) profiles for the 6 tonal functions with respect to the C-tonality. Figure 2 shows the corresponding user interface, which is called the Function Scale Matrix.

Figure 2:

Function Scale Matrix. The user has to specify six tone profiles for the tonal functions with respect to the C tonality.

In accordance with the homogenity assumption we obtain the (normalized) formula:

F unctionScale(0,f,m)(h - 7kmod 12) locusP rof ile(k,f,m)(h) := -- V~ s um 11------------------------ t=0FunctionScale(0,f,m)(t)2

- 411 -

Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory