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

i.e. in this case chords are understood as the finite subsets of the space $TON ES$ . Another possibility (to be discussed in the subsequent subsection) models chords as tone profiles or >fuzzy tone sets<.

In Hugo Riemann’s terms the semantic layer of harmony is consituted by tonal functions within given keys and modes (see Riemann (1887) as well as sections 3, 4). These are signified first of all by prime chords, i.e. major or minor triads. To each key, mode and function there is a »prototypical« prime chord signifying that locus (e.g. the C-major-triad signifying the C-major tonic). Other chords signifying the same locus (e.g. the A-minor triad) are studied in morphological relation to the prototypical one. (The A-Minor-triad shares consonant tones C and E with the prime chord and has an additional dissonant tones, namely A being >conceptually dissonant< with respect to the C-major-triad). Dissonant tones supporting the signification are called characteristic dissonances.

Therefore, a >Riemann-inspired< strategy to define a Riemann Logic $RL$ is to start with a map $locusChord : HARM --> CHORDS$ , and to first associate each harmonic locus $H (- HARM$ with a »prototypical« chord $X = locusChord(H)$ , which yields the highest truth value $RL(X |\ H)$ . In a second step one attempts to determine $RL(X'\| H)$ for other chords $X'$ by comparing them with $X$ on a morphological level. In other words, one may define

RL(X |\ H) := compare(X, locusChord(H))

on the basis of the two maps

locusChord : HARM --> CHORDS, compare : CHORDS × CHORDS --> [0, oo ),

To directly edit the individual values $RL(X |\H)$ for every tone set and every locus $H$ >by hand< is practically impossible. Even in the simple case of a 12-elemented tone set and 72 harmonic loci there are $(212- 1).72 = 294840$ values to be specified. Of course, one may object that most of these values seem counter-factual with respect to a selected corpus of musical works. But even for the actually used chords one needs either statistical methods or a computational model. The latter involve formal components which extrapolate a Riemann Logic from a small selection of parameters.

2.2 Tone Profiles

In this subsection we discuss a linear extrapolation approach which is based on calculations with tone profiles. Our goal is to specify the map $compare$ . First consider the real vector space $TONES R$ freely generated over the set $T ON ES$ . This space has an independent one-dimesional subspace for each tone $t (- TON ES$ . Tone profiles are defined as vectors

TONES (xt)t (- TONES (- R

with the following properties:

$xt >= 0$ for all $t (- TON ES$ and $xt > 0$ for only finitely many t,