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

differently. In the first case we allow non-negative values as well as negative ones. Positive values express >inhibitions< for the corresponding transitions, while nagative values express >attractions<. In the second case we allow non-negative values only. We use the following notation and terminology:

A map $ht : HARM × HARM --> R$ is called a harmonic tensor. In this case we have in mind that its values quantify transitions in terms of inhibitions (non-negative values) and attractions (negative values). We speak of para-pseudo-distance, if all $ht$ is symmetric (i.e. $ht(H1 ~ H2) = ht(H2 ~ H1)$ , for all loci $H1,H2$ ) and positive semi-definite, i.e. $ht(H1 ~ H2) >= 0$ and $ht(H ~ H) = 0$ for all loci $H1,H2, H$ . A para-pseudo-distance is called a pseudo-distance if further the triangle-inequality $ht(H1 ~ H2) + ht(H2 ~ H3) >= ht(H1 ~ H3)$ is satisfied for all $H1,H2,H3 (- HARM$ . It is called a para-distance, if it is positive definite (i.e. $ht(H1 ~ H2) = 0$ if and only if $H1 = H2$ ). Finally, $ht$ is called a distance (or a metric), if it is a pseudo-distance and a para-distance.
A map $HT : HARM ×HARM --> [0, oo )$ is called a harmonic transition value map. In this case we have in mind that its values directly quantify transitions in a monotonous way. If the image of $HT$ is actually $[0,1]$ we call it a para-probability map. If further $sum HT (H ,H) = 1 H( - HARM 1$ for all $H (- HARM 1$ we speak of a semi-probability-map and if in addition we have $sum HT (H,H ) = 1 H (- HARM 1$ for all $H (- HARM 1$ we speak of a probability-map.

We use the exponential/logarithmic functions in order to formally translate the two kinds of quantification into one another: Suppose, we are given a harmonic tensor $ht : HARM × HARM --> R$ . Its associated harmonic transition value map $HT = e-ht$ is defined by

HT (H1 ~ H2) = exp(- ht(H1 ~ H2)).

Conversely, if we are given a harmonic transition value map $HT : HARM × HARM --> [0, oo )$ , then its associated transition value map $ht = - log(HT )$ is defined by

ht(H1 ~ H2) = -log(HT (H1 ~ H2)).

Non-negative harmonic tensors (including para-pseudo-distances) formally correspond to para-probability-maps. However, we do not intend to project an ontological interpretation onto this correspondence.

1.4 Evaluation of Harmonic Analyses

We now discuss a numeric evaluation method for harmonic analyses $S |\ H$ as well a suitable algorithm for the determination of best analyses for a fixed chord sequence $S$ and varying pathways $H$ . The algorithm is called Viterbi algorithm and is used in the context of Hidden Markov Models in order to calculate a most probable process in accordance with a sequence of observations. Readers which are familiar with such models will notice that such a probabilistic interpretation can be seen