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

These maps are useful for computer-aided explorative analyses, as we will see below. With these settings we define:

transV alk : [0, oo ) × HARM × HARM --> [0, oo ) locusV alk : [0, oo ) × HARM --> [0, oo ) eval0 : HARM --> [0, oo )

by the formulas

transValk(v,Hk -1,Hk) := v .HT (Hk-1 ~ Hk), locusV alk(v,Hk) := v .c.RL(Xk |\ Hk).rk(Hk). eval0(H0) := locusVal0(1,H0) = c.RL(X0 |\H0).r0(H0).

Multiplication with positive numbers is order-preserving as well as zero-preserving. Multiplication with zero is also allowed and corresponds to the disqualification of those paths passing through the loci (or transitions) with zero-evaluations at $k$ . These formulas completely specify the data which is necessary for the best path calculation. In case of a harmonic tensor $ht : HARM × HARM --> R$ we consider the corresponding maps $transValk(v,Hk-1,Hk) := v .exp(- ht(Hk -1 ~ Hk))$ (c.f. Subsection 1.3).

In the normal case we have constant restriction maps $rk(Hk) = 1$ , which actually are not restrictive. But in explorative applications one is sometimes interested to force the pathways to pass though a certain locus $H$ at index $k$ . In this case one uses a restriction map of the kind

{ rk(Hk) = 1 f or Hk = H 0 else

forcing any path $H$ to pass trough $H$ at position $k$ .

2 Riemann Logics

In this section we discuss some specifications of the very general setting, namely to define a map

RL : CHORDS × HARM --> [0, oo ).

In this general situation chords are simply elements of the abstract set $CHORDS$ , i.e. they are not necessarily composed of tones or intervals.

2.1 General Morphology of Chords

Chord Morphology--in a broad sense--is the internal investigation of a chord vocabulary $CHORDS$ in preparation of the study of harmonic signification. A special--but music-theoretically central--case is the study of chords as tone sets. This implies the consideration of a tone space $TON ES$ with

TONES CHORDS = Fin(TON ES) (_ 2 ,