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

An altered diatonic locus is a triad within an altered diatonic scale, i.e. a pair $(ttype(k),Dia(n)S)$ , such that $|(ttype(k)|< |Dia(n)S|$ . As an example consider $Va = (tMaj(4),Dia(0)(0,0,1,0,0,0,0))$ denoting the E major triad within the corresponding alteration of the C-major/a-minor diatonic collection. The sets of all diatonic and altered diatonic loci are denoted by $Ldia$ and $Lalt$ respectively. They can ce studied as harmonic configuration spaces in their own right equipped with a harmonic tensor $ht$ from which Lerdahl derives the desired elementary values.

The Lerdahl stratification of an (altered) diatonic locus $(t,D)$ is the chain of three set inclusions $Strat(t,D) := {t[1]} < {t[1],t[2]}< |t|< |D |$ , where $t[1]$ and $t[2]$ denote the first and the second element of the sequence $t$ respectively. Given two diatonic loci $(t1,D1)$ and $(t2,D2)$ we define the difference vector

( ) #({t2[1]}\{t1[1]}) Dif f((t1,D1),(t2,D2)) := #({t2[1],t2[2]}\{t1[1],t1[2]}) #(| t2| \|t1| ) #(| D2 |\|D1|)

The difference vector counts the number differences between the two diatonic loci at all four levels of the corresponding Lerdahl stratifications separately. In order to give the possibility to give different weight to the four levels we introduce a fixed user defined stratification profile $pstrat = (p1,p2,p3,p4)$ with nonnegative real entries $p1,p2,p3,p4$ and define the (weigthed) difference sum

diff((t1,D1),(t2,D2)) := <pstrat,Diff((t1,D1),(t2,D2))>

The default setting for the stratification profile is $p = (1,1,1,1) strat$ (c.f. Lehrdahl, 2001, p. 55, 60).

Furthermore Lerdahl takes the diatonic distances between the triads and the distances between the diatonic collections into account. In order to weight their contribution to the harmonic tensor $ht$ we fix another pair $(q ,q ) 1 2$ of nonnegative real weights and define:

dist((ttype1(k1),Dia(n1)S1),(ttype2(k2),Dia(n2)S2) := q1 .ddia(k1,k2) +q2 .d(n1,n2)

Again, the default setting for the distance weigths is $qdist = (q1,q2) = (1,1)$ . The harmonic tension between two (altered) diatonic loci $X1 = (t1,D1)$ and $X2 = (t2,D2)$ is calculated by adding their distance to the difference sum:

ht(X1 ~ X2) = d(X1,X2) := dist(X1,X2) + diff(X1,X2)

Proposition 2 The harmonic tension $ht = d$ is a pseudo-distance on the space $Lalt$ of altered diatonic loci. It is a distance, if the two profile parameters $p3$ and $p4$ do not vanish.