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

In terms of graph theory we may say, that the extended graph

* GWeber = (Regions,Kin |~| KinL |~| KinS)

is colored by the distance labels $D = 7,9,10$ . Figure 6 shows two local subgraphs namely the stars of the nodes $C$ and $a$ , while figure 7 (upper part) displays the extended graph without edge labels. NB: This graph is not drawable on a torus without edge crossings.

PICT

Figure 6:

Stars of the nodes $C$ and $a$ within the extended regional kinship graph $* GWeber$ with direct distance labels $D = 7,9,10$ .

On other region pairs $(R1,R2)$ the metric $D$ is defined in terms of indirect kinship. To each sequence $s = (R1,...,Rn+1)$ one can attribute its length $sum n lengthD(s) = k=1D(Rk, Rk+1)$ and defines

' D(R, R ) = s=m(iRn,...,R')lengthD(s)

PICT

Figure 7:

Extended regional kinship graph $G*W eber$ (above) and corresponding distances $D(C, R)$ for varying region $R$ (below)

The lower part of Figure 7 displays the distances $D(C, R)$ from the fixed C-major Region to all 24 regions. We obtain one deviation from Lehrdahl (2001) p. 69, namely $D(C, Gb) = 28$ instead of $D(C,F#) = 30$ , which is not minimal.

5.3 Hybrid Space: Hierarchy versus Shortest Path

How to combine the intra-regional and the inter-regional distances? A solution is to just identify the regional loci in Weber Space with the corresponding regional centers in the disjoint union of all 24 regional spaces:

|_| |_| H168 := M aj(k) |~| M in(k). k= 0,...,11 k= 0,...,11

If we dont specify the mode of a regional locus $R$ we write $M ode(R)$ for the corresponding region and $I/R$ for its center. The intra-regional distances $dMode(R)$ together with the inter-regional distance $D$ partially define a metric which can be canonically extended to a metric $dstrict : H168 --> H168$ . Consider two loci $X1/R1, X2/R2$ in different regions. Their strict distance is then