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

dstrict(X1/R1,X2/R2) = dR (X1/R1,I/R1)+ D(R1, R2)+ dR (I/R/|-, X2/R2). 1 2

However, what Lerdahl actually proposes is not this strict hierarchy, but a weakened version of it, where each locus is directly connected with six regional centers instead of just its own one. These regions are called pivot regions. Figure 8 displays the chart of the six pivot regions for the loci in the $C$ -major region Maj(C). The same six regions serve also as pivot regions for loci of the a-Minor region Min(a). The proposal resembles the quasi-hierarchical organisiation of public transport. Local trains (or busses) move between peripherical loci and several regional centers while inter-regional trains connect regional centers but do not stop at peripherical loci.

PICT

Figure 8:

Chart of the six pivot regions for the loci in the C-major region. The edges represent the kinship relations of the second degree $ii/C$ to its pivot region centers

The pivot region chart of an arbitrary major region $R = k+$ is the set

P ivot(k+) = {(k+ 2)-,(k+ 1)-,(k+ 4)-,(k- 1)+,k+,(k+ 1)+}.

The same pivot region chart is associated with the minor region $(k+ 3)-$ relative to $R$ , i.e. we have $Pivot(k+) = Pivot((k +3) ). -$ The local pivot distances

dpivot,k+ : M aj(k)× Pivot(k+) --> [0, oo ) dpivot,k- : M in(k)× Pivot(k- )-- > [0, oo )

are specified in table 5.3.

Table 3:

Local pivot distances of chordal loci in the C-major and the a-minor region to their common pivot regions


$d pivot$	$i/d$	$i/a$	$i/e$	$I/F$	$I/C$	$I/G$

$I/C$	10	7	9	7	0	7
$ii/C$	2	5	10	9	8	7
$iii/C$	10	5	2	10	7	9
$IV/C$	9	7	10	2	5	10
$V/C$	7	8	9	10	5	2
$vi/C$	7	0	7	9	7	10
$viio/C$	9	8	7	7	8	9

$i/a$	7	0	7	9	7	10
$iio/a$	9	8	7	7	8	9
$III/a$	10	7	9	7	0	7
$iv/a$	2	5	10	9	8	7
$V/a$	11	6	4	11	9	11
$VI/a$	9	7	10	2	5	10
$#viio/a$	8	9	11	11	8	6

This map is extended to a map $t168 : H168 × H168-- > [0, oo )$ which is calculated as follows. For loci in the same region $R$ $t168$ coinsides with the intra-regional distance $dR$ . For loci in different regions $t168(X1/R1,X2/R2)$ is the minimum length of the $36$ indirect possible pathways between these loci via their pivot regions, i.e.

t (X /R ,X /R ) 168 1= 1min 2 (d2 (X /R ,S) + D(S,T) +d (X /R ,T )) , S,T pivot,S 1 1 pivot,R2 2 2

where $S (- P ivot(R1),T (- P ivot(R2)$ vary through the pivot regions of $R1$ and $R2$ respectively. Lerdahl calls $t168$ chordal/regional distance without discussing the