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

5.1 Intra-Regional Distance

According to Lerdahl’s hierarchical viewpoint each tonal region - as an isolated space - is constituted by seven chord loci, namely the triadic degrees $o M aj = {I,ii,iii,IV,V,vi,vii}$ in major regions and $o o M in = {i,ii,III,iv,V,V I,#vii}$ in minor regions. He considers the metrics $dMaj : M aj× M aj --> [0, oo )$ and $dmin : M in× M in-- > [0, oo )$ which are given by the two tables below:

Table 1:

Distance-Matrix of a Major Region


$d Maj$	$I$	$ii$	$iii$	$IV$	$V$	$vi$	$viio$

$I$	0	8	7	5	5	7	8
$ii$	8	0	8	7	5	5	7
$iii$	7	8	0	8	7	5	5
$IV$	5	7	8	0	8	7	5
$V$	5	5	7	8	0	8	7
$vi$	7	5	5	7	8	0	8
$viio$	8	7	5	5	7	8	0

Table 2:

Distance-Matrix of a Minor Region


$d Min$	$i$	$iio$	$III$	$iv$	$V$	$VI$	$#viio$

$i$	0	8	7	5	6	7	9
$iio$	8	0	8	7	6	5	8
$III$	7	8	0	8	9	5	8
$iv$	5	7	8	0	9	7	6
$V$	6	6	9	9	0	9	7
$VI$	7	5	5	7	9	0	9
$#viio$	9	8	8	6	7	9	0

In subsection 5.6 we recapitulate Lerdahl’s motivation of these concrete quantities in terms of diatonic triad stratifications. Meanwhile we consider them as arbitrary settings. In the next two subsections we are concerned with the study of 24 such regions. The regions of each mode are parametrized by the circle of fifths ( $Fenh$ ). Therefore we specify them by tones of reference or names, i.e. we write Maj(0) = Maj(C), Maj(1) = Maj(G), etc. as well as Min(0) = Min(c), Min(1) = Min(g), etc.

5.2 Inter-Regional Distance

Apart from these regions as separate spaces we consider an abstract regional space $Reg$ , i.e. a space whose 24 elements are abstract region loci. To specify them by