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

if $t(h1,g) = t(h2,g)$ and $t(g,h1) = t(g,h2)$ for all loci $g (- H$ , different from $h1$ and $h2$ . Finally, they are called indistinguishable, if they are both, extensionally and intensionally indistinguishable.

The following proposition provides a construction for a reduction of a given space $H$ with respect to the intensional indistinguishabilities of a transition value map $t$ .

Proposition 1 For symmetric transition value maps $t : H × H --> R$ , with $t(h1,h2) = t(h1,h2)$ for all $h1,h2 (- H$ intensional indistinguishability is an equivalence relation and $t$ induces a transition value map on the equivalence classes.

Proof: Reflexivity and Symmetry are evident. As to the transitivity, let $h 1$ be intensionally indistinguishable from $h 2$ and let $h 2$ be intensionally indistinguishable from $h 3$ . If $g (- H$ is any locus different from $h 2$ we have $t(g,h ) = t(g,h ) = t(g,h ) 1 2 3$ as well as $t(h ,g) = t(h ,g) = t (h ,g) 1 2 3$ . In the case of $g = h 2$ we have $t(h ,h ) = t(h ,h ) = t(h ,h ) = t(h ,h ) 2 1 3 1 1 3 2 3$ . The last statement follows from the definition of intensional indistinguishability. $[]$

Now we consider our particular cases $(H ,t ) 168 168$ and $(H ,d ) 168 168$ . If two loci are intensionally indistinguishable with respect to all the partial intra-regional, pivotal and inter-regional distances then this property is inherited by both $t 168$ and $d 168$ according to their construction. As we will see in subsection 5.6 all these partial distances are calculated on the basis of diatonic stratifications of triads. As certain loci in modally relative regions have the same stratifications and they are intensionally indistinguishable. In the case of C-major and a-minor regions we have:

I/C ~ III/a, ii/C ~ iv/a, IV/C ~ VI/a, vi/C ~ i/a, viio/C ~ iio/a

Each of the twelve pairs of modally relative regions consists of 9 equivalence classes (c.f. Figure 9). The resulting reduced chordal/regional space consists of 108 loci and is denoted by $H108$ . It can be equipped either with a para-distance $t108$ (inherited from $t168$ ) or with a distance $d108$ (inherited from $d168$ ).

Figure 9:

Classes of intensionally indistinguishable loci in modally relative regions

5.5 Lerdahl versus Lerdahl: The Faith-motive

The following example shows that Lerdahl did not strongly apply his theoretical principles from the chapter 2 of his book to the examples discussed in chapter 3 of his book. For the discussion of the harmonic pathway analysis of the Faith motive from Wagner’s Parsifal (bars 45 - 55) we may assume that the space $CHORDS = TRIADS$ consists just of the 36 major, minor and diminished triads. The analysis