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

Proof: We first check the triangle inequality $d(X1,X2) + d(X2,X3) >= d(X1, X3)$ for all $X1, X2,X3 (- Lalt$ . We show that it independently holds for all four components of the difference vector as well as for the distances $d$ and $ddia$ . As to the differences suppose one is given three finite sets $A,B, C$ . One has

card(B\A) = card(B\(A U C)) + card(B U (C\A)) card(C\B) = card(C\(B U A)) + card(C U (A\B)) card(C\A) = card(C\(A U B)) + card(C U (B\A))

The triangle inequality immediately follows from $C U (B\A) = B U (C\A)$ . The triangle inequality for $d$ is evident because it is a metric on $Z$ . Its additive invariance implies that $ddia$ is a pseudometric (to each triangle in the diatonic classes $Z7$ one finds a triangle in $Z$ representing the diatonic distances.)

The symmetry condition $d(X1,X2) = d(X2,X1)$ for all $X1$ and $X2$ holds separately for all components of the difference vector due to the fact that the cardinalities $1,2,3,7$ of the stratification levels are fixed and hence the same for $X1$ and $X2$ . For any two finite sets $A$ and $B$ of equal cardinality one always has $card(A\B) = card(B\A)$ . The symmetry of $d$ and $ddia$ follows from the fact, that they are a metric and a pseudometric respectively. So far we have shown that $d$ is a pseudometric on $Lalt$ .

As to the metric, suppose $X1 = (t1,D1)$ and $X2 = (t2,D2)$ do not coincide. In the case $t1 /= t2$ we have $d(X1,X2) > p3$ and in the case $D1 /= D2$ we have $d(X1,X2) > p2$ .

Remark 7 Lerdahl’s hierarchical tonal pitch space model is not explicitely based on the space $F$ of note names, but -- according to his own explanations -- on the space $Hoct$ of the twelve octave classes of pitch height. He studies twelve diatonic collections $* dia (k) := k+ {0,2,4,5,7,9,11}< Hoct$ for $k = 0,...,11$ as 7-elemented sets of pitch classes. However, close examination shows that the diatonic distance $ddia$ between roots of triads cannnot be properly defined in $Hoct ~= Z12$ . In other words, the chord distance rule (c.f. p. 60) is not properly defined and Lerdahl’s argument on page 63, denying a »fault not of the the rule« is not correct. Nevertheless the desired intra-regional, inter-regional and local pivot distances can be properly defined by choosing small subdomains of $F$ without enharmonic ambiguities of the diatonic distance.

References

AGON, CARLOS AUGUSTO (2003). Mixing Visual Programs and Music Notation in OpenMusic. In LLUIS PUEBLA, EMILIO ET AL. (ed.), Perspectives of Mathematical and Computer-Aided Music Theory. epOs Music, Osnabrück.

CAREY, NORMAN and CLAMPITT, DAVID (1989). Aspects of Well-Formed Scales. Music Theory Spectrum, 11(2):187-206.

CHEW, ELAINE (2000). Towards a Mathematical Model of Tonality. MIT Press.