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

quantitative comparison seems possible at all may, however, be more important for these first steps into a new field than the preliminary musicological consequences .

The following propositions answer the only general question concerning the SR treated here: How close can the SR of a melody be to one? In other words, are there melodies (almost) all of whose 3-element subsets are symmetry patches? For a given paradigmatic group and a fixed integer $k$ , a more specific mathematical problem consists in either computing or at least finding estimates for the supremum of the SR over all melodies of length $k$ . For paradigmatic groups containing retrogrades (reflections in vertical lines) or pitch inversions (reflections in horizontal lines) the answer is trivial.

Proposition 2 If the paradigmatic group $G$ contains retrogrades or pitch inversions, then there exist melodies with SR equal to one.

Proof: If $G$ contains pitch inversions, then any “melody” whose tones all have the same pitch (i.e., they all lie on one horizontal line) is a symmetry patch itself. Since the whole melody is point-wise fixed by the reflection in the line determined by the common pitch, all of its subsets are symmetry patches. In particular, this holds for the 3-element subsets. On the other hand, if the retrogrades belong to $G$ , then any self-retrograde melody (or palindrome, as they are also called) has the required property.

In the case of the group $T$ of translations the answer is a little more complicated. We will describe two situations in which the SR tends to one. The first is the “melody” formed by the successive application of one and the same translation to a single tone, and the second uses the idea of repeating a melody over and over again.

Proposition 3

Let $x (- Z2$ be a tone and $T (- Aff (R2)$ any integral translation with positive first component. Denote by $Mk(x, T)$ , $k (- N$ , the melody defined as $(x,T(x),T2(x),...,Tk(x))$ . Then we have $srT(Mk(x, T))-- > 1 for k-- > oo .$
Let $M < Z2$ be any melody. Denote by $M kp$ the melody formed by “repeating this melody $k$ times with intervals of length $p$ ”, $k,p (- N$ , $k,p > 0$ . To define $M kp$ , let $T$ denote the horizontal right translation by $p+ (en- e0)$ , where $x0 = (e0,h0)$ is the first and $xn = (en,hn)$ the last tone of $M$ , and set $M 1p = M$ , $M kp = Mpk-1 U Tk-1(M )$ . Then, again we have $k srT (M p) --> 1 for k --> oo .$