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

Proof:

First, note that for $k >= 2$ the $k$ -element subsets $(x,T (x),...,T k-1)$ and $(T(x),T2(x),...,Tk(x))$ of $M (x,T) k$ are symmetry patches, since the translation $T$ maps one into the other. Hence, for $k >= 3$ , there are more than $( k ) 3$ symmetry 3-patches. This implies the following estimate for the translational symmetry rate: $( k ) 3 1 >= srT (Mk(x, T)) >= (------)-= k(k--1)(k--2) = k--2. k+ 1 (k+ 1)k(k- 1) k+ 1 3$ This rough estimate suffices to prove the first statement.
Claim: Each subset of $M kp$ not containing one tone from $M = M 1p$ as well as one from $Tk-1(M )$ is a symmetry patch with respect to $T$ . Every subset of $M kp$ not containing a point of $Tk-1(M )$ is transformed by $T$ into another subset of $M kp$ . A similar argument using $T- 1$ applies to each subset disjoint from $M = M 1p$ . Thus, only subsets of $M kp$ containing at least one point from $M = M 1p$ as well as one from $Tk-1(M )$ can be symmetry patches for $T$ . In the case of 2-symmetry patches this implies that there are at most $n2(kn- 2)$ 3-element subsets of $M kp$ which are no symmetry patches. For $1- sr (M k) T p$ we thus have the following bounds

$2 2 2 0 < 1- srT(M k) < n((kn--)-2)= ----n-(kn---2)---- = ---6n----. kn kn(kn--1)(kn--2)- kn(kn -1) 3 6$ For $k --> oo$ this quotient tends to 0. Hence we obtain $srT(M k)-- > 1$ .

Remark 2 Perhaps the dullest and at the same time highly translationally symmetric melody $Mk(x,(1,0))$ with $T$ -symmetry rate $1- ---6--- k(k- 1)$ is the natural candidate to realize the least upper bound for the $T$ -symmetry rates of all $(k+ 1)$ -tone melodies. In addition, the proofs of both propositions show that the same results also hold in the case of higher symmetry rates.

The SR discussed above only took into account the number of elements in $S2G(M )$ , ignoring any additional structure. The new numerical characteristic to be defined and discussed below is based on the orbit structure in this set.

Definition 6 Let $M$ be a melody with $n$ tones, $G$ a paradigmatic group, and $k$ a positiv integer. Two symmetry patches $A$ and $B$ from $M elk(M )$ , $k < n- 1$ , belong to the same $G$ -orbit if there exists a transformation $g$ in $G$ taking $A$ into $B$ under its action in $M elk$ .