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

The Euclidean distance between the two motives $Motif1$ and $Motif2$ is

( 2 2 2)1/2 V~ - dCont(Motif1,Motif2) = (-1 - 0) + (- 1- - 1) +(1 -- 1) = 5

and the Euclidean distance between their respective gestalts is

GdCP (GesP (Motif),GesP(Motif)) = min{V ~ 5, V~ 5, 3,1}= 1. Cont t 1 t 2

Finally, given an $e > 0$ , the $e$ -neighborhood of $Motif1$ is the set of all motives $M$ in $M OT (S)$ of cardinality 3 or 4 such that:

${ d3(Cont(M ),(- 1,- 1,1)) ,d3(Cont(M ),(1,1,- 1)), } min d3(Cont(M ),(- 1,1,1)) ,d3(Cont(M ),(1,-1,-1)) < e,$
if $card(M ) = 3$ ;
${ } d3(Cont(M '),(-1,-1,1)) ,d3(Cont(M '),(1,1,-1)), min d3(Cont(M '),(-1,1,1)) ,d3(Cont(M '),(1,-1,- 1)) < e,$
for a suitable submotif $' M$ of cardinality 3, of $M$ with $card(M ) = 4$ .

By definition of $e$ -neighborhoods, motives with cardinality $2$ cannot be in the neighborhood of $Motif 1$ .

This completes the setup of the motivic composition space $M OT (S)$ of the score $S$ .