Tools for Music Information Retrieval and Playing - Intelligent Music Information Systems: Tools and Methodologies - page 127

its retrogradation R is the operator that maps S
into S -1 ={s n ,s n-1 ,…,s 1 }, s i ∈ Z 12 .
The retrogradation of the theme in example
shown in Figure 9 is shown in Figure 12.
The class of nonstandard rigid musical trans-
formations contains obviously all the automor-
phism of the pitch class set.
If we consider the 1-cycle permutation (A,E)
the theme in example shown in Figure 9 becomes
the theme shown in Figure 13.
The goal is to study a musical object “up to
transformations” which preserve its inner struc-
ture, that is, the connections among notes (viewed
as pitch, rhythm, or accent sequences).
A very useful trick in those situations is to
find quantities related to the object that are in-
variant under particular transformation groups
(Pinto, 2003).
In this context, we consider a structured set
of themes (a database of indexes) and a theme
(the query) which has to be compared with every
set-element. The idea is to exploit the typical
mathematical structure apt to formalize rela-
tions between objects, graphs, in order to build
a representative graph for every theme and then
work with graphs instead of themes.
In this way, is it possible to recognize a lot of
relevant structural musical similarities and also
to reduce the number of themes which should be
compared.

So to summarize, the first step is the classifi-
cation of themes by their representative graphs.
The next step is mining graphs in a database
isomorphic to the given one, and this can be done
through graph invariants in order to reduce the
computational cost of this operation; for a treat-
ment of the isomorphism problem see for example
Bollobas (1998) and Godsil and Royale (2001).
In Haus (2005) the graph-construction is
throughout discussed for purely melodic sequenc-
es. A simple sketch of the graph construction is
shown in Figure 14.
The notion of similarity function between
graphs, which provide an estimation of the “dis-
tance” between two musical objects, is the fol-
lowing (for a throughout discussion see Buckley
& Harary, 1990; Haus, 2005; Pinto, 2003).
Let M and M' be two music objects (for example
melodic sequences) with representative graphs
G=G(M) and H=H(M) . Given r ∈ N the r-order
similarity function is

Figure 12. Retrogradiation of theme

Figure 13. Theme after 1-cycle permutation

Figure 9. Theme

Figure 14. Graph representation of a melodic
sequence

Figure 10. Theme transposition

Figure 11. Tonal inversion of theme