Tools for Music Information Retrieval and Playing - Intelligent Music Information Systems: Tools and Methodologies - page 126

In Figure 6 we provide an example of analysis
based on the incipit of Mozart's Symphony in G
Minor KV 550.
Grammar rules allow analysis such that
sketched in Figures 7.a and 7.b.
Both analyses are grammatically correct but
no musician would subscribe them.
The main difficulty with Lerdahl's and
Jackendoff's work is that their view of music
is overly simplified. Even with melodic themes
numerous examples can be pointed out where
different components of music form independent
structures that overlap and cannot fit in only one
hierarchy.
A musical theme does not submit to just one
interpretation, and a piece of music with artistic
value may call forth several independent and even
contradictory sets of intuitions in an experienced
listener.
The problem is that the preference rules cannot
be used at all without a judgment by the analyst,
and even the well-formed rules contain terms
that are not formalized, for example “cadence”
and “consonant”.
The end result of this is that the theory cannot
be expressed as a computer program that would
analyze pieces by itself without any additional
input by a human analyst.

Figure 8. Identically structured themes

A motif can be transformed in a great number
of ways: the most important ones are rigid transfor-
mations, that is to say, one to one correspondences
of the set of notes. However a theme could surely
undergo transformations due, for example, to in-
sertion of notes (passing tones, embellishments,
etc.), as in the previous example.
Just to sketch the problem, look at the three
themes in Figure 8. They sound actually similar
even if, viewed as functions, they are quite dif-
ferent.
Nevertheless, from a structural point of view,
they are undistinguishable as they are obtained
by a simple permutation of notes.
So is evident that such similarities could not
be looked at just like “mistakes”.
Themes can undergo a lot of musical transfor-
mations: (rigid) standard ones are transpositions,
inversions and retrogradations.
Now we sketch those transformations in the
simple case, when the space of musical notes is
the Z 12 ring, in order to provide a brief tutorial to
nonmusical experts.
Transposition (T) is the adding of a con-
stant $n$ to a given pitch class sequence: T n
[x]=[x+n] .
Given the theme in Figure 9, one of its trans-
positions can be seen in Figure 10.
Inversion (I) consists in a sign changement:
I[x]=[-x] . The transposed inversion will be T n I
[x]=[-x+n] .
The (tonal) inversion of the theme shown in
Figure 9 is shown in Figure 11.
Retrogradation (R) is the inversion of the tem-
poral flow: given a pitch sequence S={s 1 ,s 2 …,,s n } ,

the structural approach

Functional approaches are useful in a lot of dif-
ferent contexts but not, for example, for structural
similarity queries, because they induce a lot of
musically false-true due to the lack of an intrinsic
structural coding.
In fact, how may be possible to find similarities
among the melodic sequences of Figure 8?
In many contexts of musicological analysis,
structural features are essential in order to find
similarities among musical data. This raised
the need for music information retrieval models
oriented to structural information.
Here, the most important concept we are deal-
ing with is transformations of themes.