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the ratio of the number of notes in the pattern to
the total number of notes in the piece that occur
within the region spanned by the pattern. The
“region spanned by a pattern” is defined in a
number of different ways: either as the smallest
“time segment” that contains the pattern, or as
the “bounding-box” or the “convex hull” spanned
by the pattern in the bi-dimensional representa-
tion (time/pitch) of the score. See Figure 1 for
an illustration of these geometrical objects. The
underlying assumption is that typically at least
one occurrence of the pattern should have a high
compactness value, even if the other occurrences
are highly embellished.
For instance, in Mozart's Theme and Varia-
tions on
Twinkle, Twinkle, Little Star
(K. 300e),
the basic theme common to all the variations does
not contain the pitch repetition and is therefore
reduced to only one note per bar (circled in the
figure). During the exposition of the theme (first
occurrence), its compactness with respect to the
corresponding “time segment” (between crochets)
is very low: 8 notes out of 31, or 26%. The results
are better for “bounding box” (gray rectangle, 8
notes out of 16, or 50%) and for “convex hull”
(black triangle, 8 notes out of 14, or 57%) but
not particularly high. Hence the main theme is
likely to be rejected by the heuristics, unless the
minimum compactness threshold is fixed to a low
value, below 57%. But a low threshold value may
dramatically increase the proportion of false posi-
tives, without necessarily insuring a significant
reduction of false negatives.
(Saussure, 1916). Yet, in general, music takes the
form of a complex polyphonic flux that cannot
easily be reduced to a single string of events. For
instance, the Mozart example, as shown in Figure
2, features two strings, one for each voice
2
. The
adaptation of this approach to polyphony will
require extensive subsequent works.
Following the string paradigm, each voice is
considered separately, and motives are formalized
as
substrings
of the monodic string that contain
all
the notes of the string between two given
limits. In its very simple version, this approach
can only formalize exact repetition of patterns,
transpositions, or similar simple transformations.
Ornamented motives on the contrary,—like in
the Mozart example, where
secondary notes
are
added in the temporal neighbourhood of the
pri-
mary notes
(i.e., the notes forming the reduced
theme itself) -, are beyond the compass of this
simple framework. Primary notes may be retrieved
through an automated filtering of secondary notes,
for instance, by focusing on metrically strong posi-
tions (Conklin & Anagnostopoulou, 2001; Eerola,
Järvinen, Louhivuori, & Toiviainen, 2001). This
heuristic does not, however, work correctly for
appoggiaturas, where secondary notes, instead
of primary notes, are placed on downbeats. For
instance, the pattern repetition of the theme could
not be detected in this way in the first variation of
the
Twinkle
theme (Figure 2, staves 2 and 3).
Another solution (Mongeau & Sankoff, 1990;
Rolland, 1999) consists of tolerating gaps, using
alignment strategies based on dynamic program-
ming. The idea is that a substring may be com-
pared to another substring through a successive
sequence of transformations, such as insertion of
notes from the second substring, suppression of
notes in the initial first substring, modification of
one note, and so forth. With each transformation
is associated a cost, whose summation defines
a global dissimilarity distance between the two
motives, from which identification decision can
be inferred. As explained in the next section, this
decision is based on a dissimilarity threshold,
string-Based formalization
The alternate approach is focused on monodic
sequences, which are formalized as
strings
of
notes. This follows the idea that music sequences
are perceived as a succession of events closely
connected one after the other. This hypothesis
can be associated with the phenomenological
idea of primary retention (Husserl, 1905) and
the linguistic concept of syntagmatic relation
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