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which implies that a substring will be aligned to
another substring (or a pattern) only if the total
dissimilarity does not exceed a certain threshold.
For instance, in Figure 2, the alignment of the
beginning of the theme exposition (stave 1) with
the reduced theme (circled notes) requires the
deletion of one note per retained note. But the
alignment of the beginning of the first variation
(staves 2 and 3) with the same reduced theme
requires the deletion of
seven
notes per retained
note. If the dissimilarity threshold is lower than
such high value, the second occurrence will not
be detected.
distance-Based matching
In some approaches of automated motivic analysis,
the instances are compared along one or several
given musical dimensions, and a numerical simi-
larity (or dissimilarity) distance is computed. The
matching is inferred when the similarity distance
is lower than a prespecified threshold (or if the
dissimilarity distance is higher that a pre-specified
threshold). For instance, concerning the melodic
dimension, candidates are compared along the
chromatic pitch interval
dimension, which con-
sists of the intervals in semitones between the
successive notes of the candidates. The taking
into consideration of intervals, instead of exact
pitch values, enables the detection of transposed
repetitions as well. Chromatic pitch-interval se-
quences are
δ-approximately repeated
when each
corresponding pair of chromatic interval values
are not more distant than δ semitones (Cambou-
ropoulos, Crochemore, Iliopoulos, Mouchard, &
Pinzon, 2002; Cope, 1996). Chromatic interval
sequences are
γ-approximately repeated
if the
summation of differences between all successive
couples of chromatic interval values is not higher
than γ semitones (Cambouropoulos et al., 2002).
Concerning δ-approximation, a threshold of δ=1
semitone is commonly used in order to identify a
melody played in both major and minor keys (Cope,
1996; Cambouropoulos et al., 2002). However,
this threshold also tolerates other transformations
that are not directly related to the major/minor
configuration (for instance, the dilatation of a
major third into a perfect fourth). Users may be
offered the option to fix the dissimilarity thresh-
old by themselves (Cope, 1996; Rolland, 1999).
However, no study has shown how it could be
possible in this way to find a good threshold
value for each analysis. Numerical distance may
also be used in order to detect rhythmic patterns.
Here also, different patterns can be identified if
the successive values forming their description
are sufficiently similar, with respect to a given
similarity threshold.
syntagmatic-graph approach
In order to solve this problem, we hypothesize
that the extraction of the reduced theme from
the highly ornamented phrase may be explained
by the listeners' ability to perceive not only one
single monodic line, as previously shown in Figure
2, but also a more complex network of intercon-
nections between notes that are not immediately
successive, as shown in Figure 3. In other word,
from the “
syntagmatic chain
” (Saussure, 1916)
that form the monodic string may be constructed
a
syntagmatic graph
of relations of succession.
The direct application of the pattern extraction
algorithm to this syntagmatic graph would hence
enable the detection of ornamented repetitions.
matchIng strategIes
The discovery of a repeated pattern results from
the
matching
between different candidate subsets
or substrings of the score. If the matching suc-
ceeds, the candidates form occurrences of a new
pattern. The recognition of new occurrences of
the pattern results from a matching between new
candidates and the pattern description. This sec-
tion discusses the different matching strategies
considered in the literature.
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