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Figure 4. Motivic analysis of the beginning of Beethoven's Fifth Symphony (piano reduction), using
the numerical distance paradigm. Under the score are indicated the chromatic pitch interval values
in semitones between successive notes. Chains represents pattern occurrences, where nodes represent
successive notes and transition parameters describe the chromatic pitch interval values. The first line
of chains below each stave shows patterns featuring exact descriptions. Following lines are related to
patterns with approximate descriptions.
For instance, in Figure 4, the analysis of the
beginning of Beethoven's Fifth Symphony , along
the chromatic dimension, may show different
pattern classes depending on the tuning of the
similarity threshold. For a threshold of zero
semitone, corresponding to a search for exact
repetition, we obtain several possible melodic
patterns depending on the values of the last pitch
interval of the motive: either major third (0,0,-4),
minor third (0,0,-3), or minor second (0,0,-1). The
other possible interval values such as (0,0,-5), since
not repeated several times in the example, do not
form patterns. For a threshold of one semitone,
minor and major third can be combined into one
pattern (0,0,-3±1). For a threshold of two semi-
tones, minor second intervals can be included
in the pattern notated (0,0,-3±2). For a threshold
of four semitones, the last motive, containing a
fifth-interval can be included in pattern (0,0,-3±4).
Pattern (0,0±1,0±2) is problematical since its class
and description are not very intuitive. Another
obscure example is pattern (0,-1±1,-1±2): due to
its description, its class includes the motive at bar
14 (0,-2,-2), the minor second motive (0,0-1), the
minor due to its description, the motive includes
the minor third motive (0,0,-3), but excludes the
major third motive (0,0,-4).
According to the numerical distance paradigm,
a pattern class can accept occurrences that are
not completely identical one with each other.
The description of the pattern is therefore not
explicitly given and requires further processing.
For example, in Figure 4, the pattern (0,0,-3±1)
could be described as (0,0,-4±1) as well for the
particular excerpts, but the choice of the descrip-
tion may lead to a significant divergence of results
when analyzing the whole piece. One method of
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