A Pattern Extraction Algorithm for Abstract Melodic Representations that Allow Partial Overlapping of Intervallic Categories

This paper proposes an efficient pattern extraction algorithm that can be applied on melodic sequences that are represented as strings of abstract intervallic symbols; the melodic representation introduces special “don’t care” symbols for intervals that may belong to two partially overlapping intervallic categories. As a special case the well established “step-leap” representation is examined. In the step-leap representation, each melodic diatonic interval is classified as a step (±s), a leap (±l) or a unison (u). Binary don’t care symbols are introduced to represent the possible overlapping between the various abstract categories e.g. ∗ = s, ∗ = l and # = −s, # = −l. For such a sequence, we are interested in finding maximal repeating pairs and repetitions with a hole (two matching subsequences separated with an intervening non-matching symbol). We propose an O(n + d(n − d) + z)-time algorithm for computing all such repetitions in a given sequence x = x[1..n] with d binary don’t care symbols, where z is the output size.