Point-set algorithms for pattern discovery and pattern matching in music David Meredith Goldsmiths College University of London

Uses of musical pattern discovery algorithms Indexing Store themes, motives and other memorable patterns in index to enable sub-linear retrieval times Transcription and music analysis Beat tracking and metrical structure analysis - similar patterns have similar metrical structure Grouping and phrasing - “parallellism” (Lerdahl and Jackendoff, 1983) most important factor in grouping Composer’s assistant, automatic improvisation Cure composer’s block by suggesting new material based on patterns discovered in music already written Automatically create new music that develops themes discovered in music already played

Importance of repeated patterns in music analysis and cognition Schenker (1954. p.5): repetition “is the basis of music as an art” Bent and Drabkin (1987, p.5): “the central act” in all forms of music analysis is “the test for identity” Lerdahl and Jackendoff (1983, p.52): “the importance of parallelism [i.e., repetition] in musical structure cannot be overestimated. The more parallelism one can detect, the more internally coherent an analysis becomes, and the less independent information must be processed and retained in hearing or remembering a piece”

Most musical repetitions are neither perceived nor intended

Interesting musical repetitions are structurally diverse Want to discover all and only interesting repeated patterns Class of interesting repeated patterns is structurally diverse because patterns vary widely in structural characteristics many ways of transforming a musical pattern to give another pattern that is perceived to be a version of it e.g., truncated, augmented, diminished, inverted, embellished and even reversed

Example of repeated motive

Example of thematic transformation

String-based algorithms for discovering musical patterns Most previous approaches assume music represented as strings each string represents a voice or part each character represents a note or an interval between two consecutive notes in a voice Similarity between two patterns measured in terms of edit distance calculated using dynamic programming see, e.g., Lemstrom (2000), Hsu et al. (1998), Rolland (1999)

Problems with the string-based approach - Edit distance B is an embellished version of A If both patterns represented as strings each symbol represents pitch of note then edit distance between A and B is 9 If allow pattern with 9 differences to count as a match, then get many spurious hits

Problems with string-based approach - Polyphony If searching polyphonic music and do not know voice to which each note belongs (e.g., MIDI format 0 file); or interested in patterns containing notes from 2 or more voices then combinatorial explosion in number of possible string representations if don’t use all possible representations then may not find all interesting patterns

Using multidimensional point sets to represent music (1)

Using multidimensional point sets to represent music (2)

SIA - Discovering all maximal translatable patterns (MTPs) Pattern is translatable by vector v in dataset if it can be translated by v to give another pattern in the dataset MTP for a vector v contains all points mapped by v onto other points in the dataset O(kn 2 log n) time, O(kn 2 ) space O(kn 2 ) average time with hashing (Lemstrom)

SIATEC - Discovering all occurrences of all MTPs Translational Equivalence Class (TEC) is set of all translationally invariant occurrences of a pattern

Absolute running times of SIA and SIATEC SIA and SIATEC implemented in C run on a 500MHz Sparc on 52 datasets (6≤n≤3456, 2≤k≤5) < 2 mins for SIA to process piece with 3500 notes 13 mins for SIATEC to process piece with 2000 notes

Need for heuristics to isolate interesting MTPs 2 n patterns in a dataset of size n SIA generates < n 2 /2 patterns => SIA generates small fraction of all patterns in a dataset Many interesting patterns derivable from patterns found by SIA BUT many of the patterns found by SIA are NOT interesting 70,000 patterns found by SIA in Rachmaninoff’s Prelude in C# minor probably about 100 are interesting => Need heuristics for isolating interesting patterns in output of SIA and SIATEC

Heuristics for isolating musical themes and motives Cov=6 CR=6/5 Cov=9 CR=9/5 Comp = 1/3Comp = 2/5Comp = 2/3

COSIATEC - Data compression using SIATEC Start Dataset SIATEC List of pairs Print out best pattern, P, and its translators Remove occurrences of P from dataset Is dataset empty? End No Yes

Using COSIATEC for finding themes and motives in music First iterationSecond iteration

SIAM - Pattern matching using SIA (Finding maximal matches) O(knm log(nm)) time O(knm) space O(knm) average time with hashing Query pattern Dataset

Improving SIAM - Ukkonen, Lemström & Mäkinen (2003) Use sweepline-like scanning of the dataset (Bentley and Ottmann, 1979) Generalized to approximate matching of sets of horizontal line-segments Improved running time to O(mn log m) (without hashing) and working space to O(m) Implemented as algorithm P2 on C-BRAHMS demo web site

Improving SIAM - MSM (Clifford et al., 2006) Finding size of maximal match is 3SUM hard (i.e., O(n 2 ) ) Reduce problem of multi-dimensional point-set matching to 1d binary wildcard matching Random projection to 1D Length reduction by universal hashing Binary wildcard matching using FFTs Find best match and check in O(m) time exactly how many points match at the location that can be inferred from this match Reduces time complexity to O(n log n)

Evaluating MSM: Precision-Recall Compared with OMRAS (Pickens et al., 2003) Test set of 2338 documents, 480 used as queries All score encodings in strict score time Queries had notes deleted, transposed and inserted

Evaluating MSM: Running time Run on prefixes of various sizes of first movement of Beethoven’s 3rd Symphony Each prefix matched against itself Compared with largest common subset algorithm of Ukkonen, Lemström and Mäkinen (2003) MSM nearly 2 orders of magnitude faster (log scale)

Future work Compare SIA algorithms with methods developed in other more mature fields (e.g., computer vision, graph matching) Improve time complexity of SIA and SIATEC by using randomization techniques Adapt algorithms for approximate matching and scaling (matching at different tempi) Adapt SIA and SIATEC for early pruning of uninteresting patterns