Music Processing Algorithms

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Music Processing Algorithms

• David Meredith
• Department of Media TechnologyAalborg University

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Recent projects

• Musical pattern matching and discovery
• Finding occurrences of a query pattern in a work
• Finding works that are similar to a query work
• Discovering themes in a work
• Pitch spelling
• Predicting the pitch names (e.g., C4, B_at_3) of
  notes in a piano-roll representation (e.g.,
  MIDI)
• Essential for transcription from MIDI (or audio)
  to notation

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Algorithms for pattern matching and pattern
discovery in music
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Uses of musical pattern discovery algorithms

• In content-based music retrieval
• Creating an index of memorable patterns to enable
  faster retrieval
• For music analysts, performers and listeners
• A motivic/thematic analysis can assist
  understanding and appreciation
• In transcription
• Helps with inferring beat and metrical structure
• similar patterns have similar metrical structure
• Helps with inferring grouping and phrasing
• parallellism (Lerdahl and Jackendoff, 1983)
  most important factor in grouping
• In composition and improvisation
• Cure composers 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
• Use analysed thematic structure as a template for
  a new work

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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

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Most musical repetitions are neither perceived
nor intended
Rachmaninoff, Prelude in C sharp minor, Op.3,
No.2, bars 1-6
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Interesting musical repetitions are structurally
diverse

• Want to discover all and only interesting
  repeated patterns
• i.e., themes and motives
• 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., we can transpose it, embellish it, change
  tempo harmony, accompaniment, instrumentation,
  etc.

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Example of repeated motive
Barber, Sonata for Piano, Op.26, 1st mvt, bars 1-4
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Example of thematic transformation
J.S.Bach, Contrapunctus VI from Die Kunst der
Fuge, bars 1-5
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String-based algorithms for discovering musical
patterns

• Most previous approaches assume music represented
  as strings
• each string represents a voice or part
• each symbol 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)

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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

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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 dont use all possible representations then
  may not find all interesting patterns

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Using multidimensional point sets to represent
music (1)
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Using multidimensional point sets to represent
music (2)
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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(kn2 log n) time, O(kn2) space where k
is no. of dimensions n is no. of points O(kn2)
average time with hashing
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SIATEC - Discovering all occurrences of all MTPs
Translational Equivalence Class (TEC) is set of
all translationally invariant occurrences of a
pattern
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Absolute running times of SIA and SIATEC

• SIA and SIATEC implemented in C
• run on a 500MHz Sparc on 52 datasets
• 6n3456, 2k5
• lt 2 mins for SIA to process piece with 3500 notes
• 13 mins for SIATEC to process piece with 2000
  notes

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Need for heuristics to isolate interesting MTPs

• 2n patterns in a dataset of size n
• SIA generates lt n2/2 patterns
• gt 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 Rachmaninoffs
  Prelude in C minor
• probably about 100 are interesting
• gt Need heuristics for isolating interesting
  patterns in output of SIA and SIATEC

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Heuristics for isolating musical themes and
motives
Cov6 CR6/5 Cov9 CR9/5 Comp 1/3 Comp 2/5 Comp 2/3
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COSIATEC - Data compression using SIATEC
Start
Dataset
SIATEC
List of ltPattern, Translator_setgt pairs
Print out best pattern, P, and its translators
Remove occurrences of P from dataset
Is dataset empty?
No
Yes
End
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Using COSIATEC for finding themes and motives in
music
First iteration
Second iteration
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SIAM - Pattern matching using SIA
Query pattern
Dataset

• k dimensions
• n points in dataset
• m points in query
• O(knm log(nm)) time
• O(knm) space
• O(knm) average time with hashing

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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
• However, restricted to 2-dimensional
  representations (unlike SIA-family)
• Improved complexity to
• O(mn log m n log n m log m) running time
  (without hashing)
• O(m) working space
• Implemented as algorithm P2 on C-BRAHMS demo web
  site
• lthttp//www.cs.helsinki.fi/group/cbrahms/demoengin
  e/gt

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Improving SIAM - MSM(Clifford et al., 2006)

• Finding size of maximal match is 3SUM hard (i.e.,
  O(n2) )
• 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)

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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

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Evaluating MSMRunning time

• Run on prefixes of various sizes of first
  movement of Beethovens 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)

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Pitch spelling algorithms
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A pitch spelling algorithmtakes this...
Chromatic pitch
Time
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...and computes this
Diatonic pitch
Time
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Why are pitch spelling algorithms useful?

• In transcription, for generating a correctly
  notated score from a MIDI or audio file
• In content-based music retrieval
• For representing better the perceived tonal
  relationships between notes
• Allows us to find occurrences that sound like the
  query but contain different chromatic intervals
• For better understanding the cognitive processes
  that underlie the perception of tonal music

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Why is the same sound spelt differently in
different contexts?
1
3
2
4
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Comparative analysis of pitch spelling algorithms

• Algorithms analysed, evaluated and (in some
  cases) improved
• Longuet-Higgins (1976, 1987, 1993)
• Cambouropoulos (1996,1998, 2001, 2003)
• Temperley (2001)
• Chew and Chen (2003, 2005)
• Meredith (2003, 2005, 2006)
• Test corpus
• 195972 notes, 216 movements, 8 baroque and
  classical composers
• almost exactly equal number of notes (24500) for
  each composer

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The PS13s1 algorithm
Initial pitch name class
Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F C G D A
2 9 4 11 6 1 8 3 10 5 0 7 2 9 4 11 6 1 8 3 10
1 T
T1
T 1
2 T
T 1
1 T
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The PS13s1 algorithm
Initial pitch name class
Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F C G D A
2 9 4 11 6 1 8 3 10 5 0 7 2 9 4 11 6 1 8 3 10
T1
T 1
T 1
T 1
T 1
T 2
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Evaluation criteria and performance metrics

• Evaluation criteria
• Spelling accuracy - how well an algorithm
  predicts the pitch names
• Style dependence - how much spelling accuracy
  depends on style
• Performance metrics
• Note error rate - proportion of notes in corpus
  spelt incorrectly
• Style dependence - standard deviation of note
  error rates over 8 composers
• Robustness to temporal deviations
• Best versions of algorithms also run on version
  of test corpus in which onsets and durations were
  randomly adjusted
• To evaluate how well algorithms would work on
  files generated directly from performances

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Results for algorithms that were most accurate
over clean corpus
Algorithm Clean corpus Clean corpus Noisy corpus Noisy corpus
Algorithm NER SD NER SD
PS13s1x 0.56 0.49 0.61 0.54
Temperley 0.70 1.13 3.32 3.91
Chew and Chen 0.85 0.35 0.99 0.55
Cambouropoulos 0.85 0.47 0.93 0.53
Longuet-Higgins 1.79 1.79 1.75 1.71
Fixed LOF Range (Eb-G) 4.38 1.47 4.38 1.47
xKpre 10, Kpost 42 Two-pass, half tempo
corpus, without enh. change (MH2PX2) New
optimized versions (CamOpt and CCOP01-06) Only
when music processed a voice at a time (LH1V)
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Some perceptual and cognitive implications

• PS13s1 performs best when it uses a substantial
  post-context containing 23-42 notes
• None of the other algorithms use a post-context
  larger than about 3 or 4 notes
• Suggests that whether or not a pitch class is
  perceived to be the tonic at a point depends to
  some extent on notes that immediately follow it
  in the music
• PS13s1 with only a relatively small local context
  including a post-context performed better than
  Chew and Chens algorithm which uses all the
  music preceding the note to be spelt
• Suggests that perceived tonic is much more
  dependent on local context than global context
• In agreement with a concatenationist view of
  music perception (Tillmann and Bigand, 2004
  Gurney, 1966 Levinson, 1997)
• Best context sizes for PS13s1 contained from 50
  to 58 notes
• With music at a natural tempo, this corresponds
  to an average duration of 5.03 5.81 seconds
• Corresponds well with estimates of the duration
  of the perceptual present
• Fraisse around 5 s Clarke 3-8 s
• Events within perceptual present are directly
  perceived
• Can therefore be particularly easily
  re-interpreted in the light of events that occur
  later in the perceptual present
• Therefore feasible that notes occurring up to 4
  seconds after the one to be spelt may influence
  its interpretation and therefore its spelling

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Future work
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Further development of SIA family of algorithms

• Compare SIA algorithms with methods developed in
  other more mature fields (e.g., computer vision,
  graph matching)
• Improve time complexity of SIA algorithms with
  techniques such as ones used in MSM
• Adapt algorithms for approximate matching and
  scaling (matching at different tempi)
• Adapt SIA and SIATEC for early pruning of
  uninteresting patterns

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Further work on PS13s1

• Incorporate PS13s1 into complete MIDI-to-notation
  transcription system
• Incorporate PS13s1 into Sibelius notation
  software
• Use PS13s1 for key-tracking and harmonic analysis
• Use PS13s1 for feature extraction on audio data

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References

• On pattern-matching and pattern-discovery
• Meredith, D., Lemström, K. and Wiggins, G. A.
  (2002) "Algorithms for discovering repeated
  patterns in multidimensional representations of
  polyphonic music". Journal of New Music Research,
  31(4), 321-345.http//taylorandfrancis.metapress.
  com/link.asp?idyql23xw0177lt4jd
• Meredith, D. (2006) "Point-set algorithms for
  pattern discovery and pattern matching in music".
  In Content-Based Retrieval, Dagstuhl Seminar
  Proceedings, 06171.http//drops.dagstuhl.de/opus/
  volltexte/2006/652
• On pitch-spelling algorithms
• Meredith, D. (2006) The ps13 Pitch Spelling
  Algorithm. Journal of New Music Research, 35(2),
  121-159.http//taylorandfrancis.metapress.com/lin
  k.asp?idq679l61r31m18460
• Meredith, D. (2007) Computing Pitch Names in
  Tonal Music A Comparative Analysis of Pitch
  Spelling Algorithms, DPhil dissertation,
  University of Oxford.http//www.titanmusic.com/pa
  pers/public/meredith-dphil-final.pdf

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The end

• Thanks!