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Applications Statistical Graphical Models in Music Informatics
Yushen HanFeb 10 2011
I548 Presentation
Statistical Graphical Models• graph-based representation for a probabilistic
distribution in high-dimensional space while specifying conditional independence structure– directed acyclic graph(DAG) - Bayesian Network
– undirected graph(UG) - Markov Random Field
– Mixed graph
Markov Condition on a Bayesian Network
Markov Condition: XA and XB are conditionally independent given XS whenever S separates A and B
From www.eecs.berkeley.edu/ewainwrig/
probabilistic distribution
High-dimensional space
Conditional independence
Musical Application: Bayesian Identification of Chord from Audio
tuning
octave rootChroma (pitch-class)
modekey
Peak frequencies and corresponding amplitude
Attributes
Observation:
factorization of the joint by Bayesian Inference
Application I: Score Following/Alignment
Music score (given)
Performance audio of this piece (given)
Establishing a correspondence between the two above
Score Following:Online (real time)
Score Alignment:Offline (no need to be in real time)
Application I: Score Following
Best “guess” of the current location given what the computer had heard SINCE THE BEGINNING UP TO THE MOMENT
Score Following• Difficulties
– Tempo rubato (expressive and rhythmic freedom)– Pitch / amplitude vibrato ( )– Polyphony music– Noise– (occasional) wrong notes etc.– Realtime computational requirement
• Solutions– Assuming tempo change is smooth (mostly desirable)– Robust probabilistic data model on normalized semigram– Training to learn a prior (e.g. note length distribution)– Optimized particle filtering for 2-D State-space model
Proposed Solution with 2-D State-space Model (Bayesian Network)
• Assuming smooth tempo change for tempo rubato• Two-dimensional state-space model
• Proposing a unit of tempi:• S(t) - Musical time elapse per audio frame at frame t• Interpretation: during one audio frame of fixed length
(roughly 64ms, 512 samples at 8000Hz sampling rate), how much musical time (in terms of 1/384 notes) is elapsing
• In another word, how much of the score the performer covers every 64ms (not precisely the conventional tempo)
2-D State-space Model
State variables in one audio frame (notice conditional independence)
Current “speed” at this frame
0.64ms 0.64ms
Accumulative “speed” up to this frame –location
Physical Analogy – Integration of the Speed
with stochastic components
1.0 1.1 1.05 0.95
1.0 2.12 3.19 4.11
speed
location
Observation error also involved
Speed fluctuation
Physical Analogy - to make a tractable problem
1.0 1.1 1.05 0.95
1.0 2.12 3.19 4.11
speed
location
Observation error also involved
Speed fluctuation
Assuming discrete state transition with {-0.1, -0.05, 0, 0.05, 0.1} with prob distribution {0.1, 0.2, 0.4 ,0.2, 0.1} for smoothness.
Assuming Gaussian noise in observation
Score Following – Data Model
• Data model using semigram• Regarding discretized observation as a
histogram
Chord template (pre-learned)
Application II:Graph Model to Estimate Expert Pianists’ Perceptual Present
- with the help of audio-score alignment technique
Background
• Curtain eras of classical music – no improvisation, no wrong notes etc.
• For a certain piece of music, performance varies in tempo, dynamic, articulation, vibrato etc. , depending on the interpretation of the performer
• (This research) focuses on the tempo change of piano music
Chopin Mazurka Op. 30, No.2
http://www.youtube.com/watch?v=PjYV7lJezvchttp://www.youtube.com/watch?v=vGAQONeLnXkhttp://www.youtube.com/watch?v=qJmaz1OEGTU
RubinsteinHorowitzMichelangeli
Motivation
• Musical Perceptual Present– Recent studies in diverse fields of inquiry,
including music philosophy and psychology, lend converging evidence that musical attention of both performers and listeners is primarily focused successively small “chunks” of material (hypothetically 2–10 seconds in the past) rather than larger formal relationships.
Motivation
• Instead of individual style, we are in search of a “common interpretation” shared among a collection of expert pianists
• Focus purely on tempo change per beat (since the attack of piano note is easy to capture).
Data
• Human corrected accumulative time per beat which is equivalent to IBI Inter Beat Interval• N = 32 performances ( include different
performances of the same pianist )• For the existence of the MLE, we proceed a
small chunk of data at a time I = { 7, 8, 9, 10 }
Data cont. - preprocessing• # !!!performance-id: pid9062-19• # !!!title: Mazurka in B minor, Op. 30, No.
2• # !!!trials: 1• # !!!date: 2007/02/15/• # !!!reverse-conductor: Craig Stuart Sapp• # !!!performer: Idil Biret• # !!!performance-date: 1990• # !!!label: Naxos 8.550359• # !!!label-title: Chopin: Mazurkas (Complete)• # !!!offset: 0• 0.578 0:3• 1.398 1:1• 1.708 1:2• 2.228 1:3• 2.668 2:1• 3.088 2:2• 3.748 2:3• 4.336 3:1• 4.588 3:2• 4.998 3:3• 5.498 4:1• 5.828 4:2• 6.428 4:3• 7.028 5:1• 7.355 5:2• 7.968 5:3• 8.428 6:1• 8.838 6:2• 9.418 6:3• 9.878 7:1• 10.158 7:2• 10.678 7:3
• 0.720 • 0.855 • 0.745 • 0.800 • 0.490 • 0.610 • 0.530 • 0.540 • 0.550 • 0.540 • 0.530 • 0.570• …
Original Data: Accumulative time per beat
InterBeatInterval(IBI)
IBI differencebetween beats
•0.135 •-0.110•0.055•-0.310•0.120•-0.080•0.010•0.010•-0.010• …
Data cont. - preprocessing• “Normalization”– Since the overall duration of each performance
varies significantly E.g. Mazurka Op. 30. No. 2
We “stretch” the overall duration of each performance to line up with the median of all performances - can be problematic
Sec.
Performance index
Model
• For each “trunk” of timing data X of I dimensions (beats) across N performances:
• N performances are considered i.i.d. repetitions• we assume: X ~ Ν( 0, Σ )• where the difference in IBI equals to 0 suggests that
the tempo is nearly constant “on average” (of course, but could be problematic)
• We study the structure of I by I covariance matrix Σ• Can obtain an estimate of
Graph Models
• A toy example for I = 4 case -> 3 hypotheses– H: Fully saturated model (I-1=3 order Markov chain)
– H0: A smaller model (I-2=2 order Markov chain)
– H00: The smallest model (I-3=1 order Markov chain)
diff. IBI
Graph Models cont.
• Conditional independence in the graph suggests different structures in the covariance matrix
?
? ?
To apply reconstruction algorithm
Graph Models cont. - Testing
• Testing each pair of hypotheses -2Log(Q) ~
• Accepting the result only when every single pair of hypotheses of the smallest difference between the alternative and the null hypotheses are not rejected (as small step as possible)
• Apply an appropriate degree of freedom ( = difference in number of edges between 2 graphs )
Results – Interpretation
• “smoothed” results by using a sliding window of different lengths
• A “voting” mechanism
• Room to interpret …