A Parse-based Framework for Coupled Rhythm Quantization and Score - - PowerPoint PPT Presentation
A Parse-based Framework for Coupled Rhythm Quantization and Score - - PowerPoint PPT Presentation
A Parse-based Framework for Coupled Rhythm Quantization and Score Structuring Francesco Foscarin Masahiko Sakai Florent Jacquemard Philippe Rigaux supported by: Automated Music Transcription symbolic generation MIDI (CAC) device rhythm
Automated Music Transcription piano roll unquantized onsets quantized pitches (MIDI) audio recording score (XML file) MIDI device symbolic generation (CAC) MIDI-to-score quantized piano roll
rhythm quantization score engraving sound processing
audio-to-MIDI
MIDI to score transcription with independent subtasks
- 3. Combination: Interface between the 2 subtasks?
- complex rhythm, deep nesting
- mixed tuplets
- rests, grace notes…
unquantized MIDI →
quantized MIDI
sequential data
→
sequential data quantized MIDI
→
XML score file sequential data
→
1/2 structured data
Markov models of note values [Raphael 2001], [Sagayama et al 2002], [Nakamura et al 2016]
- 1. Rhythm Quantization
sequential model of durations (HMM)
- 2. Score Engraving
delegated to functionality of a score editor (MIDI import)
MIDI to score transcription with coupled subtasks (our proposal) integrated approach to MIDI-to-score transcription coupling Rhythm Quantization & Score Production based on:
- a hierarchical model of notation (tree-structured)
similar to OpenMusic’s Rhythm Trees [Agon et al 2002] expressive, accurate for complex rhythms
- quantitative parsing techniques
for context-free grammars weighted in semirings efficient & modular (focus on rhythm transcription)
unquantized MIDI → tree representations
~
XML score file sequential data
→
1/2 structured data 1/2 structured data
Implementation, Results
https://gitlab.inria.fr/qparse/qparselib https://qparse.gitlabpages.inria.fr
- riginal score
transcription
- f MIDI
recording with qparse transcription: MIDI recording to XML/MEI
Beethoven, Trio for violin, cello and piano, op. 70 n.2 (2d mov)
Implementation, Results
- riginal score
transcription
- f MIDI
recording with Finale.
- ptions:
mixed rhythms, tuplets smallest note = 32nd The time signature and the tempo are given.
transcription: MIDI recording to MusicXML
Beethoven, Trio for violin, cello and piano, op. 70 n.2 (2d mov)
Hierarchical Structure of Music Notation
bar beat subbeat 1 3 2 4 5
1.1 1.2 1.3 2.1 2.2 2.3 3.1 3.2 3.3 4.1 4.2 4.3 5.1 5.2 5.3
1.1.1 1.1.2 2.1.1 2.1.2 3.1.1 3.1.2 4.1.2 3.3.1 3.3.2 5.1.1 5.1.2 5.2.1 5.2.2 4.1.1 4.2.1 4.2.1
Polonaise in D minor from Notebook for Anna Magdalena Bach BWV Anh II 128 The notation gives clues (to player) of the metric structure Tree representation of proportional rhythmic notation
durations: 1 2 1 4 1 4 1 16 1 16 3 4 1 16 1 16 3 4 0 1 2 1 4 1 4 2 1 2 1 4 1 4 1 16 1 16 3 4 1 2 1 4 1 4 1 2 1 4 1 4 1 2 1 4 1 4 1 16 1 16 3 4 1 2 1 6 1 6 1 6 1 2 1 2 01
Languages of rhythmic notation
q − → q0 q − → q0 q q0 − → q1 q2 q0 − → q1 q2 q2 q0 − → − q0 − →
- q0
− →
- 1
q0 − →
- 2
q1 − → − q1 − →
- q1
− →
- 1
q1 − →
- 2
q2 − → q3 q3 q2 − →
- q3
− → q4 q4 q3 − → − q3 − →
- q4
− →
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→
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<latexit sha1_base64="QWox5jdMD/RXbSLJ+yOGaGxA3Vk=">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</latexit>- 3
- Context-Free Grammar (CFG): finite set of context-free production rules.
example: (very simple) CFG for 1
4 bar sequences.
non terminal symbols: q (bar seq.), q0 (1 bar = 1 beat), q1, q2,. . . terminal symbols: − (continuation),
- (1 note),
- 1 (1 grace-note + 1 note),
- 2 (2 grace-notes + 1 note),. . .
Languages of rhythmic notation
q − → q0 q − → q0 q q0 − → q1 q2 q0 − → q1 q2 q2 q0 − → − q0 − →
- q0
− →
- 1
q0 − →
- 2
q1 − → − q1 − →
- q1
− →
- 1
q1 − →
- 2
q2 − → q3 q3 q2 − →
- q3
− → q4 q4 q3 − → − q3 − →
- q4
− →
- <latexit sha1_base64="ouQkr9DkxBYIURieBsyPiG5g4Q=">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</latexit>
Context-Free Grammar (CFG): finite set of context-free production rules. example: (very simple) CFG for 1
4 bar sequences.
non terminal symbols: q (bar seq.), q0 (1 bar = 1 beat), q1, q2,. . . terminal symbols: − (continuation),
- (1 note),
- 1 (1 grace-note + 1 note),
- 2 (2 grace-notes + 1 note),. . .
Languages of rhythmic notation
q − → q0 q − → q0 q q0 − → q1 q2 q0 − → q1 q2 q2 q0 − → − q0 − →
- q0
− →
- 1
q0 − →
- 2
q1 − → − q1 − →
- q1
− →
- 1
q1 − →
- 2
q2 − → q3 q3 q2 − →
- q3
− → q4 q4 q3 − → − q3 − →
- q4
− →
- <latexit sha1_base64="ouQkr9DkxBYIURieBsyPiG5g4Q=">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</latexit>
Context-Free Grammar (CFG): finite set of context-free production rules. example: (very simple) CFG for 1
4 bar sequences.
non terminal symbols: q (bar seq.), q0 (1 bar = 1 beat), q1, q2,. . . terminal symbols: − (continuation),
- (1 note),
- 1 (1 grace-note + 1 note),
- 2 (2 grace-notes + 1 note),. . .
Parse trees
q0→q1 q2 q1→• q2→•
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<latexit sha1_base64="t1Qw1+8vJ4yWIqlkYj3nlydo=">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</latexit>q0→q1 q2 q2 q1→•1 q2→• q2→•
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- 3
- parse
tree serialization corresponding notation
1 2 1
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<latexit sha1_base64="ef32ucSD9vRDmWtn1C3sEtsjfyg=">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</latexit>a parse tree is a tree labeled by grammar’s production rules, with tiling of non-terminals.
More parse trees
q0→q1 q2 q1→• q2→q3 q3 q3→− q3→•
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- a word (sequence of terminal symbols) is parsed by a CFG
if it is the yield of a parse tree Properties:
- languages of (parsed) words are not closed under intersection.
- languages of parse trees = regular tree languages
are closed under intersection. tree languages: → capture regularity of rhythm → grammar composition (Cartesian product) combination of grammar representation of different aspects
Objectives (informal definition) → find good compromise between precision and complexity of notation (multicriteria optimisation) → quantitative parsing input
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0.11
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0.29
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0.55
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1 8 1 4 3 8 1 2 9 16 5 8 3 4 7 8 1
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- a grammar G describing rhythmic notation preferences
- an input sequence of events σ
return:
- a parse tree t of G that fits well with σ
questions:
- How well? How to avoid complex parse trees (overfitting).
Semirings
A semiring S = hS, , 0, ⌦, 1i is a structure with
- a domain S = dom(S)
- two associative binary operators and ⌦
with neutral elements 0 and 1; and such that
- is commutative
- ⌦ distributes over : 8x, y, z 2 S, x ⌦ (y z) = (x ⌦ y) (x ⌦ z),
- 0 is absorbing for ⌦: 8x 2 S, 0 ⌦ x = x ⌦ 0 = 0
Intuitively, is for selection of a best value ⌦ is for composition of values
<latexit sha1_base64="j3RaT0udc4pnyupmGFtZ7/sGng=">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</latexit>Semirings
semiring domain ⊕ ⊗ 1 natural ordering Boolean {0, 1} ∨ ∧ 1 1 ≤S 0 Viterbi [0, 1] ⊂ R+ max . 1 x ≤S y iff x ≥ y min-plus R+ ∪ {+∞} min +∞ + x ≤S y iff x ≤ y max-plus R ∪ {−∞} max −∞ + x ≤S y iff x ≥ y These semirings are commutative: ⊗ is commutative idempotent: ∀x, x ⊕ x = x have an induced total natural ordering ≤S defined by: ∀x, y, x ≤S y iff x ⊕ y = x monotonic wrt ≤S: ∀x, y, z, x ≤S y implies x ⊕ z ≤S y ⊕ z x ⊗ z ≤S y ⊗ z
<latexit sha1_base64="2XeTv7QXWrhvkjg1CrTFal4KvNA=">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</latexit>Weighted CF Grammars model with the Viterbi Semiring = PCFG (Probabilistic Context-Free Grammars) every production rule is attached a weight value in a semiring here, min-plus semiring: weight values are penalties (costs) for 1/4 bars:
q − → q0 q q − → q0 q0 q0 − →
6
q1 q2 q0 − − →
12
q1 q2 q2 q0 − − →
15
− q0 − →
1
- q0
− − →
79
- 1
q0 − − − →
102
- 2
q1 − →
2
− q1 − →
1
- q1
− − →
25
- 1
q1 − − →
64
- 2
q2 − − →
10
q3 q3 q2 − →
2
- q3
− − →
11
q4 q4 q3 − →
4
− q3 − →
1
- q4
− →
1
- <latexit sha1_base64="4TYAS1s7siwB3RHYOJ5PNjKVKBU=">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</latexit>
s
3
- 3
- parse
tree serialization corresponding notation
1 2 1
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q0−
→
6 q1 q2
q1−
→
1 •
q2−
→
2 •
<latexit sha1_base64="hOt4O7G97eCPaADhfj0ymvKmqEc=">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</latexit>9
q0−
− →
12 q1 q2 q2
q1−
→
1 • q2−
→
2 • q2−
→
2 •
<latexit sha1_base64="LeNFoahzxI2yVRAvghlq+whlyw=">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</latexit>17
q0−
− →
12 q1 q2 q2
q1−
− →
25 •1
q2−
→
2 • q2−
→
2 •
<latexit sha1_base64="aBTP9Fjnoj7txDz0q1ks4kzbYpg=">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</latexit>41 = product with ⊗ of weights of rules involved
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1 2 3 4 1
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→
6 q1 q2
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1 • q2−
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10 q3 q3
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4 −q3−
→
1 •
<latexit sha1_base64="D6MHwB8j9KF1Aob4YVjNm9HiLo4=">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</latexit>q0−
→
6 q1 q2
q1−
→
1 • q2−
− →
10 q3 q3
q3−
→
4 − q3−
− →
11 q4 q4
q4−
→
1 • q4−
→
1 •
<latexit sha1_base64="2uaJiFCNRhFSIU1H2nmW9DbEZVQ=">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</latexit>q0−
→
6 q1 q2
q1−
→
1 • q2−
− →
10 q3 q3
q3−
→
4 − q3−
− →
11 q4 q4
q4−
→
2 −q4−
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1 •
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35 35 notational complexity
Time & Tempo
time units: Real Time Unit (RTU), performance time, in seconds Musical Time Unit (MTU), score time, in bars A tempo curve is a monotonically increasing function θ : Q+ ! Q+ θ : RTU value 7! MTU value A tempo model M is a set of tempo curves, with restrictions (piecewise linear)
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- !
serialization
event sequence ktk timestamped in MTU
<latexit sha1_base64="wGxEVnIE3U/PTQuAHA4sjw/tvhs=">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</latexit>given:
- a weighted CFG G over a semiring S,
describing rhythmic notation preferences
- an input sequence of events σ
- a tempo model M
return:
- a tempo curve θ 2 M
- a parse tree t of G minimizing weightG(t) ⌦ fit
- σ, θ−1(ktk)
- wrt S
Optimization Problem We want t that optimizes a combination (with ⌦) of the weight of t wrt G weightG(t) the fitness of t to the input σ fit
- σ, θ−1(ktk)
- with a Viterbi semiring, ⌦ is a probability product to maximize.
- with a min-plus semiring ⌦ is a sum to minimize.
This is similar to scalarization by weighted sum in multi-criteria optimiza- tion.
<latexit sha1_base64="tzlb1FqwULUS/CvdYZqG8qAHs48=">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</latexit>1-best parsing goal: find a parse tree t of a weighted CFG over S with minimal weight wrt ≤S.
<latexit sha1_base64="/Dx8U0rfe8nDrzT8q6iaArWAtE=">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</latexit>Hypotheses for the semiring S ⊗ commutative ⊕ idempotent S monotonic wrt ≤S ≤S is total: ∀x, y, x ⊕ y = x or x ⊕ y = y
<latexit sha1_base64="lkTLo2PETGiyEpzdhJsptBJlIHg=">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</latexit>... q0 − →
6
q1 q2 q0 − − →
12
q1 q2 q2 q0 − − →
15
− q0 − →
7
- q0
− − →
79
- 1
q0 − − − →
102
- 2
...
<latexit sha1_base64="uMSAlh+PtT2uYBRTSeuH+mwmc8=">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</latexit>best(q0) = min≤S 6 ⊗ best(q1) ⊗ best(q2), 12 ⊗ best(q1) ⊗ best1(q2) ⊗ best1(q2), 15, 7, 79, 102
<latexit sha1_base64="HrJe/s8huled1apk9jwtZ8SQHqc=">AnRXic5Vpbc9u4FZa3t616S7Zv7QuntDKbrqOR5DhxZrvpZnOrJ3HWtZ1NZkyvhdIQg0SNAja1nL5W/ra/pL+hv6IvnX62h4AlEQCoO06aQzVWIHwPnOwcEHnHNAKkFKcMYHg7+ufCd737v+z/48IfdH/34Jz/92bXrH32V0ZyF6GVICWvAz9DBCfoJcecoNcpQ34cEPQqOHko5K/OEMswTQ75PEXHsT9N8ASHPoeh8fW1j7yjrhf7fIZ5EaCMlx+fjgc3nc+cxWCMk3JceASdjr0DFGOGk2npdGFgwr0CGgGa4qTwGfPnZUHK7h3HoxzHKHMw8ObS5kuGt3c8LzucPT2yuOhVG9RrKTS+NaGcxf+3twhoNR10NJVLnd9RiezrgHjeNud3zNHfQH8uOYjWHVcDvVZ298vfcL6JhHqOEh8TPsqPhIOXHYJzjkCwmco9cMTf4qOoJn4OdxIXevdHowEjkTyuAn4Y4c7XZ7spc5vkN8J4AdRQy8RAk6D2kc+0nkZMITfyc8CKdnZWacOLHmMwXgBW27NadKfw4E2SJ/WyOZvM40EaDgJKo7PY0pHSzscQi47HP5kwHi5n84EIb3U0OYC5KmiZmPotpMgd2nAzHKUFOnGc4VKQAOz1JGOwePc+6+iScUgIuLVBAixP7IaN5EUGLdicbre5P3yfVzgJM05SkK1PZOcOJw6IoCcCDMUcjKHBtjEsMNOCN76IYcwE/u29/TJhnOIT7p9m69q0+TsnQ6SQkVrKvhqktwID3IvI5EgNRgG1AT+KMtFKaISyDUeM5ZwCb35TgcMS1IhoVcaPsQ/gcOrjxcBaE0ZzZMoM5QKzhCyDFe+LKbfSGmGRVoSGUYtrY5OfQ4UJ5lFlM38VEwg6N9/Gp/53DnxdN3Sb8H4ePtoygPhX/u0B25m+7toqdyDUOk8JiPMwTJAV14s4BeCNkZp2nhYZ/gaVK4rheykMGwRA5WyHVvNsHEy+A0pTzjc4IKz3GHnlNKwXq5UEyoMgUNnAXIulnJzgte1UCk4SCrmSxpXcGBlulGVZlF8X7iaUBfd2CaZgRxRDz/+XGRIF620IehMfl9LRk3zMKD25WO3MkTs8dkcFMh8GYyARW3Bv3RAMWlpyELdwS2u0tDtuPXq0z9AbwHjmsmN7fApEn+6P2Qv970c9109K3P73+0PQtS7ay2kro1UL4qhra3rkzm7fcb6/8eV4IqxRTcKMOZIKiQBwpW6g7LI3d0rIosQ+dQCJGTQd3JxIA3gaEFoe6We8e9624rUzZJ0fUgnwM9g8qltw7jnsXeHc7XWxYiUeauLRQownEY4d7zwa3IdfQ8djxIfdUL+FiqjeQly4W+IOJd8lgoSr4gj9Bwx930hBPABvEvpEAQprbYcW8L4UB4gwj021wYShfqkwyrSRp29EkaHohJlgcZEG1YL3iVjRX5KrNCmcUw+mVxItE2mlGPaHEN/i35H8F3wq6sejFIegipfLyFKENUxDXIz+a6a3hsLkYoq6Xc8wZ17V5lA7QBhgvcNseVA53KTemJEwx5Aa7wASKiO3ICyuG2M5YhXI17q3t7URkrj0bHxSp5y41deul96t2H3/fhz6fi/DWygSwZpkEetpiEHFDITHC52fWvf71umFbrfaOv0qwy3qjBhj1F29XWLl2UwdW0aeFDJSlKIJN96MZDfvwESHpn1EcOTMcRfAQRs+qtFmLp+h9BxSY/j8KMvZjWwnIrp6OIgPHIAYZ+HyOVBsL4ZN5OiCY17gPurDBax6IpRSNF1KUX+qS8ZX4qhvZKqi985oSt10dGN83BlHWJzKW2ufwJn8SJTj8cQHkmWMyRo8B5hHx6Po9ICL1U5gaJe014XMdDAPsgjTEe7O492yuLBaLfmfx0l5KMspAz2aHd0UHNTrOI37vA+bJKfTOH53BVlVzY1r1IfM7WDCik2bsMRW2fHPyirOPNJ8UA/EV/UhF/owoc14UNd+KgmfKQLH1fCICge67InK9kTXfa0ZvSpLtypCXdKndpnK7PdM1nNU0prL/bKby9mnhP191vnJQVcN+Irn3WCh3LdjYpWGKoHbZoHRrI57C9LeDnBni3BblrIF9YkMDiCwP4usXka7PyxCkIpzxZpid+QxutJjQxIifIKJ8sYM4Kfq6RSxfi1YIiPIMchI3QAhieAkRHR0gitsKIXs6RCa8FUZ1NVCUsxVCdPTWC1HvkHVdWls4VHYAYE+ETCW2dFSYsJG1qm7qaToVqGDodjKiUGPmvFZ1Z81OJ6lJnYAzyNfd26HNRip6l2OEPcUJODl4Uco5Sndt+EyMDjBNdOhOzpeUgrCWRA8PF4fl0VCk7ID54YlK7rKpx6LIBlUhSxplsI4CiSwANf+hCIzlG4W6AQmk+UdgPqBVHG62BzR0tOE5LaMNBIVxDZ0xDy+4MlQvbMIK9DsA2Titq59AQ6upw2XKWmBUgdHDUyCTesnKNmFqj6+n0quDyXsOANeQubDVfoWPUevEKYgVRX2PT1UytEZmJdMNL1UqwuYblklVJtdmV1q438e2pJwKd80mIPBMb0+SULy+0Ly7WF5dZCoUZjWVBabQtxi6J/cakiM21tFKZ26jMG1TmFirFWPsKqFVqdX7SqgrcSjntHWqpdSu1jbZUmowRbRDTxan3o5ChDQjqGmNadaY1ZqBsgRjdW+31j8pMhTQtJZmoGPIGZYFewmRfb20xDwv5ZVqVojm6smnsYAViNVBWpVafOHcrFUNzDeI0dVdeqCLaYJW0qHhbEK5T9LZJf/6SRPwiaN07G4+hu25JemK3ZETz9i4n7F/XrliSzZPITnV4xqV7bFgH5tPUNJPSNXfeOsJCe1k3Kiy+O8ZkJ0dH1U3TZ0w/tHCNSC3/V1cyoQVncv5UXim+rC4C2pPpksqeTowqguJ6IKk4oA5+i0pEXBc2vE5wkSx7h3N+K8JnhvIZREC0zsLisZQLoGY/etRw+wUYC/zLntTP8ZVl9Vzmb81nsHIqvR6Frd6sevlnsE8Ih6sRzcy5uLFfS0l+4OJ+YN0ZTLRTPBLYTp+GCnBDELb4wdTsimMyfkrmIdBjI+pvmAZO4B+Ge32rEksQsN3iOjYSHE3GinZ6zm6h3Ic5v9ZjUdG4Zt5qU5NqjAazacu/OSN6Sb0sLPZl65rLAI9uzjAGkTevztS7tdhZ/VqNcq30FeoLyObLhwqmDozGJKNY52nTYmKBs9k4zX2uV63itsXIEmizEuLkVLOxZbFRwWwWMqyTcdiQKGs+ijVeC/u2gwomM3CLMe6hW2LhQpms5CgXLu9FvcsFiqYzUKeNJOCN9ZfnO3b3jfBEX7MmHr3Or7mDvX/O2U2vhr1h5v90e9H7ufb1f+r+rDzy86vOh93hp27nc87v+vsdV52wrX52h/X/rT25xt/ufG3G3+/8Q8F/WCt0vl5p/G58c9/AVqpu+E=</latexit>etc for acyclic grammars. for cyclic ones: generalisation of Dijkstra algorithm to hypergraphs.
1-best parsing goal: find a parse tree t of a weighted CFG over S with minimal weight wrt ≤S.
<latexit sha1_base64="/Dx8U0rfe8nDrzT8q6iaArWAtE=">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</latexit>Hypotheses for the semiring S ⊗ commutative ⊕ idempotent S monotonic wrt ≤S ≤S is total: ∀x, y, x ⊕ y = x or x ⊕ y = y
<latexit sha1_base64="lkTLo2PETGiyEpzdhJsptBJlIHg=">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</latexit>... q0 − →
6
q1 q2 q0 − − →
12
q1 q2 q2 q0 − − →
15
− q0 − →
7
- q0
− − →
79
- 1
q0 − − − →
102
- 2
...
<latexit sha1_base64="uMSAlh+PtT2uYBRTSeuH+mwmc8=">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</latexit>best(q0) = min≤S 6 ⊗ best(q1) ⊗ best(q2), 12 ⊗ best(q1) ⊗ best1(q2) ⊗ best1(q2), 15, 7, 79, 102
<latexit sha1_base64="HrJe/s8huled1apk9jwtZ8SQHqc=">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</latexit>etc best(q) = M
ρ0=q−
− →
w0
a
ρ0 ⊕ h M
ρ=q−
→
w q1...qn
ρ
- best(q1), . . . , best(qn)
i
<latexit sha1_base64="cMERnlZMPlzFv7aTbdhqj0byUWQ=">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</latexit>for acyclic grammars. for cyclic ones: generalisation of Dijkstra algorithm to hypergraphs.
Transcription by 1-best parsing principle: apply a 1-best algorithm to a weighted CFG G ⇥ Gσ combining G and a representation of the input σ
- ex. steady tempo, all-left alignments.
I extend q0
- !
12
q1 q2 q2 of G into q0.[0, 1[
- !
12⊗δ(σ,[0,1[)
q1.[1, 1 3[ q2.[1 3, 2 3[ q2.[2 3, 1[ where δ(σ, [0, 1[) = 1 if σ|[0,1[ 6= ; and δ(σ, [0, 1[) = 0 otherwise (+1 for min-plus) I extend q1 !
1
- of G into q1.[1, 1
3[
- !
1⊗ε(σ,[1, 1
3 [)
- where
- if σ has 1 point x in the first half of [1, 1
3[ and none in the second half,
ε(σ, [1, 1
3[) is a normalized distance between x and 1,
- otherwise ε(σ, [1, 1
3[) = 0
For efficiency, the combined grammar is computed on-the-fly
<latexit sha1_base64="riUPjfRrU03d4w5q2zNpEC+NC34=">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</latexit>Transcription by 1-best parsing
q.[0, 2[ q0.[0, 1[ q1.[1, 1
3[
q2.[ 1
3, 2 3[
q2.[ 2
3, 1[
q3.[ 2
3, 5 6[
q3.[ 5
6, 1[
q0.[1, 2[ q1.[1, 4
3[
q2.[ 4
3, 5 3[
q2.[ 5
3, 2[
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1 1 3
x0
2 2 3
x0
3 5 6
x0
4 1
x0
5 4 3
x0
6 5 3 2
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x1
0.07 1 3 1 3
x2
0.72 5 6
x3
0.91 1
x4
1.05 4 3
x5
1.36 5 3
x6
1.71 2
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3
- 4
1
- ex.1: steady tempo, all-left alignments
Transcription by 1-best parsing input best parse tree serialization
x1
0.07 1 2 3 4
x2
0.72 7 8
x3
0.911
x4
1.05 4 3
x5
1.36 5 3
x6
1.71 2
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x0
6 5 3 2
<latexit sha1_base64="DC2kJCx1/h2SW+/P5z1zKEA5nA=">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</latexit>- 3
- 4
1
- ex.2: another extension for transcription with steady tempo, alignments to left or right
Transcription by 1-best parsing if we reduce the penalty for grace-notes, q1 − →
7
- 1
the best parse tree corresponds to:
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x0
1 1 2
x0
2 3 4
x0
3, x0 4 1
x0
5 4 3
x0
6 5 3 2
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0.07 1 2 3 4
x2
0.72
x3
0.911
x4
1.05 4 3
x5
1.36 5 3
x6
1.71 2
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- 3
- 4
1
Implementation, Results
https://gitlab.inria.fr/qparse/qparselib https://qparse.gitlabpages.inria.fr
- riginal score
transcription
- f MIDI
recording (100 ms) by qparse with generic grammar MEI output, display with Verovio
C++ library. MIDI input, XML/MEI or Lilypond output
Polonaise in D minor from Notebook for Anna Magdalena Bach BWV Anh II 128
Conclusion Rhythm transcription based on language theoretic models
- conversion of linear data (MIDI) into structured data (scores)
= parsing
- quantitative (based on semiring)
- Symbolic Tree Automata (more general than CFG)
- labels for inner nodes
- infinite set of labels for leaves
- guards in transitions
- Training rhythm grammars from corpus (Francesco, SMC 2019)
- Piano scores (process onsets & offsets)
https://gitlab.inria.fr/qparse/qparselib https://qparse.gitlabpages.inria.fr