a parse-based framework for coupled rhythm quantization

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A Parse-based Framework for Coupled Rhythm Quantization and Score Structuring Florent Jacquemard Masahiko Sakai Francesco Foscarin Philippe Rigaux supported by:

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Page 1: A Parse-based Framework for Coupled Rhythm Quantization

A Parse-based Framework for Coupled Rhythm Quantization

and Score Structuring Florent Jacquemard Masahiko SakaiFrancesco Foscarin

Philippe Rigaux

supported by:

Page 2: A Parse-based Framework for Coupled Rhythm Quantization

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

Page 3: A Parse-based Framework for Coupled Rhythm Quantization

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 Engravingdelegated to functionality of a score editor (MIDI import)

Page 4: A Parse-based Framework for Coupled Rhythm Quantization

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

Page 5: A Parse-based Framework for Coupled Rhythm Quantization

Implementation, Results

https://gitlab.inria.fr/qparse/qparselib https://qparse.gitlabpages.inria.fr

original score

transcription of MIDI

recording with

qparse

transcription: MIDI recording to XML/MEI

Beethoven, Trio for violin, cello and piano, op.70 n.2 (2d mov)

Page 6: A Parse-based Framework for Coupled Rhythm Quantization

Implementation, Results

original score

transcription of MIDI

recording with Finale.

options: 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)

Page 7: A Parse-based Framework for Coupled Rhythm Quantization

Hierarchical Structure of Music Notation

barbeat

subbeat

1 32 4 51.1 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 5.31.1.1 1.1.2 2.1.1 2.1.2 3.1.1 3.1.2 4.1.23.3.1 3.3.2 5.1.1 5.1.2 5.2.1 5.2.24.1.1 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: 12

14

14

116

116

34

116

116

34

012

14

14 2

12

14

14

116

116

34

12

14

14

12

14

14

12

14

14

116

116

34

12

16

16

16

12

12 01

Page 8: A Parse-based Framework for Coupled Rhythm Quantization

Languages of rhythmic notation

q �! q0 q �! q0 q

q0 �! q1 q2 q0 �! q1 q2 q2q0 �! � q0 �! • q0 �! •1 q0 �! •2q1 �! � q1 �! • q1 �! •1 q1 �! •2q2 �! q3 q3 q2 �! •q3 �! q4 q4 q3 �! � q3 �! • q4 �! •

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! !!!3

! !!

Context-Free Grammar (CFG): finite set of context-free production rules.

example: (very simple) CFG for 14 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),. . .

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Page 9: A Parse-based Framework for Coupled Rhythm Quantization

Languages of rhythmic notation

q �! q0 q �! q0 q

q0 �! q1 q2 q0 �! q1 q2 q2q0 �! � q0 �! • q0 �! •1 q0 �! •2q1 �! � q1 �! • q1 �! •1 q1 �! •2q2 �! q3 q3 q2 �! •q3 �! q4 q4 q3 �! � q3 �! • q4 �! •

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Context-Free Grammar (CFG): finite set of context-free production rules.

example: (very simple) CFG for 14 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),. . .

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Page 10: A Parse-based Framework for Coupled Rhythm Quantization

Languages of rhythmic notation

q �! q0 q �! q0 q

q0 �! q1 q2 q0 �! q1 q2 q2q0 �! � q0 �! • q0 �! •1 q0 �! •2q1 �! � q1 �! • q1 �! •1 q1 �! •2q2 �! q3 q3 q2 �! •q3 �! q4 q4 q3 �! � q3 �! • q4 �! •

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Context-Free Grammar (CFG): finite set of context-free production rules.

example: (very simple) CFG for 14 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),. . .

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Page 11: A Parse-based Framework for Coupled Rhythm Quantization

Parse trees

q0!q1 q2

q1!• q2!•<latexit sha1_base64="o4sg1YV35B4XTzCD6o+L065KMzI=">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</latexit>

q0!q1 q2 q2

q1!• q2!• q2!•<latexit sha1_base64="t11Qww1+8vJ4yWIqlkYj3nlydoo=">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</latexit>

q0!q1 q2 q2

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3

! !!!! !! !

3

!"

parse tree

serialization

corresponding notation

0 12

1<latexit sha1_base64="5mVrpfs781m7C449GrFRhkivAXE=">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</latexit>

0 13

23

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a parse tree is a tree labeled by grammar’s production rules, with tiling of non-terminals.

Page 12: A Parse-based Framework for Coupled Rhythm Quantization

More parse trees

q0!q1 q2

q1!• q2!q3 q3

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Page 13: A Parse-based Framework for Coupled Rhythm Quantization

Closure of languages

‣ a word (sequence of terminal symbols) is parsed by a CFGif it is the yield of a parse tree

Properties:

‣ languages of (parsed) words are not closed under intersection.

‣ languages of parse trees = regular tree languagesare closed under intersection.

tree languages: → capture regularity of rhythm → grammar composition (Cartesian product) combination of grammar representation of different aspects

Page 14: A Parse-based Framework for Coupled Rhythm Quantization

Objectives (informal definition)

→ find good compromise between precision and complexity of notation (multicriteria optimisation)→ quantitative parsing

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34

78

1<latexit sha1_base64="YzcWZzVWC8LUTlM21tbh5DRQpsM=">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</latexit>

given:

- a grammar G describing rhythmic notation preferences

- an input sequence of events �

return:

- a parse tree t of G that fits well with �

questions:

- How well? How to avoid complex parse trees (overfitting).<latexit sha1_base64="L6nJEFJkpxLWChm7We4zn8hO3P4=">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</latexit>

Page 15: A Parse-based Framework for Coupled Rhythm Quantization

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="j3RaT0udc4pnyuupmGFtZ7/sGng=">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</latexit>

Page 16: A Parse-based Framework for Coupled Rhythm Quantization

Semirings

semiring domain � 0 ⌦ 1 natural orderingBoolean {0, 1} _ 0 ^ 1 1 S 0Viterbi [0, 1] ⇢ R+ max 0 . 1 x S y iff x � y

min-plus R+ [ {+1} min +1 + 0 x S y iff x ymax-plus R [ {�1} max �1 + 0 x S y iff x � y

These semirings are

commutative: ⌦ is commutative

idempotent: 8x, x� x = x

have an induced total natural ordering S defined by:

8x, y, x S y iff x� y = x

monotonic wrt S : 8x, 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>

Page 17: A Parse-based Framework for Coupled Rhythm Quantization

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 �!0 q0 q q �!0 q0 q0

q0 �!6 q1 q2 q0 ��!12 q1 q2 q2

q0 ��!15 � q0 �!1 • q0 ��!79 •1 q0 ���!102 •2q1 �!2 � q1 �!1 • q1 ��!25 •1 q1 ��!64 •2q2 ��!10 q3 q3 q2 �!2 •q3 ��!11 q4 q4 q3 �!4 � q3 �!1 • q4 �!1 •

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Page 18: A Parse-based Framework for Coupled Rhythm Quantization

s

3

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3

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parse tree

serialization

corresponding notation

0 12

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0 13

23

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0 13

23

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weight = notational complexity

q0�!6 q1 q2

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9

q0��!12 q1 q2 q2

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0 12

34

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q4�!2 �q4�!1 •<latexit sha1_base64="9TtR6l9xUDGrAFz0lIqFTzW0aPc=">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</latexit>

!"! ! !! " !""!

22 35 35notational complexity

Page 20: A Parse-based Framework for Coupled Rhythm Quantization

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✓ : Q+ ! Q+

✓ : RTU value 7! MTU value

A tempo model M is a set of tempo curves, with restrictions(piecewise linear)

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Page 21: A Parse-based Framework for Coupled Rhythm Quantization

Objective

function fit: pointwise comparison

input: event sequence � timestamped in RTU

tempo curve ✓ ✓�1(ktk) (RTU)

parse tree t of G �������!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 �

- a tempo model M

return:

- a tempo curve ✓ 2 M

- a parse tree t of G minimizing weightG(t)⌦ fit��, ✓�1(ktk)

�wrt S

<latexit sha1_base64="i6fCvs3bLST4WhWIcqobFXx9tX0=">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</latexit>

Page 22: A Parse-based Framework for Coupled Rhythm Quantization

Optimization Problem

We want t that optimizes a combination (with ⌦) ofthe weight of t wrt G weightG(t)the fitness of t to the input � fit

��, ✓�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.

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Page 23: A Parse-based Framework for Coupled Rhythm Quantization

1-best parsing

goal: find a parse tree t of a weighted CFG over Swith minimal weight wrt S .

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Hypotheses for the semiring S

⌦ commutative

� idempotent

S monotonic wrt S

S is total: 8x, y, x� y = x or x� y = y<latexit sha1_base64="lkTLo2PETGiyEpzdhJsptBJlIHg=">AAAnVHic5Vpbcxu3FaaSpk3ZNrXbt/YF7VJxJ5U1JGXZStO4cXyLx5ajSnLsGa2i2QtIosIuVgBWIrPZ9/6avra/pDP9L33oAbAkd7GgrLTuuDOlxxKA8+Hg4DsXYJcKM0qE7Pf/sfbOu9977/s/eP+H3R/9+Ccf/PTa9Z99JVjOI/wiYpTxV2EgMCUpfiGJpPhVxnGQhBS/DE/vK/nLc8wFYemhnGX4OAnGKRmRKJAwdHJ97Vd+ykga41SiL2YZkxMssEAjxhE0kcAJ4SQdI9TzD6p2r+uHeEzSIsYi4iRTisquTyROjo4BxyRJsOihiCVJLmGdc1yXZjQHIYlxAqvBsjXZYgWUsJRJlpII+Rdcgojis5O6BYs5TQEiAkkmA/o71O35sIuAUjTdQLPeBupNkVkdzdCnaNpDsMfm2AwU4zRubqx7cs3rb/b1B7Ubg6rhdarP3sn19V/4MYvyBHYX0UCIo0E/k8dFwCWJKAaucoGzIDoNxvgImmkAfB0X2pklWoeRWDtgxMAperTbXdc9gQJEAxSCgzHv+hyn+ELRHKSxL0YxHgU5lUU2OS8t4ShICJ3NAUssbK9mTBEkIgnkpET2qJgloTUahozGZXfdQmozG1sshEwCPuM2WK0UhFNrdDc9gLUYbaqYBBxCYgbsIEGSjGKU5AKiQ68G7KxrwgLO2YXo2otIxiiYNEcBLSgJIs6ayKmAFjin2236R452jguSZjkEa2TcM8opRBlS+YRiwnEk6QwaoJOAh1EE1gaRhKxTftt7/GgDHZLTb7rrN9/Up0lZNh5llCnWzXDVpSTkwHsRBxKrgSajgNqA/4Yy1UoZhP0GUmO5ZMBb0JwgYQtmRLUq5UciDU4heO3xIoRZY87yNBatSYXkGDuGK1vmy29kTBCVg5DY1dbq6CyQQHEqHCIxCTK1gKJ//+HL/SeHT54/fpP0+5A+/j6O80jZ5w28obfl3SrWu76KN45p4fOACAzFAU/9ScimSnYuWVb4JKBknBae50c84jCskf0lsudPRoT6pgQJOaO48JE38FGpBb1yPjFlRpWu4MCFOgPEKclKpZSMJ1ITClPolWZcyYxhy4yyLIvy68LbKk8K71YJqsAjhqFn/8sMUTz6TgS9jo9L6VjXfEwYO50uPXPkDY69YVH4VAn8iS5Axc3+5rCPk9K3kIU3BN3dhSJX+K1Xqv4E1gPHNZVb26CyTf7w7ZDfa9rZaxv6neP3P3LPnFQ3qytJ3e4bWw1DO9tXJvPW2831f48rRZVhCi6Y0UQRVOiAgp16g/LIGx6bQ5bjCzgI4fII545QA/4IhuaEetvebe+Ot2NUuSRF14d6DvT0P61Y8m4j7w7wgrydntqxEQ8s8XAuJqOYJHBzjPt34ccA+ZwG4A3zU01Rp7cSF962CgF0yWcxQeMNcZRdYI68LV8ZAWzQYKoFijDjYuTdUsK+sgZT6K8yYaBNqC8yqBZp6LEXaVigFhkRFciKasV6xa1qLsk1zoomjED0auKV21RZKQabA8hv9Xuof4NNRT08ShUEVb5cRpYhrKEa8mL4X1O9PVAq50vUVb/hBW5/fJUFjAcoV7xvKJdXDXymnbKuIhjqAlzhQ0xVd4hCJuG2c6JTuBr3l/f2olJWHg2Pi2Xx1o5dWOl/4t+Fn3fh3ycq/hrVQB8ZbYUyWqESakChK8Hlantff9RrqTb7fa2tWq1R3jiDW/oMbVfbuzZRJ1dTp4MPU6SMohicHsQTFm3CR6VkcM5IjCYkjuEhjJ1XZbOWT/HbTihQ/X+UUI7YjV0REV89HdQHAiAhIlo8Byr3EnCixFNJZEE28SZcwKonQi3F44UUb45t6QWXCzG0l1Jz8bugbDlddWzlMlpqh9xcSJv7H0EsToV5PIb0SEXOsaLBf0ACeDyOSwe8NMcJHOq12T2VAw3svTwmbLj75MGTsrg33K3ZX0cp+VBEjIOPdocHNTPVLn7vDe6Ck4J0DM/nnjp2ddOyKgsINx40SOW4DaRc58bfK6s8C2hxz46Iz2vCz23h/Zrwvi18UBM+sIUPK2EYFg9t2aOl7JEte1xT+tgWPqkJn5Q2tU+Xap/aM5/WZmph/d1O4e/VxHv23P1GpCyB+63s2ucroSe6LUYFT1vTDlfMOmwhn4F7V4CftcC7K5C7LeRzBxJYfN4Cvlqh8lX75EkyKEREyGaanQccbrSE6teCTfeFMZNzD5K02LQ1Ev3KtEJAlguoSbIFwpDDC4jq2AB1uC0RumdDdMFbYkzXAsU5XyJUx47Gajv6Ta49lyUOHpUeENgLAWPCjdaSNpyuQuvS3TQyMtuw4RCmWtLCi5V44cTHK0yPRRt7QMZJYGvXg1buNKcdTrBsTdODl6UcZ0xmbtuUqIUnKalFhO7ZR8tBVCsiBy07nx+WRwNVskMeRKemuOumnYuqGlQHWdo4BusokOgDoGY/HAIn+g3HCugIFtPHOwDtgDR5OneO6tlHThOSuTDQyJYQ3bMQIa4r0b12ktchxIXJ1Nm5sAQ6tpw1TGVtDVA6JG5UEtnScoGbVaDq2/ep8PJawsPX1BG4sNVshU7rrFevIJYQ07V9fGaKoTMzK5mteDGlUjyH2ZpNQXXp1dXV1tqov2eOAnwmRyv0gaC1fH7JxnL3xnJrY7nzoDCjiT5QVupW4hUTg+mlE0Hc3stKKnMXlXmDytxBpRpbvYNK6Jy00vpKaE+S6qvGlUstpO5pqxZbSFtMUSvo6Tzq3ShMaTODmtq4pY07tbVQjmSs7u3O80+LWhPwuFZmoNOSc6IP7AVE9+2jJZF5qa9Uk0I1l08+jQ0sQbwOsk6p6nte66xqYL7BnC3v0n1bzFK8lA5axqbqy+Nscsn9fzzK06hJ4/hEXf1buvSXpkt2VM8OMXW/kkH95Ikd1TyC51eCa1e2+YB9bT3Hab0iV/1WrKSntUg5teVJXlOhOvZ8XHe67tlBOyOY1tLfdC01ZlAf7t/qC8W31QXA2lJ9Md2zyTEHoLqeqFOcMg42xSXSFwXLrlOSpgseIe5vxuS8ZbyFMRCrMvCkrFUC6LUevWs1fERaBfzLXNZi+Muy+q5yMpOTBB2qr0eh6zarnr4iCSiVkHXquTlXN5YrzbJfuKDftm+M7WmReiZwRZyFC3NKsXTYws3tiBI6e0xnKtNhQGxutQNM4+5Fe8FKJY4i5rjBS9IqeCRVEY3W0W5q3oWgP9g5ac252brVqD8YaWJg1457t6D5inpbOugR5pnLAY9dzzItMIy8fnfteTfnnrWv1Ti3jr7CfAHZfPlQwUzAWExyRmyethwq5jiXjrM8kPapVdxyKFkAXVoikp5ZOrYdOiqYS4MgNhm3HQoMyjkfZxbvxR2XAgNzaZjkxNaw49BQwVwaUpxbt9fiY4eGCubSkKfNouCf2C/O9l3vmyCEH3Ju3r2eXPMG9t9OtRtfDTcHW5vDPw69z3aqv6t6v/PLzq87v+kMOnc6n3W+6Ox1XnSitT+v/WXtr2t/+/DvH/7zxrs33jPQd9aqOT/vND43PvgXwmu/uQ==</latexit>

...

q0 �!6 q1 q2 q0 ��!12 q1 q2 q2

q0 ��!15 � q0 �!7 • q0 ��!79 •1 q0 ���!102 •2...

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best(q0) = minS

8<

:

6⌦ best(q1)⌦ best(q2),12⌦ best(q1)⌦ best1(q2)⌦ best1(q2),15, 7, 79, 102

9=

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etc

for acyclic grammars. for cyclic ones: generalisation of Dijkstra algorithm to hypergraphs.

Page 24: A Parse-based Framework for Coupled Rhythm Quantization

1-best parsing

goal: find a parse tree t of a weighted CFG over Swith minimal weight wrt S .

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Hypotheses for the semiring S

⌦ commutative

� idempotent

S monotonic wrt S

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...

q0 �!6 q1 q2 q0 ��!12 q1 q2 q2

q0 ��!15 � q0 �!7 • q0 ��!79 •1 q0 ���!102 •2...

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best(q0) = minS

8<

:

6⌦ best(q1)⌦ best(q2),12⌦ best(q1)⌦ best1(q2)⌦ best1(q2),15, 7, 79, 102

9=

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etcbest(q) =M

⇢0=q��!w0 a

⇢0 �h M

⇢=q�!w q1...qn

⇢�best(q1), . . . , best(qn)

�i

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for acyclic grammars. for cyclic ones: generalisation of Dijkstra algorithm to hypergraphs.

Page 25: A Parse-based Framework for Coupled Rhythm Quantization

Transcription by 1-best parsing

principle: apply a 1-best algorithm to a weighted CFG G ⇥ G�

combining G and a representation of the input �

ex. steady tempo, all-left alignments.

I extend q0 ��!12 q1 q2 q2 of G into

q0.[0, 1[���������!12⌦�(�,[0,1[)

q1.[1,1

3[ q2.[

1

3,2

3[ q2.[

2

3, 1[

where �(�, [0, 1[) = 1 if �|[0,1[ 6= ; and �(�, [0, 1[) = 0 otherwise(+1 for min-plus)

I extend q1 �!1 • of G into q1.[1,13 [��������!

1⌦"(�,[1, 13 [) •where

- if � has 1 point x in the first half of [1, 13 [ and none in the second half,

"(�, [1, 13 [) is a normalized distance between x and 1,

- otherwise "(�, [1, 13 [) = 0

For efficiency, the combined grammar is computed on-the-fly<latexit sha1_base64="riUPjfRrU03d4w5q2zNpEC+NC34=">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</latexit>

Page 26: A Parse-based Framework for Coupled Rhythm Quantization

Transcription by 1-best parsing

q.[0, 2[

q0.[0, 1[

q1.[1,13 [ q2.[

13 ,

23 [ q2.[

23 , 1[

q3.[23 ,

56 [ q3.[

56 , 1[

q0.[1, 2[

q1.[1,43 [ q2.[

43 ,

53 [ q2.[

53 , 2[

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x01

0 13

x02

23

x03

56

x04

1

x05

43

x06

53

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input

best parse tree

serialization

0

x1

0.07 13

13

x2

0.72 56

x3

0.91 1

x4

1.05 43

x5

1.36 53

x6

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!!

3 3

!41 ! ! !!

ex.1: steady tempo, all-left alignments

Page 27: A Parse-based Framework for Coupled Rhythm Quantization

Transcription by 1-best parsing

input

best parse tree

serialization

0

x1

0.07 12

34

x2

0.72 78

x3

0.911

x4

1.05 43

x5

1.36 53

x6

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q.[0, 2[.0.0

q0.[0, 1[.0.0

q1.[1,12 [.0.0 q2.[

12 , 1[.0.0

q3.[12 ,

34 [.0.1 q3.[

34 , 1[.1.0

q4.[34 ,

78 [.1.0 q4.[

78 , 1[.0.0

q0.[1, 2[.0.0

q1.[1,43 [.0.0 q2.[

43 ,

53 [.0.0 q2.[

53 , 2[.0.0

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x01

0 12

x02

34

x03

78

x04

1

x05

43

x06

53

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! !

3

!!"41 ! !

ex.2: another extension for transcription with steady tempo, alignments to left or right

Page 28: A Parse-based Framework for Coupled Rhythm Quantization

Transcription by 1-best parsing

if we reduce the penalty for grace-notes, q1 �!7 •1the best parse tree corresponds to:

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input

x01

0 12

x02

34

x03, x

04

1

x05

43

x06

53

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0

x1

0.07 12

34

x2

0.72

x3

0.911

x4

1.05 43

x5

1.36 53

x6

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Page 29: A Parse-based Framework for Coupled Rhythm Quantization

Implementation, Results

https://gitlab.inria.fr/qparse/qparselib https://qparse.gitlabpages.inria.fr

original score

transcription of 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

Page 30: A Parse-based Framework for Coupled Rhythm Quantization

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