Symbolic music features: Rhythm and meter

Transcription

Symbolic music features:
Rhythm and meter
Anja Volk
Sound and Music Technology,
May 26, 2016
1
Today
!  Main modules
A.  Sound and music for games
B.  Analysis, classification, and retrieval of sound and music for
media
C.  Generation and manipulation of sound and music for games
and media
2
Today
!  Main modules
media
• 
• 
Representation of musical data: symbolic vs. audio data
Main topic: Symbolic music feature extraction: Rhythm and meter
– 
– 
– 
– 
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9
What is rhythmic and metric information in music?
Computational models for rhythm and meter extraction
Inner Metric Analysis
Application in Music Analysis, Music Cognition and Music Information
Retrieval
Student Presentation: paper by Andre Holzapfel
Module B: Analysis, classification, and
retrieval of sound and music for media
!  Music Information Retrieval
!  Symbolic vs. audio data
10
11
12
Representation of musical data:
symbolic vs. audio data
!  Audio is most natural representation of music
!  ideally, this is what the MIR user works with
!  Important digital audio formats
!  PCM (Pulse Code Modulation), CD quality:
•  sample rate 44.100 Hz
•  amplitude resolution 16 bits
•  2 channels
!  MP3
•  lossy, perceptual compression
•  perceptual masking
•  suppresses inaudible components
•  ‘bit rate’ can be set by creator
•  many different MP3s from one
recording
13
!  Musical audio is analysed
in 2 domains:
!  time domain
•  change of amplitude
over time
!  frequency domain
•  within certain time
interval
•  Fast Fourier Transform
•  magnitude of
frequency components
!  Musical tones have a
periodic spectrum
(f0, 2f0 , 3f0 , 4f0 ,…)
!  harmonics
trumpet
14
!  Sample audio features
!  frequencies
!  fundamental frequency
!  spectral energy
!  Cepstral coefficients, MFCC
!  amplitude envelope
!  attack
!  decay
!  zero crossing rate
!  spectral centroid
!  standard deviations of the above
!  etc. etc.
!  Low-level features: most of these not directly observable by
listener
!  Used in instrument recognition, genre classification, etc.
15
16
!  MIDI: Musical Instrument Digital Interface
!  Event messages: sound events
!  Standardized in 1983
!  Protocol for communicating with electronic instruments
17
Sound event perception
MIDI
pitch
low-high: c. 90 categories
octave equivalence
pitch number
duration
long-short: multiples of 2
and 3
Onset and offset
times
loudness soft-loud: continuous
Velocity number
timbre
MIDI Instrument
tone quality
sound source
18
MFile 1 2 1024
MTrk
0 TimeSig 2/4 24 8
0 KeySig 0 major
!  MIDI:
0 Tempo 500000
!  Event messages: sound events 16385 Meta TrkEnd
TrkEnd
MTrk
0 Meta TrkName "Acoustic Grand
Piano"
0 PrCh ch=1 p=0
0 On ch=1 n=72 v=64
0 On ch=1 n=48 v=64
1024 Off ch=1 n=72 v=0
1024 On ch=1 n=72 v=64
1024 Off ch=1 n=48 v=0
1024 On ch=1 n=60 v=64
2048 Off ch=1 n=72 v=0
2048 Off ch=1 n=60 v=0
2048 On ch=1 n=79 v=64
2048 On ch=1 n=64 v=64
3072 Off ch=1 n=79 v=0
3072 On ch=1 n=79 v=64
3072 Off ch=1 n=64 v=0
3072 On ch=1 n=60 v=64
4096 Off ch=1 n=79 v=0
piano roll notation
MIDI in action:
https://www.youtube.com/watch?v=gGdSxEKujDE
19
piano roll notation
Origin: piano roll
20
piano roll notation
Origin: piano roll
21
!  Symbolic data
!  MIDI
!  Kern
!  MusicXML
!  …
!  Beyond MIDI. The Handbook of Musical Codes. ed. Eleanor
Selfridge-Field (1997)
!  c. 27 music encoding systems, detailed descriptions
!  non-proprietory formats only
22
!  Symbolic data
!  MIDI
!  Kern
!  MusicXML
!  …
!  Beyond MIDI. The Handbook of Musical Codes. ed. Eleanor
Selfridge-Field (1997)
!  c. 27 music encoding systems, detailed descriptions
!  non-proprietory formats only
23
!  Symbolic data
!  David Huron, c. 1990
!  http://csml.som.ohio-state.edu/
Humdrum/
!  syntax for developing (light-weight)
encoding systems
!  data organised in parallel ‘spines’
!  humdrum representation for CMN: kern
!  humdrum tools (UNIX)
24
**kern
*staff2
=1*clefF4
*k[]
*M2/4
*^ *
2ryy
.
=2
2ryy
.
=3
2ryy
.
=4
2ryy
.
=5
2ryy
.
=6
2ryy
.
=7
**kern
*staff1
=1*clefG2
*k[]
*M2/4
4C
4c
=2
4e
4c
=3
4f
4c
=4
4e
4c
=5
4d
4B
=6
4c
4A
=7
4cc
4cc
=2
4gg
4gg
=3
4aa
4aa
=4
4gg
4gg
=5
4ff
4ff
=6
4ee
4ee
=7
Today
!  Main modules
media
• 
• 
– 
– 
– 
– 
– 
25
Retrieval
Student Presentation: Model evaluation
Symbolic music feature extraction:
Rhythm and meter
!  What is rhythm and meter?
!  Fundamental to human perception and cognition of music
26
Rhythm and meter
!  “Rhythm and pitch are the two primary parameters of musical
structure (Meyer, E 1973). … If pitch is concerned with the
disposition of the frequencies of musical notes, then rhythm is
concerned with the description and understanding of their duration
and durational patternings. These durations may be more or less
regular, may or may not give rise to a sense of beat or tempo,
and may be more or less continuous, but as all music involves
duration(s), all music necessarily has some manner of rhythm.
Grove Music Online
27
Rhythm and meter
Height (Pitch)
Time
28
Rhythm and meter
!  Listening experiment
29
Dancing
Picture from hcs.harvard.edu/~hma-bdc
30
Dancing
Mambo !
Merengue !
31
… even 7 months old babies are experts
Phillips-Silver/Trainor: Feeling the Beat: Movement Influences Infant Rhythm Perception, 2005
32
Rhythm
Spontaneous reaction: tapping
33
Rhythm: Meter
1
34
2
3
1
2
3
Rhythm: Meter
1
35
2
3
4
1
2
3
4
Rhythm: Meter
Accent Pattern
36
Theories of rhythm and meter
Yeston, Lerdahl/Jackendoff, Hasty
I define the meter of a work as the union of all layers of
motion (i.e., series of regularly recurring pulses) active
within it.
Harald Krebs
37
nested pulses
38
nested pulses
39
Today
!  Main modules
media
• 
• 
• 
Short Recapitulation: Music Information Retrieval
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40
Retrieval
Background:
Mathematical Music Theory
Simplicial complex (Algebraic Topology, Guerino Mazzola)
A. Fleischer (Volk), G. Mazzola and T. Noll (2000): Computergestützte Musiktheorie,
Musiktheorie, 15(4), 314-325. 42
Inner metric analysis: motivation
43
44
46
Anton Webern
Scott Joplin
Piano Variations Op. 27, II
Nonpareil Rag
47
Outer Metric Structure
Inner Metric Structure
48
WITCHCRAFT,
Metric Weight
49
Metric Weight
Cycles in 3
Cycles in 2
50
Metric Weight
Picture from www.ritzenhoff.de
51
Metric Weight
52
Metric Weight
53
Metric Weight
54
Metric Weight
55
Spectral Weight
56
Metric Weight
Metric Weight
57
Spectral Weight
64
Anton Webern
Scott Joplin
Piano Variations Op. 27, II
Nonpareil Rag
?
65
Webern vs. Joplin
66
Webern vs. Joplin
left hand
right hand
67
Webern vs. Joplin
68
Today
!  Main modules
media
• 
• 
• 
Short Recapitulation: Music Information Retrieval
– 
– 
– 
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69
Retrieval
Music Analysis
Symphony C Major K. 551
1. Movement (4/4)
Wolfgang A. Mozart
Excerpt from Metric weight W2,2 of the exposition (bars 1-55)
70
Music Analysis
Igor Stravinsky: The Rite of Spring
Danse Sacrale
71
Music Analysis
72
Music Analysis
73
Music Cognition
J. Snyder & C. Krumhansl: Tapping to Ragtime - Cues to Pulse Finding
!  Ragtime: Cues for pulse finding?
!  First hypothesis: pitch
!  Tapping experiments: rhythm only!
74
Music Cognition
J. Snyder & C. Krumhansl: Tapping to Ragtime - Cues to Pulse Finding
American Beauty
Sensations
Glad Cat
S: Both Hands
S: Both Hands
S: Both Hands
S: Right Hand
S: Right Hand
S: Right Hand
S: Left Hand
S: Left Hand
S: Left Hand
Anja Volk (2008), The Study of Syncopation using Inner Metric Analysis: Linking Theoretical and
Experimental Analysis of Metre in Music, In: Journal of New Music Research, Vol. 37:4, p. 259-273.
75
Music Information Retrieval
76
I taught cha cha, waltz, foxtrot, west coast swing, tango, rumba, etc. And sure, there were
music catalogues with metadata tagged in all these genres. But the songs that had
those metadata tags were all ballroom dance music. Awful, awful stuff. Stuff
nobody really wanted to listen to.
So what I really wanted was something that could find songs that were both good
to listen to, and good to dance to at the same time.
… I don't see how this could be done without looking at the content of the music.
Jeremy Pickens, MIR-list, May 8th 2011
77
Classification
all in 4/4 !
Merengue
Rumba
Tango
BossaNova
March/Anthem
Picture from www.mondolatino.it Picture from www.ritzenhoff.de
Picture from www.accomodationbsas.co.ar Picture from www.armygermany.com
78
Picture from www.danceuniverse.co.kr E. Chew, A. Volk, and C.Y Lee, Dance Music Classification Using Inner Metric Analysis, 2005
Complete Book of the Worlds’ Dance Rhythm
http://andrewjpeterswrites.com/?
tag=stephanie-spinner
Similarity
(correlation
coefficient)
80
E. Chew, A. Volk, and C.Y Lee, Dance Music Classification Using Inner Metric Analysis, 2005
Template Rhythms: Complete Book of the Worlds’ Dance Rhythm
Corpus: 21 Marches, 17 Merengues, 25 Bossa Nova, 17 Rumbas, 22 Tangos
81
Metric Weight
Metric Weight
82
Spectral Weight
Spectral Weight
83
Classification
correct genre: 80%
Merengue
Rumba
Tango
BossaNova
March/Anthem
Picture from www.mondolatino.it Picture from www.ritzenhoff.de
Picture from www.accomodationbsas.co.ar Picture from www.armygermany.com
84from www.danceuniverse.co.kr Picture
E. Chew, A. Volk, and C.Y Lee, Dance Music Classification Using Inner Metric Analysis, 2005
Ragtime
90
Ragtime
“A discussion on what constitutes ragtime could fill several web pages …”
91
What’s the problem?
!  historic sources vs. study of the musical material itself
!  not just the three main composers (Joplin, Scott, Lamb)
!  Edward Berlin: we need more balanced view on ragtime
Computational corpus-based study is the next step
107
Patterns in the RAG-collection
!  11.000 MIDI files
!  contains Ragtimes from 1890s to 2012
!  Ragtime Compendium:
!  15.000 identified ragtime compositions
!  out of these 5600 pieces have MIDI files.
!  probably the most complete listing that exists, organized
by the Internet community of ragtime lovers
108
Some numbers from the compendium…
!  128 websites for Ragtime MIDI’s
!  101 cited sources of data (Libraries, societies, individuals)
!  38 contributors for the compendium creation
109
ragtime
Syncopation
24
Œ
œœ œ
!  1-2-1
a)
2
b) 4
2
c) 4
7
2
b) 4
!
I"I"O"I""."."."."".".".".""."."."." I"I"O"I""."."."."".".".".""."."."."
untied
2
c) 4
2
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
.".".".""I"I"O"I"".".".".""."."."." .".".".""I"I"O"I"".".".".""."."."."
42
2
e) 4
d)
110
!
a)
Œ œœ œ ‰ œœ œœ‰
2
e) 4
tied
.".".".""."."I"I""O"I".".""."."."." .".".".""."."I"I""O"I".".""."."."."
11
ragtime
Syncopation
24
Œ
œœ œ
!  1-2-1
a)
2
b) 4
2
c) 4
7
2
b) 4
!
I"I"O"I""."."."."".".".".""."."."." I"I"O"I""."."."."".".".".""."."."."
untied
2
c) 4
2
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
.".".".""I"I"O"I"".".".".""."."."." .".".".""I"I"O"I"".".".".""."."."."
42
2
e) 4
d)
111
!
a)
2
e) 4
tied
.".".".""."."I"I""O"I".".""."."."." .".".".""."."I"I""O"I".".""."."."."
11
ragtime
Syncopation
24
Œ
œœ œ
!  1-2-1
a)
2
b) 4
2
c) 4
7
2
b) 4
!
I"I"O"I""."."."."".".".".""."."."." I"I"O"I""."."."."".".".".""."."."."
untied
2
c) 4
2
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
.".".".""I"I"O"I"".".".".""."."."." .".".".""I"I"O"I"".".".".""."."."."
42
2
e) 4
d)
112
!
a)
2
e) 4
tied
.".".".""."."I"I""O"I".".""."."."." .".".".""."."I"I""O"I".".""."."."."
11
ragtime
Syncopation
24
Œ
œœ œ
!  1-2-1
a)
2
b) 4
2
c) 4
7
2
b) 4
!
I"I"O"I""."."."."".".".".""."."."." I"I"O"I""."."."."".".".".""."."."."
untied
2
c) 4
2
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
.".".".""I"I"O"I"".".".".""."."."." .".".".""I"I"O"I"".".".".""."."."."
42
2
e) 4
d)
113
!
a)
2
e) 4
tied
.".".".""."."I"I""O"I".".""."."."." .".".".""."."I"I""O"I".".""."."."."
11
ragtime
Syncopation
24
Œ
œœ œ
!  1-2-1
a)
2
b) 4
2
c) 4
7
2
b) 4
!
I"I"O"I""."."."."".".".".""."."."." I"I"O"I""."."."."".".".".""."."."."
untied
2
c) 4
2
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
.".".".""I"I"O"I"".".".".""."."."." .".".".""I"I"O"I"".".".".""."."."."
42
2
e) 4
d)
114
!
a)
2
e) 4
tied
.".".".""."."I"I""O"I".".""."."."." .".".".""."."I"I""O"I".".""."."."."
11
Musicological hypothesis
a)
a)
a)
I"I"O"I""."."."."".".".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
42
2
I"I"O"I""."."."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
c)
2
b) 42
b) 4
2
c) 42
c) 4
2
d) 42
d) 4
2
e) 42
e) 4
2
4
I"I"O"I""."."."."".".".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
.".".".""."."I"I""O"I".".""."."."."
e)
.".".".""I"I"O"I"".".".".""."."."."
a)
2
4
.".".".""."."I"I""O"I".".""."."."."
2
.".".".""."."I"I""O"I".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
c) 4
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
I"I"O"I""."."."."".".".".""."."."."
b)
42
2
e) 4
d)
1890
1900
early Ragtime era
115
2
b) 4
1910
."."I"I""O"I"."."".".".".""."."."."
1920
late Ragtime era
…
.".".".""."."I"I""O"I".".""."."."."
Result RAG-collection:
a)
a)
a)
I"I"O"I""."."."."".".".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
42
2
I"I"O"I""."."."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
d) 4
c)
2
b) 42
b) 4
2
c) 42
c) 4
2
d) 42
d) 4
2
e) 42
e) 4
2
4
I"I"O"I""."."."."".".".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
.".".".""."."I"I""O"I".".""."."."."
e)
.".".".""I"I"O"I"".".".".""."."."."
a)
2
4
.".".".""."."I"I""O"I".".""."."."."
2
.".".".""."."I"I""O"I".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
c) 4
."."I"I""O"I"."."".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
I"I"O"I""."."."."".".".".""."."."."
b)
42
2
e) 4
d)
1890
1900
early Ragtime era
116
2
b) 4
1910
."."I"I""O"I"."."".".".".""."."."."
1920
late Ragtime era
…
.".".".""."."I"I""O"I".".""."."."."
a)
proportion of syncopation patterns per bar
a)
1890
1900
early Ragtime era
117
2
42
2
d) 4
e)
1910
1920
late Ragtime era
I"I"O"I""."."."."".".".".""
.".".".""I"I"O"I"".".".".""
42
2
."."I"I""O"I"."."".".".".""
d) 4I"I"O"I""."."."."".".".".""."."."."
c)
b) 4
c)
2
b) 4
e)
2
.".".".""."."I"I""O"I".".""
4.".".".""I"I"O"I"".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
42
.".".".""."."I"I""O"I".".""."."."."
…
a)
a)
1890
1900
early Ragtime era
118
2
42
2
d) 4
e)
1910
1920
late Ragtime era
I"I"O"I""."."."."".".".".""
.".".".""I"I"O"I"".".".".""
42
2
."."I"I""O"I"."."".".".".""
d) 4I"I"O"I""."."."."".".".".""."."."."
c)
b) 4
c)
2
b) 4
e)
2
.".".".""."."I"I""O"I".".""
4.".".".""I"I"O"I"".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
42
.".".".""."."I"I""O"I".".""."."."."
…
?
1890
1900
1910
Ragtime era
119
1920
…
modern times
a)
1890
1900
I"I"O"I""."."."."".".".".""."."."
42
2
d) 4
c)
e)
1910
Ragtime era
120
2
b) 4
1920
.".".".""I"I"O"I"".".".".""."."."
."."I"I""O"I"."."".".".".""."."."
42
.".".".""."."I"I""O"I".".""."."."
…
modern times
a)
1890
1900
*
e)
Ragtime era
121
1920
I"I"O"I""."."."."".".".".""."."."
42
2
d) 4
c)
*
1910
2
b) 4
.".".".""I"I"O"I"".".".".""."."."
."."I"I""O"I"."."".".".".""."."."
42
.".".".""."."I"I""O"I".".""."."."
…
modern times
Result RAG-collection
a)
*
1890
1900
e)
1910
Ragtime era
122
1920
I"I"O"I""."."."."".".".".""."."."
42
2
d) 4
c)
*
2
b) 4
.".".".""I"I"O"I"".".".".""."."."
."."I"I""O"I"."."".".".".""."."."
42
.".".".""."."I"I""O"I".".""."."."
…
modern times
Metric weight patterns in ragtime
Metric Feature
123
My Best Girl
No untied patterns, tied patterns
124
Metric weight
patterns in ragtime
a)
2
b) 4
42
2
d) 4
c)
e)
42
I"I"O"I""."."."."".".".".""."."."."
.".".".""I"I"O"I"".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
.".".".""."."I"I""O"I".".""."."."."
You can’t expect kisses from me
tied pattern 1
125
a)
2
b) 4
42
2
d) 4
c)
e)
42
A Catchy Thing
126
high number
I"I"O"I""."."."."".".".".""."."."."
lower number
.".".".""I"I"O"I"".".".".""."."."."
."."I"I""O"I"."."".".".".""."."."."
.".".".""."."I"I""O"I".".""."."."."
And further
!  Automatic meter finding
!  W.B. de Haas & A.Volk: Meter detection in symbolic music
using Inner Metric Analysis, just accepted to ISMIR 2016
!  Application to Audio
210
And further
!  Automatic meter finding
!  W.B. de Haas & A.Volk: Meter detection in symbolic music
of the piece of length at least `, then the metric weight of
an onset, o 2 On, is defined as follows:
{
(1)
SW`,p (t) =
X
p
km
.
2
4
In this paper we have use the standard parameters p = 2,
and ` = 2. Hence, we consider all local meters that exist in
a piece. A more elaborate explanation of the IMA including
examples can be found in [28].
211
3. IMA BASED METER DETECTION
In this section we will outline the P RIMA model in a
bottom-up fashion. We start with the input MIDI data, and
0
4
(3)
3
6
{
(2)
(2)
{m2M (`):t2ext(m)}
3
4
NSW
The spectral weight is calculated in a similar fashion,
but for the spectral weight each local meter is extended
throughout the entire piece. The idea behind this is that the
metrical structure induced by the onsets stretches beyond
the region in which onsets occurs. The extension of a the
local meter m is defined as ext(ms,d,k ) = {s + id, 8i}
where i is an integer number. For all discrete metrical positions t, regardless whether it contains an onset or not, the
spectral weight is defined as follows:
(1)
NSW bin 3
{m2M (`):o2m}
{
p
km
.
NSW
X
NSW bin 0
W`,p (o) =
IMA
using Inner Metric Analysis, ISMIR 2016
NSW bin 4
NSW bin 6
Figure 1. The construction of NSW profiles for a piece in
2
4 : (1) displays IMA , (2) displays the NSW profiles derived
from IMA for a 24 and a 34 meter, and (3) shows how two
bins are selected from each profile and used to estimate the
probability of that meter. The ellipse represents the Gaussian distribution fitted to selected bins of the NSW profiles
in the training phase. Note that the 34 NSW profile does not
3
Automatic meter detection
at least `, then the metric weight of
defined as follows:
=
X
p
km
.
{
use the standard parameters p = 2,
consider all local meters that exist in
rate explanation of the IMA including
d in [28].
ED METER DETECTION
3
4
0
4
(3)
3
6
{
(2)
(2)
{m2M (`):t2ext(m)}
212
2
4
NSW
ht is calculated in a similar fashion,
weight each local meter is extended
piece. The idea behind this is that the
uced by the onsets stretches beyond
nsets occurs. The extension of a the
ned as ext(ms,d,k ) = {s + id, 8i}
number. For all discrete metrical poshether it contains an onset or not, the
ned as follows:
(1)
NSW bin 3
{m2M (`):o2m}
(1)
{
p
km
.
NSW
X
NSW bin 0
)=
IMA
Conditional probability a certain meter, given a set of onsets
NSW bin 4
NSW bin 6
Figure 1. The construction of NSW profiles for a piece in
2
4 : (1) displays IMA , (2) displays the NSW profiles derived
from IMA for a 24 and a 34 meter, and (3) shows how two
bins are selected from each profile and used to estimate the
probability of that meter. The ellipse represents the Gaussian distribution fitted to selected bins of the NSW profiles
Try out yourself?
!  Java tool: JMetro
!  Visualizes metric weights
!  Displays all local meters
!  Tutorial
!  Send email to me!
213
Literature
n 
n 
n 
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214
Elaine Chew, Anja Volk (Fleischer), and Chia-Ying Lee, Dance Music Classification Using Inner
Metric Analysis: a computational approach and case study using 101 Latin American Dances and
National Anthems. in The Next Wave in Computing, Optimization and Decision Technologies:
Proceedings of the 9th INFORMS Computer Society Conference. Bruce Golden, S. Raghavan,
Edward Wasil (eds.) Kluwer, 2005.
Haral Krebs: Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann,
Oxford University Press, 1999.
Fred Lerdahl & Ray Jackendoff: Generative Theory of Tonal Music, MIT Press, 1983.
Anja Volk (2008), The Study of Syncopation using Inner Metric Analysis: Linking Theoretical and
Experimental Analysis of Metre in Music, In: Journal of New Music Research, Vol. 37:4, p.
259-273.
Anja Volk (2008), Persistence and Change: Local and Global Components of Metre Induction
using Inner Metric Analysis, In: Journal of Mathematics and Computation in Music, Vol. 2:2, p.
99-115.
Maury Yeston: The Stratification of Musical Rhythm, Yale University Press,1976.

Symbolic music features: Rhythm and meter

Transcription

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