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Meinard Müller, Christof Weiss, Stefan Balke
Harmony Analysis
International Audio Laboratories Erlangen{meinard.mueller, christof.weiss, stefan.balke}@audiolabs-erlangen.de
TutorialAutomatisierte Methoden der Musikverarbeitung47. Jahrestagung der Gesellschaft für Informatik
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Book: Fundamentals of Music Processing
Meinard MüllerFundamentals of Music ProcessingAudio, Analysis, Algorithms, Applications483 p., 249 illus., hardcoverISBN: 978-3-319-21944-8Springer, 2015
Accompanying website: www.music-processing.de
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Book: Fundamentals of Music Processing
Meinard MüllerFundamentals of Music ProcessingAudio, Analysis, Algorithms, Applications483 p., 249 illus., hardcoverISBN: 978-3-319-21944-8Springer, 2015
Accompanying website: www.music-processing.de
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Dissertation: Tonality-Based Style Analysis
Christof WeißComputational Methods for Tonality-Based Style Analysis of Classical Music Audio RecordingsPhD thesis, Technical University of Ilmenau, 2017
Chapter 5: Analysis Methods for Key and Scale StructuresChapter 6: Design of Tonal Features
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Recall: Chroma Representations
Salie
nce
/ Lik
elih
ood
L. van Beethoven,Fidelio, Overture,Slovak Philharmonic
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Recall: Chroma Representations
Orchestra
Piano
L. van Beethoven,Fidelio, Overture,Slovak Philharmonic
Fidelio, Overture,arr. Alexander ZemlinskyM. Namekawa, D.R. Davies, piano four hands
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Recall: Chroma Representations
Orchestra
L. van Beethoven,Fidelio, Overture,Slovak Philharmonic
Gómez, Tonal Description of Polyphonic Audio, PhD thesis, Barcelona 2006
Müller / Ewert, Towards Timbre-Invariant Audio Features for Harmony-Based Music, IEEE TASLP, 2010
Mauch / Dixon, Approximate Note Transcription for the Improved Identification of Difficult Chords, ISMIR 2010
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Tonal Structures
ChordsCM GM7 Am
Global key
Local keyC major G major C major
Global key detection
Chord recognition
Music transcriptionNote level
Segment level
Chord level
Movement level C major
MelodyMiddle voices
Bass line
Local key detection
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Chord Recognition
Chord recognition
Typically: Feature extraction, pattern matching, filtering
“Out-of-the-box” solutions
Sonic Visualizer, Chordino Vamp Plugin(Queen Mary University of London)
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Chord Recognition
C G
Audiorepresentation
Prefiltering▪ Compression▪ Overtones▪ Smoothing
▪ Smoothing▪ Transition▪ HMM
Chromarepresentation
Patternmatching
Recognitionresult
Postfiltering
Majortriads
Minortriads
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Time (seconds)
C G Am F C G F C
Chord Recognition
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Chord Recognition: Basics
B
A
G
FE
D
C
G♯/A♭
D♯/E♭
C♯/D♭
A♯/B♭
F♯/G♭
C D♭ D E♭ E F G♭ G A♭ A B♭ B
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Chord Recognition: Basics
B
A
G
FE
D
C
G♯/A♭
D♯/E♭
C♯/D♭
A♯/B♭
F♯/G♭
Cm C♯m Dm E♭m Em Fm F♯m Gm G♯m Am B♭m Bm
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Chroma vectorfor each audio frame
24 chord templates(12 major, 12 minor)
Compute for each frame thesimilarity of the chroma vector
to the 24 templates
B
A
G
F
E
D
C
G♯
D♯
C♯
A♯
F♯
C C♯ D … Cm C♯m Dm
0 0 0 … 0 0 0 …
0 0 0 … 0 0 0 …
0 0 1 … 0 0 1 …
0 1 0 … 0 1 0 …
1 0 0 … 1 0 0 …
0 0 1 … 0 0 0 …
0 1 0 … 0 0 1 …
1 0 0 … 0 1 0 …
0 0 0 … 1 0 0 …
0 0 1 … 0 0 1 …
0 1 0 … 0 1 0 …
1 0 0 … 1 0 0 …
…
Chord Recognition: Template Matching
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Cho
rdP
itch
clas
s
Chord Recognition: Template Matching
Time (seconds)
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Chord Recognition: Label Assignment
Assign to each frame the chord labelof the template that maximizes the
similarity to the chroma vector
Chroma vectorfor each audio frame
24 chord templates(12 major, 12 minor)
Compute for each frame thesimilarity of the chroma vector
to the 24 templates
B
A
G
F
E
D
C
G♯
D♯
C♯
A♯
F♯
C C♯ D … Cm C♯m Dm
0 0 0 … 0 0 0 …
0 0 0 … 0 0 0 …
0 0 1 … 0 0 1 …
0 1 0 … 0 1 0 …
1 0 0 … 1 0 0 …
0 0 1 … 0 0 0 …
0 1 0 … 0 0 1 …
1 0 0 … 0 1 0 …
0 0 0 … 1 0 0 …
0 0 1 … 0 0 1 …
0 1 0 … 0 1 0 …
1 0 0 … 1 0 0 …
…
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Chord Recognition: Label Assignment
Cho
rdC
hord
Time (seconds)
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Chord Recognition: Evaluation
Time (seconds)
C G Am F C G F C
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Chord Recognition: Evaluation Frame-wise approach: Too many / too rapid changes of output label
Improvement strategies:
Pre-filtering: Average chroma features over several frames (smoothing)
Evaluation on all Beatles songs
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
Smoothing length
F-m
easu
re
Binary templates
1 5 9 13 17 21 25
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Chord Recognition: Markov Chains Probabilistic model for sequential data
Markov property: Next state depends only on current state (no “memory”)
Consist of:
Set of states
State transition probabilities
Initial state probabilities C
FG
0.8
0.60.7
0.2
0.1
0.3
0.10.1
0.1
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Chord Recognition: Hidden Markov Models States as hidden variables
Consist of:
Set of states (hidden)
State transition probabilities
Initial state probabilities C
FG
0.8
0.60.7
0.2
0.1
0.3
0.10.1
0.1
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Chord Recognition: Hidden Markov Models States as hidden variables
Consist of:
Set of states (hidden)
State transition probabilities
Initial state probabilities
Observations (visible)
C
FG
0.8
0.60.7
0.2
0.1
0.3
0.10.1
0.1
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Chord Recognition: Hidden Markov Models States as hidden variables
Consist of:
Set of states (hidden)
State transition probabilities
Initial state probabilities
Observations (visible)
Emission probabilities
0.7
0.1
0.9 0.2
0.3
0.8
C
FG
0.8
0.60.7
0.2
0.1
0.3
0.10.1
0.1
0
00
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Chord Recognition: Hidden Markov Models Different algorithmic problems
Estimation problem: Given an observation sequence, determine the model parameters
„Training“ the HMM on given data
Baum-Welch algorithm (Expectation-Maximization)
Uncovering problem: Given the model, find the optimal hidden state sequence
Use Viterbi algorithm based on dynamic programming
Corresponds to chord estimation task!
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Chord Recognition: Hidden Markov Models Parameters: Transition probabilities Estimated from data
State αj
Sta
te α
i
Log
prob
abilit
y
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Chord Recognition: Hidden Markov Models Parameters: Transition probabilities Estimated from data
Log
prob
abilit
y
Major chords Minor chords
Maj
or c
hord
sM
inor
cho
rds
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Major chords Minor chords
Maj
or c
hord
sM
inor
cho
rds
Chord Recognition: Hidden Markov Models Parameters: Transition probabilities Transposition-invariant
Log
prob
abilit
y
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Chord Recognition: Hidden Markov Models Parameters: Transition probabilities Uniform transition matrix (only smoothing)
State αj
Sta
te α
i
Log
prob
abilit
y
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(a) Template Matching (frame-wise)
Chord Recognition: Evaluation Effect of HMM-based chord estimation and smoothing:
C Dm G C
Time (seconds)Time (seconds)
(b) HMM
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Chord Recognition: Evaluation Evaluation on all Beatles songs
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
Smoothing length
F-m
easu
re
binary
HMM
1 5 9 13 17 21 25
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Tonal Structures
ChordsCM GM7 Am
Global key
Local keyC major G major C major
Global key detection
Chord recognition
Music transcriptionNote level
Segment level
Chord level
Movement level C major
MelodyMiddle voices
Bass line
Local key detection
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Local Key Detection
Key as an important musical concept (“Symphony in C major”)
Modulations → Local approach
Diatonic Scales
Simplification of keys
Perfect-fifth relation
0 diatonic1# diatonic2b diatonic
Circle of fifths →
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Local Key Detection Example: J.S. Bach, Choral "Durch Dein Gefängnis" (Johannespassion) Score – Piano reduction
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Local Key Detection: Chroma Features Example: J.S. Bach, Choral "Durch Dein Gefängnis" (Johannespassion) Audio – Chroma features (Scholars Baroque Ensemble, Naxos 1994)
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Local Key Detection: Chroma Smoothing Summarize pitch classes over a certain time
Chroma smoothing Parameters: blocksize b and hopsize h
bb
bh
h
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Local Key Detection: Chroma Smoothing Choral (Bach)
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Local Key Detection: Chroma Smoothing Choral (Bach) — smoothed with b = 42 seconds and h = 15 seconds
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Local Key Detection: Diatonic Scales Choral (Bach) — Re-ordering to perfect fifth series
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Local Key Detection: Diatonic Scales Choral (Bach) — Re-ordering to perfect fifth series
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation (7 fifths)
4#
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation (7 fifths)
5#
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation: Multiply chroma values*
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation: Multiply chroma values
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation
4 #(E major)
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Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation: Shift to global key
4 #(E major)
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Local Key Detection: Diatonic Scales Choral (Bach) — 0 ≙ 4#
Weiss / Habryka, Chroma-Based Scale Matchingfor Audio Tonality Analysis, CIM 2014
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Local Key Detection: Examples L. v. Beethoven – Sonata No. 10 op. 14 Nr. 2, 1. Allegro — 0 ≙ 1
(Barenboim, EMI 1998)
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Local Key Detection: Examples R. Wagner, Die Meistersinger von Nürnberg, Vorspiel — 0 ≙ 0
(Polish National Radio Symphony Orchestra, J. Wildner, Naxos 1993)