2017 weiss lecturemusicprocessing harmonyanalysis · harmonic figuration: broken chords (arpeggio)...

Music Processing

Christof Weiß and Meinard Müller

Lecture

Harmony Analysis

International Audio Laboratories Erlangen{christof.weiss,meinard.mueller}@audiolabs-erlangen.de

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

5.1 Basic Theory of Harmony5.2 Template-Based Chord Recognition5.3 HMM-Based Chord Recognition5.4 Further Notes

In Chapter 5, we consider the problem of analyzing harmonic properties of apiece of music by determining a descriptive progression of chords from a givenaudio recording. We take this opportunity to first discuss some basic theory ofharmony including concepts such as intervals, chords, and scales. Then,motivated by the automated chord recognition scenario, we introducetemplate-based matching procedures and hidden Markov models—a conceptof central importance for the analysis of temporal patterns in time-dependentdata streams including speech, gestures, and music.

Chapter 5: Chord Recognition

Dissertation: Tonality-Based Style Analysis

Christof WeißComputational Methods for Tonality-Based Style Analysis of Classical Music Audio RecordingsPhD thesis, Ilmenau University of Technology, 2017

Chapter 5: Analysis Methods for Key and Scale StructuresChapter 6: Design of Tonal Features

Chromatic circle Shepard’s helix of pitch

Recall: Chroma Features Human perception of pitch is periodic Two components: tone height (octave) and chroma (pitch class)

Time (seconds)

Chr

oma

Recall: Chroma Features

Sal

ienc

e/

Like

lihoo

d

Harmony Analysis: Overview

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

Western music (and most other music): Different aspects of harmony Referring to different time scales

Chord Recognition

Source: www.ultimate-guitar.com

Time (seconds)

Chord Recognition

Chord Recognition

Chord Recognition

C G

Audiorepresentation

Prefiltering

▪ Compression▪ Overtones▪ Smoothing

▪ Smoothing▪ Transition▪ HMM

Chromarepresentation

Patternmatching

Recognitionresult

Postfiltering

Majortriads

Minortriads

Chord Recognition: Basics Musical chord: Group of three or more notes

Combination of three or more tones which sound simultaneously

Types: triads (major, minor, diminished, augmented), seventh chords…

Here: focus on major and minor triads

C

→ C Major (C)

Chord Recognition: Basics Musical chord: Group of three or more notes

Combination of three or more tones which sound simultaneously

Types: triads (major, minor, diminished, augmented), seventh chords…

Here: focus on major and minor triads

Enharmonic equivalence: 12 different root notes possible → 24 chords

C Major (C)

C Minor (Cm)

Chord Recognition: BasicsChords appear in different forms:

Inversions

Different voicings

Harmonic figuration: Broken chords (arpeggio)

Melodic figuration: Different melody note (suspension, passing tone, …)

Further: Additional notes, incomplete chords

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

Templates: Major Triads

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

Templates: Minor Triads

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

Chord Recognition: Template Matching Similarity measure: Cosine similarity (inner product of normalized

vectors)

Chord template:

Chroma vector:

Similarity measure:

Cho

rdC

hrom

a

Chord Recognition: Template Matching

Time (seconds)

Chord Recognition: Label Assignment

Assign to each frame the chord labelof the template that maximizes the

similarity to the chroma vector

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 …

…Chroma vector

for each audio frame24 chord templates(12 major, 12 minor)

Compute for each frame thesimilarity of the chroma vector

to the 24 templates

Chord Recognition: Label Assignment

Cho

rdC

hord

Time (seconds)

Chord Recognition: Evaluation

Time (seconds)

C G Am F C G F C

Chord Recognition: Evaluation “No-Chord” annotations: not every frame labeled

Different evaluation measures:

Precision:

Recall:

F-Measure (balances precision and recall):

Without “No-Chord” label:

#TP#TP #FP

#TP#TP #FN

2 ⋅ ⋅

Chord Recognition: Smoothing Apply average filter of length ∈ :

Time (seconds)

Chord Recognition: 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

Markov Chains Probabilistic model for sequential data

Markov property: Next state only depends on current state(no “memory”)

Consist of:

Set of states (hidden)

State transition probabilities

Initial state probabilitiesC

FG

0.8

0.60.7

0.2

0.1

0.3

0.10.1

0.1

Markov ChainsNotation:

C

FG

0.8

0.60.7

0.2

0.1

0.3

0.10.1

0.1

A α1 α2 α3

α1

α2

α3


Initial state probabilities

C α1 α2 α3

States for ∈ 1:

for ∈ 1:

Markov Chains Application examples:

Compute probability of a sequence using given a model (evaluation)

Compare two sequences using a given model

Evaluate a sequence with two different models (classification)

C

FG

0.8

0.60.7

0.2

0.1

0.3

0.10.1

0.1

Hidden Markov Models

C

FG

0.8

0.60.7

0.2

0.1

0.3

0.10.1

0.1

States as hidden variables

Consist of:




Hidden Markov Models

C

FG

0.8

0.60.7

0.2

0.1

0.3

0.10.1

0.1

States as hidden variables

Consist of:




Observations (visible)

Hidden Markov Models States as hidden variables

Consist of:





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

Hidden Markov ModelsNotation:

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

A α1 α2 α3

α1

α2

α3

B β1 β2 β3

α1

α2

α3




C α1 α2 α3

States for ∈ 1:

Observation symbols for ∈ 1:

Markov Chains Analogon: the student‘s life

Consists of:



Initial state probabilities0.5

0.30.4

0.2

0.2

codingsleep

eatingsocialactivity

0.7

0.1

0.2

0.1 0.10.3 0

0.1

0.2

0.4

0.2

Hidden Markov Models Analogon: the student‘s life

Consists of:






0.5

0.30.4

0.2

0.2

codingsleep

eatingsocialactivity

0.7

0.1

0.2

0.1 0.10.3 0

0.1

0.2

0.4

0.2

0.9

Smell

NoiseLight

0.5

0.30.4

0.3

0.7

Hidden Markov Models Only observation sequence is visible!

Different algorithmic problems:

Evaluation problem Given: observation sequence and model

Calculate how well the model matches the sequence

Uncovering problem: Given: observation sequence and model

Find: optimal hidden state sequence

Estimation problem („training“ the HMM):

Given: observation sequence

Find: model parameters

Baum-Welch algorithm (Expectation-Maximization)

Uncovering problem Given: observation sequence , … , of length ∈ and

HMM (model parameters)

Find: optimal hidden state sequence ∗ ∗, … , ∗

Corresponds to chord estimation task!

Observation sequence

β1 β3 β1 β3 β3 β2

, , , , ,


β1 β3 β1 β3 β3 β2

, , , , ,

α1 α1 α1 α3 α3 α1

Hidden state sequence ∗ ∗ , ∗ , ∗ , ∗ , ∗ , ∗






β1 β3 β1 β3 β3 β2

, , , , ,

C C C G G C

Hidden state sequence ∗ ∗ , ∗ , ∗ , ∗ , ∗ , ∗





Uncovering problem Optimal hidden state sequence?

“Best explains” given observation sequence

Maximizes probability , Θ

Straight-forward computation (naive approach):

Compute probability for each possible sequence

Number of possible sequences of length ( number of states):

Prob∗ max , Θ

∗ argmax , Θ

· ·… ·

factorscomputationally infeasible!

Viterbi Algorithm Based on dynamic programming (similar to DTW)

Idea: Recursive computation from subproblems

Use truncated versions of observation sequence

Define , as the highest probability along a single state sequence, … , that ends in state

Then, our solution is the state sequence yielding

Prob∗ max∈ :

,

1: ≔ ,… , , length ∈ 1:

, max,… ,

1: , , … , , Θ

Viterbi Algorithm : matrix of size

Recursive computation of , along the column index

Initialization:

1

Truncated observation sequence: 1 Current observation:

, 1 ⋅ for some ∈ 1:



Recursion:

∈ 2:

Truncated observation sequence: 1: , … , Last observation:

, ⋅ ∗ ⋅ 1: 1 , , … , ∗ Θ for ∈ 1:

, ⋅ ∗ ⋅ ∗, 1

must be maximal!



Recursion:

∈ 2:

Truncated observation sequence: 1: , … , Last observation:

, ⋅ ∗ ⋅ 1: 1 , , … , ∗ Θ for ∈ 1:

, ⋅ ∗ ⋅ ∗, 1

must be maximal!

must be maximal (best index ∗)

, ⋅ max∈ :

⋅ , 1

Viterbi Algorithm given – find optimal state sequence ∗ ∗, … , ∗ ≔ ,… ,

Backtracking procedure (reverse order)

Last element:

Optimal state:

argmax∈ :

,



Further elements:

1, 2, … , 1 Optimal state:

argmax∈ :

⋅ ,



Further elements:

1, 2, … , 1 Optimal state:

Simplification of backtracking: Keep track of maximizing index in

Define 1 matrix :

argmax∈ :

⋅ ,

, ⋅ max∈ :

⋅ , 1

, 1 argmax∈ :

⋅ , 1

Viterbi AlgorithmSummary

1

1

2

3

4

5

6

7

8

…, 1

1

States∈ 1:

Sequenceindex ∈ 1:

Initialization


1

1

2

3

4

5

6

7

8

……

………

…

……

…

…

…

n – 1 n

… …

, 1

1

States∈ 1:


Initialization Recursion


1

1

2

3

4

5

6

7

8

……

………

…

……

…

…

…

n – 1 n

,

… …

, 1

1

States∈ 1:




1

1

2

3

4

5

6

7

8

……

………

…

……

…

…

…

n – 1 n

, 1

,

… …

, 1

1

States∈ 1:



……

…… …

……

…

…

…

…

…

……

…

…


1

2

3

4

5

6

7

8

n – 1 n N

, 1

,

,

… …

, 1

1

States∈ 1:


Initialization Recursion Termination

1

……

…… …

……

…

…

…

…

…

……

…

…


1

1

2

3

4

5

6

7

8

n – 1 n N

, 1

,

,

… …

, 1

1

States∈ 1:


Initialization Recursion Termination

Backtrackingmatrix

Viterbi Algorithm: Example

A α1 α2 α3

α1

α2

α3

B β1 β2 β3

α1

α2

α3

State transition probabilities Emission probabilities Initial state probabilities

C α1 α2 α3

Statesfor ∈ 1:

Observation symbolsfor ∈ 1:

HMM:


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:


β1 β3 β1 β3 β3 β2

O = (o1,o2,o3,o4,o5,o6)

Input


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1

α2

α3

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1

α2

α3


β1 β3 β1 β3 β3 β2

O = (o1,o2,o3,o4,o5,o6)

Input Viterbi algorithm


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1

α2

α3

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1

α2

α3

Viterbi algorithm

, 1 ⋅

Initialization


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1 0.4200α2 0.0200α3 0

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1

α2

α3

Viterbi algorithm

, ⋅ max∈ :

⋅ , 1

, 1 argmax∈ :

⋅ , 1

, 1 ⋅

Initialization

Recursion

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1 0.4200 0.1008 0.0564 0.0135 0.0033 0α2 0.0200 0 0.0010 0 0 0.0006α3 0 0.0336 0 0.0045 0.0022 0.0003

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1 1 1 1 1 1α2 1 1 1 1 3α3 1 3 1 3 3

Viterbi algorithm


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1

α2

α3

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1

α2

α3

Viterbi algorithm

,

argmax∈ :

,

Backtracking

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1 0.4200 0.1008 0.0564 0.0135 0.0033 0α2 0.0200 0 0.0010 0 0 0.0006α3 0 0.0336 0 0.0045 0.0022 0.0003

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1 1 1 1 1 1α2 1 1 1 1 3α3 1 3 1 3 3

i6 = 2

Viterbi algorithm

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1 0.4200 0.1008 0.0564 0.0135 0.0033 0α2 0.0200 0 0.0010 0 0 0.0006α3 0 0.0336 0 0.0045 0.0022 0.0003

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1 1 1 1 1 1α2 1 1 1 1 3α3 1 3 1 3 3

Viterbi algorithm


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:

D o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1

α2

α3

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1

α2

α3

Viterbi algorithm

,

argmax∈ :

,

Backtracking


A α1 α2 α3

α1 0.8 0.1 0.1α2 0.2 0.7 0.1α3 0.1 0.3 0.6

B β1 β2 β3

α1 0.7 0 0.3α2 0.1 0.9 0α3 0 0.2 0.8

C α1 α2 α3

0.6 0.2 0.2


Statesfor ∈ 1:


HMM:

Optimal state sequenceD o1= β1 o2= β3 o3= β1 o4= β3 o5= β3 o6= β2

α1 0.4200 0.1008 0.0564 0.0135 0.0033 0α2 0.0200 0 0.0010 0 0 0.0006α3 0 0.0336 0 0.0045 0.0022 0.0003

E o1= β1 o2= β3 o3= β1 o4= β3 o5= β3

α1 1 1 1 1 1α2 1 1 1 1 3α3 1 3 1 3 3

i6 = 2


β1 β3 β1 β3 β3 β2

S* = (α1,α1,α1,α3,α3,α2)O = (o1,o2,o3,o4,o5,o6)

Input OutputViterbi algorithm

(a) Template Matching (frame-wise)

HMM: Application to Chord Recognition Effect of HMM-based chord estimation and smoothing:

C Dm G C

Time (seconds)Time (seconds)

(b) HMM

HMM: Application to Chord Recognition Parameters: Transition probabilities Estimated from data

State αj

Sta

te α

i

Log

prob

abilit

y

Parameters: Transition probabilities Estimated from data

Log

prob

abilit

y

Major chords Minor chords

Maj

or c

hord

sM

inor

cho

rds

HMM: Application to Chord Recognition

Major chords Minor chords

Maj

or c

hord

sM

inor

cho

rds

HMM: Application to Chord Recognition Parameters: Transition probabilities Transposition-invariant

Log

prob

abilit

y

HMM: Application to Chord Recognition Parameters: Transition probabilities Uniform transition matrix (only smoothing)

State αj

Sta

te α

i

Log

prob

abilit

y

HMM: Application to Chord Recognition 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

A

Cm

Em

Am

C

C

E G

E♭

B

E G

Em

C

B

CCmaj7

Chord Recognition: Further Challenges Chord ambiguities

Acoustic ambiguities (overtones)

Use advanced templates (model overtones, learned templates)

Enhanced chroma (logarithmic compression, overtone reduction)

Tuning inconsistency

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

Local Key Detection

G

D

E

BD♭

E♭

F

A

G♭F♯

A♭

B♭

E ♭mD♯m

B♭mFm

Cm

Gm

Dm

G♯mC♯m

F♯m

Bm

EmAm

C

♭ ♯

Key as an important musical concept (“Symphony in C major”)

Modulations → Local approach

Key relations: Circle of fifth

Local Key Detection

Key as an important musical concept (“Symphony in C major”)

Modulations → Local approach

Diatonic Scales

Simplification of keys

Perfect-fifth relation

1# diatonic

Circle of fifths →

Local Key Detection Example: J.S. Bach, Choral "Durch Dein Gefängnis" (Johannespassion) Score – Piano reduction

Local Key Detection Example: J.S. Bach, Choral "Durch Dein Gefängnis" (Johannespassion) Audio – Waveform (Scholars Baroque Ensemble, Naxos 1994)

Time (seconds)

Local Key Detection: Chroma Features Example: J.S. Bach, Choral "Durch Dein Gefängnis" (Johannespassion) Audio – Chroma features (Scholars Baroque Ensemble, Naxos 1994)

Local Key Detection: Chroma Smoothing Summarize pitch classes over a certain time

Chroma smoothing Parameters: blocksize b and hopsize h

bb

bh

h

Local Key Detection: Chroma Smoothing Choral (Bach)

Local Key Detection: Chroma Smoothing Choral (Bach) — smoothed with b = 4.2 seconds and h = 1.5 seconds

Local Key Detection: Diatonic Scales Choral (Bach) — Re-ordering to perfect fifth series

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation (7 fifths)

4#

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation (7 fifths)

5#

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation: Multiply chroma values*

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation: Multiply chroma values

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation

4 #(E major)

Local Key Detection: Diatonic Scales Choral (Bach) — Diatonic Scale Estimation: Shift to global key

4 #(E major)

Local Key Detection: Diatonic Scales Choral (Bach) — 0 ≙ 4#

Weiss / Habryka, Chroma-Based Scale Matchingfor Audio Tonality Analysis, CIM 2014

Local Key Detection: Examples L. v. Beethoven – Sonata No. 10 op. 14 Nr. 2, 1. Allegro — 0 ≙ 1

(Barenboim, EMI 1998)

Local Key Detection: Examples R. Wagner, Die Meistersinger von Nürnberg, Vorspiel — 0 ≙ 0

(Polish National Radio Symphony Orchestra, J. Wildner, Naxos 1993)

2017 weiss lecturemusicprocessing harmonyanalysis · harmonic figuration: broken chords (arpeggio)...

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