Analysis of Music Data -...
Transcript of Analysis of Music Data -...
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Analysis of Music Data
Florian Schoppmann
Department of Computer Science
International Graduate School Dynamic Intelligent SystemsUniversity of Paderborn
July 17, 2007
University of Paderborn Florian Schoppmann · 1 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Analysis of Music DataGoals:
I Measuring Similarity of Music DataI Memory researchI Music psychologyI Music analysis
(“ethnomusicology”)I Copyright issues
I Extracting high-level informationfrom general audio signals
I Restoration of musical sourcesI Music transcription
I Sound characteristics of orchestra instrumentsI classicifaction of the register of an instrument by timbre
(and not pitch)
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Music Representation
Symbolically,e.g., also MIDI:
Audio Signals, e.g., in AIFF, MP3, etc.:
MIDI Parameters: Note, Velocity,Modulation, Balance, etc.
University of Paderborn Florian Schoppmann · 3 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Technical Terms
Pitch Perceived fundamental frequency of a sound
Fundamental Frequency f0 lowest frequency in a harmonic series
Harmonic Series (Besides maths. . . ) Multiples of f0Overtone/Partial sinusoidal component of a waveform, of greater
frequency than f0Register Relative “height” or range of a note, set of pitches,
melody, instrument, etc.
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Example
Sine wave at 440 Hz (Duration 100 ms): Play
First 100ms of piano A4: Play
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Spectrum of Sine Wave
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Spectrum of Piano A4
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Disclaimer :-)
I am not an expert in any of the following fields
I signal processing
I musicology
I statistics
Everything presented is to the best of my knowledge
Technical details sometimes omitted. Hopefully:
I Discussion about the overall picture
I Not too many mistakes
University of Paderborn Florian Schoppmann · 8 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Similarity of Melodies
Klaus Frieler (2006):Generalized N-gram Measures for Melodic Similarity.In: Data Science and Classification, Springer, pp. 289–298.http://dx.doi.org/10.1007/3-540-34416-0_31
Abstraction needed: Melody as Time Series
DefinitionLet j ≤ k ∈ N. A sequence φ over some set E is a map
φ : [j : k] → Ei 7→ φi .
φ is in normal form if j = 0.
University of Paderborn Florian Schoppmann · 9 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Melody Abstraction
DefinitionLet φ = (φi )i∈[j :k] be a sequence. An n-gram of φ is a subsequence(φi )i∈[l :m] where m − l + 1 = n and i ≤ l ≤ m ≤ k .
DefinitionLet E be an event space. A sequence µ : [0 : n − 1] → R× E,i 7→ µi =: (ti , pi ), is called melody if
i < j =⇒ ti < tj .
Assumption here: Onset and pitch are sufficient to capture the“essence” of a melody
University of Paderborn Florian Schoppmann · 10 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Similarity Measures (1/2)
Basic Choice about what to measure:
I DissimilarityMore common to mathematics (distance measures, metrics)
I SimilarityJudging similarity far more familiar for musical experts
DefinitionLet M be a set of melodies. A similarity map σ is a mapσ : M×M→ [0, 1] such that the following is fulfilled:
I Symmetry: σ(µ, µ′) = σ(µ′, µ)
I Self-identity: σ(µ, µ) = 1 and σ(ε, µ) = 0
(We denote by ε the empty melody.)
University of Paderborn Florian Schoppmann · 11 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Similarity Measures (2/2)
Further requirement for melodies: Invariance under
I (Pitch) transposition,
I Time Shift,
I Tempo Change.
General idea in the following:
I Counting common n-grams
University of Paderborn Florian Schoppmann · 12 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Further Preliminary Definitions
Let φ = (φi )i∈[j :k] be a sequence. Then:
I | · | is the length operator,
I · is the normalizing operator
such that |φ| = k − j + 1 and φi = φi+j .
For any n ∈ N:
Sn(φ) := r | r is n-gram of φ and
Sn(φ) := r | r is n-gram of φ
The frequency of an n-gram r with respect to φ is defined as:
fφ(r) = |u ∈ S|r |(φ) | r = u|
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
n-Gram Measures (1/2)
Definition1. The Count-Distinct measure (CDM) is the count of n-grams
common to both sequences:
SCDMn (φ, ϕ) := |Sn(φ) ∩ Sn(ϕ)|
2. The Sum-Common measure (SCM) is the sum of frequenciesof n-grams common to both sequences:
SSCMn (φ, ϕ) :=
∑r∈ bSn(φ)∩ bSn(ϕ)
[fφ(r) + fϕ(r)
]Normalization to get similarity measure:
σCDMn (φ, ϕ) :=
SCDMn (φ, ϕ)
12 ·
[|Sn(ϕ)|+ |Sn(φ)|
] σSCMn (φ, ϕ) :=
SSCMn (φ, ϕ)
|φ|+ |ϕ| − 2(n − 1)
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
n-Gram Measures (2/2)
Definition (continued)3. The Ukkonen measure (UM) counts the absolute differences of
frequencies of all distinct n-grams of both sequences:
SUMn (φ, ϕ) :=
∑r∈ bSn(φ)∪ bSn(ϕ)
|fφ(r)− fϕ(r)|
Transform into similarity measure:
σUMn (φ, ϕ) := 1− SUM
n (φ, ϕ)
|φ|+ |ϕ| − 2(n − 1)
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Example: Similarity of Melodies (1/2)
C major and C minor scale:
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Music engraving by LilyPond 2.10.25—www.lilypond.org
Play
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Music engraving by LilyPond 2.10.25—www.lilypond.org
Play
According to invariance w.r.t. transposition, tempo:
φ = (2, 2, 1, 2, 2, 2, 1), ϕ = (2, 1, 2, 2, 1, 2, 2)
University of Paderborn Florian Schoppmann · 16 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Example: Similarity of Melodies (2/2)
Two 6-grams for each melody:
r1 = (2, 2, 1, 2, 2, 2), r2 = (2, 1, 2, 2, 2, 1)
s1 = (2, 1, 2, 2, 1, 2), s2 = (1, 2, 2, 1, 2, 2)
We get σ?n(φ, ϕ) = 0 for all measures.
Consider only first 6 tones and 4-grams:
φ = (2, 2, 1, 2, 2), ϕ = (2, 1, 2, 2, 1)
Two 4-grams for each melody:
r1 = (2, 2, 1, 2), r2 = (2, 1, 2, 2)
s1 = (2, 1, 2, 2), s2 = (1, 2, 2, 1)
We get σ?n(φ, ϕ) = 1
2 for all measures.University of Paderborn Florian Schoppmann · 17 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Generalized Similarity Measures
Definitions of S?n (φ, ϕ) can be rewritten in terms of frequencies:
SCDMn (φ, ϕ) := |Sn(φ) ∩ Sn(ϕ)| =
∑r∈Sn(φ)
1Sn(ϕ)(r) ·1
fφ(r)
SSCMn (φ, ϕ) :=
∑r∈ bSn(φ)∩ bSn(ϕ)
[fφ(r) + fϕ(r)
]=
∑r∈Sn(φ)
1Sn(ϕ)(r) ·(
1 +fϕ(r)
fφ(r)
)...
Generalize the notion of frequency. For a similarity measure σ, let
νφ(r) :=∑
u∈S|r|(φ)
σ(u, r) ≥ |u ∈ S|r |(φ) | r = u| = fφ(r) .
University of Paderborn Florian Schoppmann · 18 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Excursion: Edit-Distance
Edit-Distance d(φ, ϕ) between two sequences in normal form:I Defined as number of deletion, insertion, and substitution steps
Typical example of dynamic programming:ε S a t u r d a y
ε 0 1 2 3 4 5 6 7 8S 1 0 1 2 3 4 5 6 7u 2 1 1 2 2 3 4 5 6n 3 2 2 2 3 3 4 5 6d 4 3 3 3 3 4 3 4 5a 5 4 3 4 4 4 4 3 4y 6 5 4 4 5 5 5 4 3
Recurrence relation for i ∈ [0 : |φ|], j ∈ [0 : |ϕ|]:
D(i , j) := min
D(i − 1, j) + 1,D(i , j − 1) + 1,D(i − 1, j − 1) + δ(φi , ϕj)
D(i , 0) := iD(0, j) := j
University of Paderborn Florian Schoppmann · 19 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Summary and Criticism
Summary:
I Goal: Similarity of monophonic melodies
I Method: Melodies as time series
I Generalizing measures based on “n-grams”, i.e.,subsequences of fixed length
Criticism (→ discussions? :-))
I Why only n-grams for arbitrary (but fixed) n’s?
I Generalization unmotivated and “arbitrary”
I burdensome yet still imprecise mathematical formalisms(tried to mitigate here)
University of Paderborn Florian Schoppmann · 20 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Evaluating Similarity Approaches
Daniel Müllensiefen, Klaus Frieler (2006):Evaluating Different Approaches to Measuring the Similarity ofMelodies.In: Data Science and Classification, Springer, pp. 299–306.http://dx.doi.org/10.1007/3-540-34416-0_32
Several studies to comparison similarity measurement for melodies:
I “Very different similarity values”
I “not clear which one is the most adequate”
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
General Idea of Similarity Algorithms1. Basic transformations (representations)
I Projections: E.g., to pitch componentI Differentiation
2. Main transformations, e.g.:I Rhythmical weighting: Assume melody is quantized, replace
pitch of duration n · T , n > 1, by sequence of n tone withduration T
I Contourization: Idea that perceptionally important notes arethe extrema→ substitute pitches in between by linear interpolation
3. Computation
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Experimental Evaluation
Subjects: Musicology students with longtime musical experience
1st test: Similarity of 84 melody pairs with constructed errors
I Generated from 14 original “western popular” melodies
I Rhythm errors, Pitch errors, etc.
2nd and 3rd test: Also completely different melodies
Only let subjects with stable judgement continue:
I 1st test: 23 out of 82
I 2nd test: 12 out of 16
I 3rd test: 5 out of 10
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Results
Homogeneity of human similarity judgement:
I High correlation between judgements of selected subjects
I Hypothesis: Objective similarity at least for “western” experts
I “Conceptual Foundation for statistical modeling”
Human notion of similarity is “adaptive”:
I 1st test with only slightly altered melodies: Pitch informationsufficient
I 2nd test: When melodies are different, subjects’ ratings bestmodeled by including rhythmical information
“Unrelated melodies that differ strongly [. . . ] are hard to relate”. :-)
University of Paderborn Florian Schoppmann · 24 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Estimate Pitch in General Audio Data
Katrin Sommer, Claus Weihs (2006):Using MCMC as a Stochastic Optimization Procedure forMusical Time Series.In: Data Science and Classification, Springer, pp. 307-314.http://dx.doi.org/10.1007/3-540-34416-0_33
Pitch estimation of monophonic sound by joint estimation ofovertones
I Based on a model by Davy and Godsill (2002)
I Estimating parameters of the model requires computingmulti-dimensional integrals
I An MCMC (Monte Carlo Markov Chain) algorithm is used
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Harmonic Model (1/2)
Basic model:
yt =H∑
h=1
ah,t cos(2πhf0t) + bh,t sin(2πhf0t)) + εt
where yt is the instantaneous amplitude at time t.
Idea:
I Tone is composed out of harmonics from H partial tones
I First partial is fundamental frequency f0I Remaining H − 1 partials are called overtones
I Amplitudes of each partial tone are time dependent
I εt is model error
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Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Harmonic Model (2/2)
Estimating the parameters of the former model requires computinga multidimensional integral of the form∫
Ωf (θ, f0, H, σ2
ε)p(θ, f0, H, σ2ε | y) dθ df0 dH dσ2
ε .
Standard numerical techniques generally inaccurate or too slow
I Therefore, use a Metropolis-Hastings MCMC algorithm
I Sufficient for this talk: Generalization of Monte CarloIntegration
University of Paderborn Florian Schoppmann · 27 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Mini-Excursion: Monte Carlo Integration
Wanted:
I =
∫ 1
0f (ω)dω
Let (Ui )i∈N be independent random variable with Ui ∼ U(0, 1).According to strong law of large numbers:
limn→∞
1n
n∑i=1
f (Ui ) = E[f (Ui )] = I
The probabilistic error bound decreases as 1/√
n.
Mainly suited for multi-dimensional integrals (Dimension = d):I Standard numerical approaches need exponentially many
samples in dI Monte Carlo approach: Error bound independent of d
University of Paderborn Florian Schoppmann · 28 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Summary and Criticism
Goal:
I Estimate pitch from general audio signals
I Davy and Godsill (2002): General model very successful
I New: Introduction of “technical” stochastical optimization
Criticism:
I Hardly (if at all) understandable without looking at originalpaper by Davy and Godsill (2002)
University of Paderborn Florian Schoppmann · 29 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Sound characteristics
Claus Weihs, Gero Szepannek, Uwe Ligges, Karsten Luebke,Nils Raabe (2006):Local Models in Register Classification by Timbre.In: Data Science and Classification, Springer, pp. 315–322.http://dx.doi.org/10.1007/3-540-34416-0_34
Follow-up paper to the following goal:
I Identify high and low musical register (Soprano, Alto vs.Tenor, Bass) by timbre
I After pitch information is eliminated from the spectrum
University of Paderborn Florian Schoppmann · 30 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
The Data
17 singers performing the song “Tochter Zion” (G.F. Händel)I Downsampled to 11,025Hz and standardized to interval [−1, 1]I Analyses are based on characteristics derived from tones
corresponding to single notes (→ suitable segmentation)I For identified notes: Derive pitch-idependent periodogram
Register Classification by Timbre 3
FF OT 1 OT 2 OT 3 OT 4 OT 5 OT 6 OT 7 OT 8 OT 9
0.0
00.0
10.0
2
Fig. 1. Pitch independent periodogram (professional bass singer).
violin-mv.Bass version: bassoon, bflute-flu, bflute-vib, cello-bv, elecbass1, elecbass5,
elecbass6, elecguitar1, elecguitar2, elecguitar4, frehorn, frehorn-m, marimba,piano-ld, piano-pl, piano-sft, tromb-ten, tromb-tenm, tuba, viola-mv.Thus, 28 high instruments and 20 low instruments were chosen together with10 high female singers and 7 male.
From the periodogram corresponding to each tone corresponding to anidentified note voice print characteristics are derived (cp. Weihs and Ligges(2003b)). For our purpose we only use the size and the shape correspondingto the first 13 partials, i.e. to the fundamental frequency and the first 12 over-tones, in a pitch independent periodogram (cp. Figure 1). In order to measurethe size of the peaks in the spectrum, the mass (weight) of the peaks of thepartials are determined as the sum of the percentage shares of those parts ofthe corresponding peak in the spectrum which are higher than a pre-specifiedthreshold. The shape of a peak cannot easily be described. Therefore, we onlyuse one simple characteristic of the shape, namely the width of the peak ofthe partials. The width of a peak is measured by the half tone distance be-tween the smallest and the biggest frequency of the peak with a spectralheight above a pre-specified threshold. Overall, every tone is characterizedby the above 26 characteristics which are used as a basis for classification.For details on the computation of the measures see Guttner (2001). Note thatpitch information is eliminated in that the frequencies corresponding to fun-damentals and overtones are ignored in the pitch independent periodogram.Mass is measured as a percentage (%), whereas width is measured in partsof halftones (pht). Figure 2 illustrates the voice print corresponding to thewhole song “Tochter Zion” for a particular singer. For masses and widthsboxplots are indicating variation over the involved tones. For the analysesof this paper we ignore halftone distance and formant intensity (cp. Weihsand Ligges (2003b)), and use the other characteristics of the voice print forindividual tones, as well as averaged characteristics over all involved tones,leading to only one value for each characteristic per singer or instrument.
3 Classification Methods
On these data we applied supervised classification methods (see, e.g., Michieet al. (1994)) trying to reproduce the pre-defined grouping by means of classi-
I Consider only size and shape of first 13 partials. Aftercropping Fourier frequencies below some threshold
I Mass: sum of the percentage sharesI Width: difference between max and min frequency
University of Paderborn Florian Schoppmann · 31 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Tochter Zion, freue Dich
Georg Friedrich Händel (1685-1759):
Georg Friedrich Händel (1685-1759)
Tochter Zion, freue Dich
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Music engraving by LilyPond 2.8.6 — www.lilypond.org
CC by-sa - http://creativecommons.org/licenses/by-sa/2.0/de/
University of Paderborn Florian Schoppmann · 32 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Classification
Common techniques: Linear Discriminant Analysis (LDA)I 2 · 13 = 26 Variables are generated for every note separatelyI Averaged over all notesI Yields one single value of mass and width per harmonic and
singer/instrument
Global Model:I Training set includes samples from all instruments
Local Mode:I Create own classification rules for each instrumentI Problem: Which classification rule to use?
I Maximum-posterior rule: k ∈ arg maxk(maxl pl(k | x))I Average-posterior rule: k ∈ arg maxk
∑l pl(k | x)
I Majority Voting: . . .University of Paderborn Florian Schoppmann · 33 / 34
Motivation Similarity Measures Evaluating Similarity Approaches Pitch Estimation Sound characteristics
Results
Classification of voices alone gives good results. Explanations:I Human mouth acts as a highpass filter
I The lower the tone the less the mass of the fundamentalfrequency compared to 1st overtone
I Therefore: Sopranos have more mass in the fundamental f.
I Local model yields improvements when there are instrumentsI Voice print of a professional bass singer:
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Halftone Distance
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Formant Intensity
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Figure 2: Voice print of professional bass singer.
distance between two Fourier frequencies is 5.38 Hertz.Overall, every tone is characterized by the above 26 characteristics which
are used as a basis for classification. For details on the computation of themeasures see Guttner (2001). Mass is measured as a percentage (%), whereaswidth is measured in Hertz.
Figure 2 illustrates the voice print corresponding to the whole song “Toch-ter Zion” for a particular singer. For masses and widths boxplots are indicat-ing variation over the involved tones. For the analyses of this paper we ignorehalftone distance and formant intensity (cp. Weihs and Ligges (2003)), anduse the other characteristics of the voice print for individual tones, as wellas averaged characteristics over all involved tones, leading to only one valuefor each characteristic per harmonic and singer or instrument.
3 Global modelling
Classification of the register of di!erent instruments and singers is performedusing two very common techniques: the classical linear discriminant analysis[LDA] (Fisher (1936)) as well as Classification trees (more specifically the
5
University of Paderborn Florian Schoppmann · 34 / 34