CS 591 S1 – Computational Audiosnyder/cs591/Lectures/Rhythm Detection.pdf · CS 591 S1 – Computational Audio ... Transcription (melody, chords, score) ... Note: “Tatum” was

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CS 591 S1 – Computational Audio

Lecture 18: Music Information Retrieval & Rhythm

Overview of Music Information Retrieval

Rhythm: Basic Notions

Onset Detection

Beat Detection

Tempo Estimation

Higher-level Rhythmic Patterns

Wayne Snyder Computer Science Department

Boston University

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Music Information Retrieval: Overview

Music Information Retrieval is an interdisciplinary science which attempts to extract interesting and useful information from musical signals using computational tools. Researchers from Electrical Engineering, Computer Science, Musicology, Psychology, and Mathematics apply a variety of techniques in their scientific study of music. Some Important Areas of Current Interest:

Automatic Feature Extraction (pitch, melody, harmony, rhythm, mood, genre, ...) Feature Tagging (automatic or manual “ground truth”) Beat Tracking and Rhythm Analysis Transcription (melody, chords, score) Multimodal Synchronization/Alignment Database Retrieval and search Fingerprinting and classification Similarity Structure Analysis Performance Analysis �

Applications include many software tools both for the professional community and for the consumer market.

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Music Information Retrieval: Rhythm

A good place to start with with Rhythm:

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We could usefully divide the subject of rhythm into a hierarchy of levels, from the fastest to the slowest divisions of time. The basic beat is called the Tactus � this is what most people would tap their foot to.

Music Information Retrieval: Rhythm

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Further Examples:

If I Had You (Benny Goodman) Shakuhachi Flute Liszt: Sonetto No. 104 Del Petrarca

Where is the beat? Can you tap your foot to it? What is the meter? How to find the underlying regular beat which is being varied by the composer and/or performer for expressive effect?

Rhythm Analysis: Introduction

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Further Examples:

If I Had You (Benny Goodman) Shakuhachi Flute Liszt: Sonetto No. 104 Del Petrarca

Where is the beat? Can you tap your foot to it? What is the meter? How to find the underlying regular beat which is being varied by the composer and/or performer for expressive effect?


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Even when rhythm is regular, there is a complicated semantic

problem: rhythm is hierarchical, consisting of many

interrelated groupings:

Pulse level: Measure

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Pulse level: Tactus (beat)

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Example: Happy Birthday to you

Pulse level: Tatum (fastest unit of division)

Note: “Tatum” was named after Art Tatum, one of the greatest of all jazz pianists, who played a lot of fast notes!

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In a sophisticated piece of music, these various levels are exploited

by the composer in complicated ways. How should it be notated

and described precisely? What is the time signature? Example:

Bach, WTC, Fugue #1 in C Major


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§  Hierarchical levels often unclear

§  Global/slow tempo changes (all musicians do this!)

§  Local/sudden tempo changes (e.g. rubato)

§  Vague information

(e.g., soft onsets, false positives )

§  Sparse information: not all beats occur!

(often only note onsets are used)

Challenges in beat tracking

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§  Onset detection §  Beat tracking§  Tempo estimation

Tasks

Introduction

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Tasks

Tasks in Rhythm Analysis

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period phase


Tasks


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Tempo := 60 / period Beats per minute (BPM)


Tasks

period


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Onset Detection

§  Finding start times of perceptually relevant acoustic events in music signal

§  Onset is the time position where a note is played

§  Onset typically goes along

with a change of the signal�s properties: §  energy or loudness §  pitch or harmony §  timbre

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Onset Detection

[Bello et al., IEEE-TASLP 2005]

§  Finding start times of perceptually relevant acoustic events in music signal

§  Onset is the time position where a note is played

§  Onset typically goes along

with a change of the signal�s properties: –  energy or loudness –  pitch or harmony –  timbre

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Steps

Time (seconds)

Waveform

Onset Detection (Amplitude or Energy-Based)

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Time (seconds)

Squared waveform

Steps 1.  Amplitude squaring (full-wave rectification of power signal)

Onset Detection (Amplitude or Energy-Based)

Computer Science Onset Detection (Amplitude or Energy-Based)

Time (seconds)

Steps 1.  Amplitude squaring (full-wave rectification of power signal) 2.  Windowing (taking mean or max in each window): “energy

envelope”

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Computer Science Onset Detection (Energy-Based)

Time (seconds)

Steps 1.  Amplitude squaring (full-wave rectification of power signal) 2.  Windowing (taking mean or max in each window) : “energy

envelope” 3.  Difference Function (using appropriate Distance Function):

captures changes in signal energy: “novelty curve.”


Time (seconds)

Steps 1.  Amplitude squaring (full-wave rectification of power signal) 2.  Windowing (taking mean or max in each window) : “energy

envelope” 3.  Difference Function (using appropriate Distance Function):

captures changes in signal energy: “novelty curve.” 4.  Half-wave Rectification (negative samples => 0.0): note onsets are

indicated by increases in energy only.

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Time (seconds)

Steps 1.  Amplitude squaring 2.  Windowing 3.  Differentiation 4.  Half wave rectification 5.  Peak picking

Peak positions indicate note onset candidates

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Energy based methods work well for percussive instruments, including piano:

Example: Bach Well-Tempered Clavier, Book 1,

Fugue #1 in C major (Glenn Gould)

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Onset Detection

§  Energy curves often only work for percussive music

§  Many instruments have weak note onsets: wind, strings, voice. §  Example: Shakuhachi Flute

§  Biggest problem: pitch or timbre changes (corresponding to note onset) may not correlate with energy changes, e.g., a singer may change the loudness without changing pitch/note, or change pitch/note without appreciable change in loudness.

§  More refined methods needed that capture changes in energy spread over the spectrum [Bello et al., IEEE-TASLP 2005]

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1.  Spectrogram Magnitude spectrogram

Freq

uenc

y (H

z)

Time (seconds)

|| X Steps: Onset Detection (Spectral-Based)

§  Aspects concerning pitch, harmony, or timbre are captured by spectrogram

§  Allows for detecting local energy changes in certain frequency ranges

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Compressed spectrogram Y

|)|1log( XCY ⋅+=

Onset Detection (Spectral-Based)

1.  Spectrogram 2.  Logarithmic compression

Steps:

§  Accounts for the human logarithmic sensation of sound intensity

§  Dynamic range compression §  Enhancement of low-intensity

values §  Often leading to enhancement

of high-frequency spectrum Time (seconds)

Freq

uenc

y (H

z)

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Spectral difference


1.  Spectrogram 2.  Logarithmic compression 3.  Differentiation

Steps:

§  First-order temporal difference

§  Captures changes of the spectral content

§  Only positive intensity changes considered

Time (seconds)

Freq

uenc

y (H

z)

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Spectral difference

t Novelty curve


1.  Spectrogram 2.  Logarithmic compression 3.  Differentiation 4.  Accumulation: spectral

differences summarized by a number.

Steps:

§  Frame-wise accumulation of all positive intensity changes

§  Encodes changes of the spectral content

Freq

uenc

y (H

z)

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Digression: Difference/Distance Metrics

One of the most important issues in analyzing data, especially, multi-dimension and/or time-series data, is understand how similar two pieces of data are (represented typically by a vector or multi-dimensional array). There are two principle methods for such comparisons: Distance Metrics: Similar data vectors are regarded as closer in a geometrical sense; the range is [0 .. ∞), where distance = 0 means the vectors are identical: Dependence Metrics: Similar data vectors exhibit dependence: they “move together” in similar ways; the range of the coefficients is [-1 .. 1]:

D( a, b ) = “distance” between a and b

b

a

-1 0 1 Inverse No Strong Dependence

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Distance Metrics

A Distance Metric obeys typical geometric laws: A set with an associated Distance Metric is called a Metric Space.

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Distance Metrics

A variety of metrics have been developed, from fields as diverse as game playing to pattern recognition, and the most important of these is as follows: Sum of Absolute Difference (Manhattan Distance): Sum of Squared Difference: Mean Absolute Error: Mean Squared Error: Euclidean Distance:

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Distance Metrics

These measures extend our common understanding of the notion of distance to complex mathematical domains (such as vector spaces) and give us tools to understand how similar or dissimilar two objects are.

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Dependence Metrics

Two common dependence metrics are as follows: Correlation (Pearson’s Product-Moment Correlation Coefficient): Correlation measures the linear dependence of two vectors or random variables X and Y. Cosine Similarity: Cosine similarity measures the cosine of the angle between two vectors of length N in N-dimensional space. NOTE that these are similar calculations, except that correlation subtracts the mean from each point. For musical signals of any length, the mean will be very close to 0, and so these are effectively the same.

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Distance Metrics

Dependence metrics can be converted (almost) into distance metrics by the simple expediency of subtracting them from 1.0: Cosine Distance = 1.0 - Cosine Similarity Pearson’s Distance = 1.0 - Correlation Coefficient Now these are in the range [0..2], with 0 indicating the strongest possible dependence; these are not actually distance metrics, since a metric of 0 does not indicate identity, but just the strongest possible linear depedence; and the cosine distance does not satisfy the triangle inequality; however, this does not prevent them from being extremely useful!!

Computer Science Onset Detection (Spectral-Based)

1.  Spectrogram 2.  Logarithmic compression 3.  Differentiation 4.  Accumulation

Steps:

Novelty curve

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Subtraction of local average


1.  Spectrogram 2.  Logarithmic compression 3.  Differentiation 4.  Accumulation 5.  Normalization

Steps:

Novelty curve


1.  Spectrogram 2.  Logarithmic compression 3.  Differentiation 4.  Accumulation 5.  Normalization

Steps:

Normalized novelty curve

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1.  Spectrogram 2.  Logarithmic compression 3.  Differentiation 4.  Accumulation 5.  Normalization 6.  Peak picking

Steps:

Normalized novelty curve

Computer Science Examples of Onset Detection:

WTC Fugue #1 (Bach)A Smooth One (Benny Goodman)Doc‘s Guitar (Doc Watson)WTC Prelude #5 (Bach)Poulenc, Valse No.114Faure, Op.15, No.1

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Beat and Tempo

§  Steady pulse that drives music forward and provides the temporal framework of a piece of music

§  Sequence of perceived pulses that are equally spaced in time

§  The pulse a human taps along when listening to the music

[Parncutt 1994]

[Sethares 2007]

[Large/Palmer 2002]

[Lerdahl/ Jackendoff 1983]

[Fitch/ Rosenfeld 2007]

What is a beat?

The term tempo then refers to the speed of the pulse.

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Beat and Tempo

§  Analyze the novelty curve with respect to reoccurring or quasi-periodic patterns

§  Avoid the explicit determination of note onsets (no peak picking)

Strategy

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Beat and Tempo

Strategy

§  Autocorrelation §  Fourier transfrom

Methods

§  Analyze the novelty curve with respect to reoccurring or quasi-periodic patterns—as if it were a musical signal and you are trying to find the component pitches (= periodic patterns of the novelty curve)

§  Avoid the explicit determination of note onsets (no peak picking)

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Definition: A tempogram is a time-tempo representation that encodes the local tempo of a music signal over time (= spectrograph of novelty curve!).

Tem

po (B

PM

)

Time (seconds)

Inte

nsity

Tempogram

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Definition: A tempogram is a time-tempo represenation that encodes the local tempo of a music signal over time (= spectrograph of novelty curve!).

§  Compute a spectrogram (STFT) of the novelty curve §  Convert frequency axis (given in Hertz) into

tempo axis (given in BPM) §  Magnitude spectrogram indicates local tempo

Fourier-based method

Tempogram (Fourier)

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Tem

po (B

PM

)

Time (seconds)

Tempogram (Fourier)

Novelty curve

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Tem

po (B

PM

) Tempogram (Fourier)

Novelty curve (local window)

Time (seconds)

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Tem

po (B

PM

)

Hann-windowed sinusoidal

Tempogram (Fourier)

Time (seconds)

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Tem

po (B

PM

)


Tempogram (Fourier)

Time (seconds)

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Tem

po (B

PM

)

Tempogram (Fourier)


Time (seconds)

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Definition: A tempogram is a time-tempo represenation that encodes the local tempo of a music signal over time (= spectrograph of novelty curve!).

§  Compare novelty curve with time-lagged local sections of itself

§  Convert lag-axis (given in seconds) into tempo axis (given in BPM)

§  Autocorrelogram indicates local tempo

Autocorrelation-based method (cf. pitch determination algorithm).

Tempogram (Autocorrelation)

Computer Science Tempogram (Autocorrelation)

Novelty curve (local window)

Lag

(sec

onds

)

Time (seconds)

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Windowed autocorrelation

Lag

(sec

onds

)


Lag = 0 (seconds)

Lag

(sec

onds

)

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Lag = 0.26 (seconds)

Lag

(sec

onds

)



Lag

(sec

onds

)

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Lag

(sec

onds

)



Lag

(sec

onds

)

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Time (seconds)

Time (seconds)

Lag

(sec

onds

)

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300

60

80

40

30

120

Tempogram (Autocorrelation)

Tem

po (B

PM

)

Time (seconds)

Time (seconds)

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600

500

400

300

200

100

Tempogram (Autocorrelation) Te

mpo

(BP

M)

Time (seconds)

Time (seconds)

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Time (seconds)

Tempogram Fourier Autocorrelation

Time (seconds)

Tem

po (B

PM

)

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Computer Science Tempogram Fourier Autocorrelation

210

70

Tem

po (B

PM

)

Tempo@Tatum = 210 BPM Tempo@Measure = 70 BPM Time (seconds) Time (seconds)

Computer Science Tempogram

Fourier Autocorrelation

Time (seconds) Time (seconds)

Tem

po (B

PM

)

Time (seconds)

Emphasis of tempo harmonics (integer multiples)

Emphasis of tempo subharmonics (integer fractions)

[Grosche et al., ICASSP 2010] [Peeters, JASP 2007]

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Computer Science Tempogram (Summary)

Fourier Autocorrelation

Novelty curve is compared with sinusoidal kernels each representing a specific tempo

Novelty curve is compared with time-lagged local (windowed) sections of itself

Convert frequency (Hertz) into tempo (BPM)

Convert time-lag (seconds) into tempo (BPM)

Reveals novelty periodicities Reveals novelty self-similarities

Emphasizes harmonics Emphasizes subharmonics

Granularity increases as tempo increases; Suitable to analyze tempo on tatum and tactus level

Granularity increases as tempo decreases; Suitable to analyze tempo on tatum and measure level

Documents

CS 591 S1 – Computational Audiosnyder/cs591/Lectures/Rhythm Detection.pdf · CS 591 S1 – Computational Audio ... Transcription (melody, chords, score) ... Note: “Tatum” was