Book: Fundamentals of Music Processing. Audio Features. Book: Fundamentals of Music Processing. Book: Fundamentals of Music Processing

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1 Book: Fundamentals of Music Processing Lecture Music Processing Audio Features Meinard Müller International Audio Laboratories Erlangen Meinard Müller Fundamentals of Music Processing Audio, Analysis, Algorithms, Applications 483 p., 249 illus., hardcover ISBN: Springer, 2015 Accompanying website: Book: Fundamentals of Music Processing Book: Fundamentals of Music Processing Meinard Müller Fundamentals of Music Processing Audio, Analysis, Algorithms, Applications 483 p., 249 illus., hardcover ISBN: Springer, 2015 Accompanying website: Meinard Müller Fundamentals of Music Processing Audio, Analysis, Algorithms, Applications 483 p., 249 illus., hardcover ISBN: Springer, 2015 Accompanying website: Chapter 2: Fourier Analysis of Signals Chapter 3: Music Synchronization 2.1 The in a Nutshell 2.2 Signals and Signal Spaces Discrete (DFT) 2.5 Short-Time (STFT) 2.6 Further Notes Important technical terminology is covered in Chapter 2. In particular, we approach the Fourier transform which is perhaps the most fundamental tool in signal processing from various perspectives. For the reader who is more interested in the musical aspects of the book, Section 2.1 provides a summary of the most important facts on the Fourier transform. In particular, the notion of a spectrogram, which yields a time frequency representation of an audio signal, is introduced. The remainder of the chapter treats the Fourier transform in greater mathematical depth and also includes the fast Fourier transform (FFT) an algorithm of great beauty and high practical relevance. 3.1 Audio Features 3.2 Dynamic Time Warping 3.3 Applications 3.4 Further Notes As a first music processing task, we study in Chapter 3 the problem of music synchronization. The objective is to temporally align compatible representations of the same piece of music. Considering this scenario, we explain the need for musically informed audio features. In particular, we introduce the concept of chroma-based music features, which capture properties that are related to harmony and melody. Furthermore, we study an alignment technique known as dynamic time warping (DTW), a concept that is applicable for the analysis of general time series. For its efficient computation, we discuss an algorithm based on dynamic programming a widely used method for solving a complex problem by breaking it down into a collection of simpler subproblems.

2 Idea: Decompose a given signal into a superposition of sinusoidals (elementary signals). Each sinusoidal has a physical meaning and can be described by three parameters: Sinusoidals Sinusoidals Signal Each sinusoidal has a physical meaning and can be described by three parameters: Each sinusoidal has a physical meaning and can be described by three parameters: Sinusoidals Signal Signal Fouier transform Example: Superposition of two sinusoidals Example: C4 played by piano

3 Example: C4 played by trumpet Example: C4 played by violine Example: C4 played by flute Example: Speech Bonn Example: Speech Zürich Example: C-major scale (piano)

4 Example: Chirp signal Each sinusoidal has a physical meaning and can be described by three parameters: Polar coordinates: Im Complex formulation of sinusoidals: Re Signal Signal Fourier representation, Fourier representation, Fourier transform Fourier transform Tells which frequencies occur, but does not tell when the frequencies occur. Frequency information is averaged over the entire time interval. Time information is hidden in the phase Short Time Idea (Dennis Gabor, 1946): Consider only a small section of the signal for the spectral analysis recovery of time information Short Time (STFT) Section is determined by pointwise multiplication of the signal with a localizing window function

5 Short Time Short Time Short Time Short Time Short Time Short Time

6 Short Time Short Time Definition Signal Window function (, ) STFT with Short Time Intuition: Short Time Intuition: is musical note of frequency, which oscillates within the translated window is musical note of frequency, which oscillates within the translated window Inner product measures the correlation between the musical note and the signal. Window Function Window Function Box window Triangle window

7 Window Function Window Function Hann window Trade off between smoothing and ringing Time-Frequency Representation Time-Frequency Representation Frequency (Hertz) Frequency (Hertz) Spectrogram Time (seconds) Intensity (db) Time (seconds) Intensity (db) Time-Frequency Representation Chirp signal and STFT with box window of length 0.05 Time-Frequency Representation Chirp signal and STFT with Hann window of length 0.05

8 Time-Frequency Localization Short Time Signal and STFT with Hann window of length 0.02 Size of window constitutes a trade-off between time resolution and frequency resolution: Large window : poor time resolution good frequency resolution Small window : good time resolution poor frequency resolution Heisenberg Uncertainty Principle: there is no window function that localizes in time and frequency with arbitrary position. Short Time MATLAB Signal and STFT with Hann window of length 0.1 MATLAB function SPECTROGRAM N = window length (in samples) M = overlap (usually ) Compute DFT N for every windowed section Keep lower Fourier coefficients Sequence of spectral vectors (for each window a vector of dimension ) Example Let x be a discrete time signal Sampling rate: Window length: Overlap: Hopsize: Example Time resolution: Frequency resolution: Let corresponds to window Hz

9 Model assumption: Equal-tempered scale MIDI pitches: Piano notes: Concert pitch: Center frequency: Hz Idea: Binning of Fourier coefficients Divide up the fequency axis into logarithmically spaced pitch regions and combine spectral coefficients of each region to a single pitch coefficient. Logarithmic frequency distribution Octave: doubling of frequency Time-frequency representation Windowing in the time domain Windowing in the frequency domain Details: Let be a spectral vector obtained from a spectrogram w.r.t. a sampling rate and a window length N. The spectral coefficient corresponds to the frequency Let be the set of coefficients assigned to a pitch Then the pitch coefficient is defined as Example: A4, p = 69 Center frequency: Lower bound: Upper bound: STFT with, Example: A4, p = 69 Center frequency: Lower bound: Upper bound: STFT with, S(p = 69)

10 Note MIDI pitch Center [Hz] frequency Left [Hz] boundary Right [Hz] boundary Width [Hz] A A# B C C# D D# E F F# G G# A Note: For some pitches, S(p) may be empty. This particularly holds for low notes corresponding to narrow frequency bands. Linear frequency sampling is problematic! Solution: Multi-resolution spectrograms or multirate filterbanks Example: Friedrich Burgmüller, Op. 100, No. 2 Spectrogram Intensity Spectrogram Pitch representation C8 C8: 4186 Hz C7: 2093 Hz Intensity C7 C6 C5 C6: 1046 Hz C5: 523 Hz C4: 261 Hz C4

11 Pitch representation C8 Example: Chromatic scale Spectrogram C7 C6 C5 MIDI pitch C4 Example: Chromatic scale Example: Chromatic scale Spectrogram Log-frequency spectrogram C8: 4186 Hz C8: 4186 Hz C7: 2093 Hz C7: 2093 Hz C6: 1046 Hz C5: 523 Hz C4: 261 Hz C3: 131 Hz C6: 1046 Hz C5: 523 Hz C4: 261 Hz C3: 131 Hz Example: Chromatic scale Example: Chromatic scale Log-frequency spectrogram Log-frequency spectrogram Pitch (MIDI note number) Pitch (MIDI note number) Chroma C

12 Example: Chromatic scale Example: Chromatic scale Log-frequency spectrogram Chroma representation Pitch (MIDI note number) Chroma Chroma C # Example: Chromatic scale Chroma representation (normalized, Euclidean) Chroma Intensity (normalized) Human perception of pitch is periodic in the sense that two pitches are perceived as similar in color if they differ by an octave. Seperation of pitch into two components: tone height (octave number) and chroma. Chroma : 12 traditional pitch classes of the equaltempered scale. For example: Chroma C Computation: pitch features chroma features Add up all pitches belonging to the same class Result: 12-dimensional chroma vector. C2 C3 C4 Chroma C

13 C # 2 C # 3 C # 4 Chroma C # D2 D3 D4 Chroma D Chromatic circle Shepard s helix of pitch perception Example: C-Major Scale Meinard Müller: Fundamentals of Music Processing Chapter 1: Music Representations, Fig. 1.3 Springer International Publishing Switzerland, 2015 C8 Pitch representation Chroma representation C7 C6 C5 MIDI pitch Chroma C4

14 Example: Beethoven s Fifth Chroma representation (normalized, 10 Hz) Chroma representation (normalized) Karajan Scherbakov Chroma Intensity (normalized) Example: Beethoven s Fifth Chroma representation (normalized, 2 Hz) Smoothing (2 seconds) + downsampling (factor 5) Karajan Scherbakov Example: Beethoven s Fifth Chroma representation (normalized, 1 Hz) Smoothing (4 seconds) + downsampling (factor 10) Karajan Scherbakov Example: Bach Toccata Example: Bach Toccata Koopman Ruebsam Koopman Ruebsam Time (sampels) Time (sampels) Time (sampels) Time (sampels) Feature resolution: 10 Hz

15 Example: Bach Toccata Example: Bach Toccata Koopman Ruebsam Koopman Ruebsam Time (sampels) Time (sampels) Time (sampels) Time (sampels) Feature resolution: 1 Hz Feature resolution: 0.33 Hz Sequence of chroma vectors correlates to the harmonic progression Example: Zager & Evans In The Year 2525 Normalization changes in dynamics makes features invariant to Further quantization and smoothing: CENS features Taking logarithm before adding up pitch coefficients accounts for logarithmic sensation of intensity How to deal with transpositions? Example: Zager & Evans In The Year 2525 Example: Zager & Evans In The Year 2525 Original: Original: Shifted:

16 Audio Features There are many ways to implement chroma features Properties may differ significantly Appropriateness depends on respective application MATLAB implementations for various chroma variants