Music Synchronization (libfmp.c3)

The FMP notebooks provide detailed textbook-like explanations of central techniques and algorithms implemented in the libfmp. The part of FMP related to this module is available at the following URL:

https://www.audiolabs-erlangen.de/resources/MIR/FMP/C3/C3.html

libfmp.c3.c3s1_audio_feature.compute_chromagram(Y_LF)

Computes a chromagram

Notebook: C3/C3S1_SpecLogFreq-Chromagram.ipynb

Parameters

Y_LF (np.ndarray) – Log-frequency spectrogram

Returns

C (np.ndarray) – Chromagram

libfmp.c3.c3s1_audio_feature.compute_spec_log_freq(Y, Fs, N)

Computes a log-frequency spectrogram

Notebook: C3/C3S1_SpecLogFreq-Chromagram.ipynb

Parameters

• Y (np.ndarray) – Magnitude or power spectrogram

• Fs (scalar) – Sampling rate

• N (int) – Window size of Fourier fransform

Returns

• Y_LF (np.ndarray) – Log-frequency spectrogram

• F_coef_pitch (np.ndarray) – Pitch values

libfmp.c3.c3s1_audio_feature.f_pitch(p, pitch_ref=69, freq_ref=440.0)

Computes the center frequency/ies of a MIDI pitch

Notebook: C3/C3S1_SpecLogFreq-Chromagram.ipynb

Parameters

• p (float) – MIDI pitch value(s)

• pitch_ref (float) – Reference pitch (default: 69)

• freq_ref (float) – Frequency of reference pitch (default: 440.0)

Returns

freqs (float) – Frequency value(s)

libfmp.c3.c3s1_audio_feature.note_name(p)

Returns note name of pitch

Notebook: C3/C3S1_SpecLogFreq-Chromagram.ipynb

Parameters

p (int) – Pitch value

Returns

name (str) – Note name

libfmp.c3.c3s1_audio_feature.pool_pitch(p, Fs, N, pitch_ref=69, freq_ref=440.0)

Computes the set of frequency indices that are assigned to a given pitch

Notebook: C3/C3S1_SpecLogFreq-Chromagram.ipynb

Parameters

• p (float) – MIDI pitch value

• Fs (scalar) – Sampling rate

• N (int) – Window size of Fourier fransform

• pitch_ref (float) – Reference pitch (default: 69)

• freq_ref (float) – Frequency of reference pitch (default: 440.0)

Returns

k (np.ndarray) – Set of frequency indices

libfmp.c3.c3s1_post_processing.log_compression(v, gamma=1.0)

Logarithmically compresses a value or array

Notebook: C3/C3S1_LogCompression.ipynb

Parameters

• v (float or np.ndarray) – Value or array

• gamma (float) – Compression factor (Default value = 1.0)

Returns

v_compressed (float or np.ndarray) – Compressed value or array

libfmp.c3.c3s1_post_processing.median_downsample_feature_sequence(X, Fs, filt_len=41, down_sampling=10)

Smoothes and downsamples a feature sequence. Smoothing is achieved by median filtering

Notebook: C3/C3S1_FeatureSmoothing.ipynb

Parameters

• X (np.ndarray) – Feature sequence

• Fs (scalar) – Frame rate of X

• filt_len (int) – Length of smoothing filter (Default value = 41)

• down_sampling (int) – Downsampling factor (Default value = 10)

Returns

• X_smooth (np.ndarray) – Smoothed and downsampled feature sequence

• Fs_feature (scalar) – Frame rate of X_smooth

libfmp.c3.c3s1_post_processing.normalize_feature_sequence(X, norm='2', threshold=0.0001, v=None)

Normalizes the columns of a feature sequence

Notebook: C3/C3S1_FeatureNormalization.ipynb

Parameters

• X (np.ndarray) – Feature sequence

• norm (str) – The norm to be applied. ‘1’, ‘2’, ‘max’ or ‘z’ (Default value = ‘2’)

• threshold (float) – An threshold below which the vector v used instead of normalization (Default value = 0.0001)

• v (float) – Used instead of normalization below threshold. If None, uses unit vector for given norm (Default value = None)

Returns

X_norm (np.ndarray) – Normalized feature sequence

libfmp.c3.c3s1_post_processing.smooth_downsample_feature_sequence(X, Fs, filt_len=41, down_sampling=10, w_type='boxcar')

Smoothes and downsamples a feature sequence. Smoothing is achieved by convolution with a filter kernel

Notebook: C3/C3S1_FeatureSmoothing.ipynb

Parameters

• X (np.ndarray) – Feature sequence

• Fs (scalar) – Frame rate of X

• filt_len (int) – Length of smoothing filter (Default value = 41)

• down_sampling (int) – Downsampling factor (Default value = 10)

• w_type (str) – Window type of smoothing filter (Default value = ‘boxcar’)

Returns

• X_smooth (np.ndarray) – Smoothed and downsampled feature sequence

• Fs_feature (scalar) – Frame rate of X_smooth

libfmp.c3.c3s1_transposition_tuning.compute_freq_distribution(x, Fs, N=16384, gamma=100.0, local=True, filt=True, filt_len=101)

Compute an overall frequency distribution

Notebook: C3/C3S1_TranspositionTuning.ipynb

Parameters

• x (np.ndarray) – Signal

• Fs (scalar) – Sampling rate

• N (int) – Window size (Default value = 16384)

• gamma (float) – Constant for logarithmic compression (Default value = 100.0)

• local (bool) – Computes STFT and averages; otherwise computes global DFT (Default value = True)

• filt (bool) – Applies local frequency averaging and by rectification (Default value = True)

• filt_len (int) – Filter length for local frequency averaging (length given in cents) (Default value = 101)

Returns

• v (np.ndarray) – Vector representing an overall frequency distribution

• F_coef_cents (np.ndarray) – Frequency axis (given in cents)

libfmp.c3.c3s1_transposition_tuning.cyclic_shift(C, shift=1)

Cyclically shift a chromagram

Notebook: C3/C3S1_TranspositionTuning.ipynb

Parameters

• C (np.ndarray) – Chromagram

• shift (int) – Tranposition shift (Default value = 1)

Returns

C_shift (np.ndarray) – Cyclically shifted chromagram

libfmp.c3.c3s1_transposition_tuning.plot_freq_vector_template(v, F_coef_cents, template_max, theta_max, ax=None, title=None, figsize=(8, 3))

Plots frequency distribution and similarity-maximizing template

Notebook: C3/C3S1_TranspositionTuning.ipynb

Parameters

• v – Vector representing an overall frequency distribution

• F_coef_cents – Frequency axis

• template_max – Similarity-maximizing template

• theta_max – Maximizing tuning parameter

• ax – Axis (in case of ax=None, figure is generated) (Default value = None)

• title – Title of figure (or subplot) (Default value = None)

• figsize – Size of figure (only used when ax=None) (Default value = (8, 3))

Returns

• fig – Handle for figure

• ax – Handle for axes

• line – handle for line plot

libfmp.c3.c3s1_transposition_tuning.plot_tuning_similarity(sim, theta_axis, theta_max, ax=None, title=None, figsize=(4, 3))

Plots tuning similarity

Notebook: C3/C3S1_TranspositionTuning.ipynb

Parameters

• sim – Similarity values

• theta_axis – Axis consisting of cent values [-50:49]

• theta_max – Maximizing tuning parameter

• ax – Axis (in case of ax=None, figure is generated) (Default value = None)

• title – Title of figure (or subplot) (Default value = None)

• figsize – Size of figure (only used when ax=None) (Default value = (4, 3))

Returns

• fig – Handle for figure

• ax – Handle for axes

• line – handle for line plot

libfmp.c3.c3s1_transposition_tuning.template_comb(M, theta=0)

Compute a comb template on a pitch axis

Notebook: C3/C3S1_TranspositionTuning.ipynb

Parameters

• M (int) – Length template (given in cents)

• theta (int) – Shift parameter (given in cents); -50 <= theta < 50 (Default value = 0)

Returns

template (np.ndarray) – Comb template shifted by theta

libfmp.c3.c3s1_transposition_tuning.tuning_similarity(v)

Compute tuning similarity

Notebook: C3/C3S1_TranspositionTuning.ipynb

Parameters

v (np.ndarray) – Vector representing an overall frequency distribution

Returns

• theta_axis (np.ndarray) – Axis consisting of all tuning parameters -50 <= theta < 50

• sim (np.ndarray) – Similarity values for all tuning parameters

• ind_max (int) – Maximizing index

• theta_max (int) – Maximizing tuning parameter

• template_max (np.ndarray) – Similiarty-maximizing comb template

libfmp.c3.c3s2_dtw.compute_accumulated_cost_matrix(C)

Compute the accumulated cost matrix given the cost matrix

Notebook: C3/C3S2_DTWbasic.ipynb

Parameters

C (np.ndarray) – Cost matrix

Returns

D (np.ndarray) – Accumulated cost matrix

libfmp.c3.c3s2_dtw.compute_accumulated_cost_matrix_21(C)

Compute the accumulated cost matrix given the cost matrix

Notebook: C3/C3S2_DTWvariants.ipynb

Parameters

C (np.ndarray) – Cost matrix

Returns

D (np.ndarray) – Accumulated cost matrix

libfmp.c3.c3s2_dtw.compute_cost_matrix(X, Y, metric='euclidean')

Compute the cost matrix of two feature sequences

Notebook: C3/C3S2_DTWbasic.ipynb

Parameters

• X (np.ndarray) – Sequence 1

• Y (np.ndarray) – Sequence 2

• metric (str) – Cost metric, a valid strings for scipy.spatial.distance.cdist (Default value = ‘euclidean’)

Returns

C (np.ndarray) – Cost matrix

libfmp.c3.c3s2_dtw.compute_optimal_warping_path(D)

Compute the warping path given an accumulated cost matrix

Notebook: C3/C3S2_DTWbasic.ipynb

Parameters

D (np.ndarray) – Accumulated cost matrix

Returns

P (np.ndarray) – Optimal warping path

libfmp.c3.c3s2_dtw.compute_optimal_warping_path_21(D)

Compute the warping path given an accumulated cost matrix

Notebook: C3/C3S2_DTWvariants.ipynb

Parameters

D (np.ndarray) – Accumulated cost matrix

Returns

P (np.ndarray) – Optimal warping path

libfmp.c3.c3s2_dtw_plot.plot_matrix_with_points(C, P=np.empty((0, 2)), color='r', marker='o', linestyle='', **kwargs)

Compute the cost matrix of two feature sequences

Parameters

• C – Matrix to be plotted

• P – List of index pairs, to be visualized on the matrix (Default value = np.empty((0, 2)))

• color – The color of the line plot (Default value = ‘r’). See https://matplotlib.org/users/colors.html

• marker – The marker of the line plot (Default value = ‘o’). See https://matplotlib.org/3.1.0/api/markers_api.html

• linestyle – The line-style of the line plot (Default value = ‘’). See https://matplotlib.org/gallery/lines_bars_and_markers/line_styles_reference.html

• **kwargs – keyword arguments for libfmp.b.b_plot.plot_matrix()

Returns

• fig – Handle for figure

• im – Handle for imshow

• line – handle for line plot

libfmp.c3.c3s3_tempo_curve.compute_score_chromagram(score, Fs_beat)

Compute chromagram from score representation

Notebook: C3/C3S3_MusicAppTempoCurve.ipynb

Parameters

• score (list) – Score representation

• Fs_beat (scalar) – Sampling rate for beat axis

Returns

• X_score (np.ndarray) – Chromagram representation X_score

• t_beat (np.ndarray) – Time axis t_beat (given in beats)

libfmp.c3.c3s3_tempo_curve.compute_strict_alignment_path(P)

Compute strict alignment path from a warping path

Notebook: C3/C3S3_MusicAppTempoCurve.ipynb

Parameters

P (list or np.ndarray) – Warping path

Returns

P_mod (list or np.ndarray) – Strict alignment path

libfmp.c3.c3s3_tempo_curve.compute_strict_alignment_path_mask(P)

Compute strict alignment path from a warping path

Notebook: C3/C3S3_MusicAppTempoCurve.ipynb

Parameters

P (list or np.ndarray) – Wapring path

Returns

P_mod (list or np.ndarray) – Strict alignment path

libfmp.c3.c3s3_tempo_curve.compute_tempo_curve(score, x, Fs=22050, Fs_beat=10, N=4410, H=2205, shift=0, sigma=np.array([[1, 0], [0, 1], [2, 1], [1, 2], [1, 1]]), win_len_beat=4)

Compute a tempo curve

Notebook: C3/C3S3_MusicAppTempoCurve.ipynb

Parameters

• score (list) – Score representation

• x (np.ndarray) – Audio signal

• Fs (scalar) – Samping rate of audio signal (Default value = 22050)

• Fs_beat (scalar) – Sampling rate for beat axis (Default value = 10)

• N (int) – Window size for computing audio chromagram (Default value = 4410)

• H (int) – Hop size for computing audio chromagram (Default value = 2205)

• shift (int) – Cyclic chroma shift applied to audio chromagram (Default value = 0)

• sigma (np.ndarray) – Step size set used for DTW (Default value = np.array([[1, 0], [0, 1], [2, 1], [1, 2], [1, 1]]))

• win_len_beat (float) – Window length (given in beats) used for smoothing tempo curve (Default value = 4)

Returns

• f_tempo (np.ndarray) – Tempo curve

• t_beat (np.ndarray) – Time axis (given in beats)

libfmp.c3.c3s3_tempo_curve.plot_measure(ax, measure_pos)

Plot measure positions

Notebook: C3/C3S3_MusicAppTempoCurve.ipynb

Parameters

• ax (mpl.axes.Axes) – Figure axis

• measure_pos (list or np.ndarray) – Array containing measure positions

libfmp.c3.c3s3_tempo_curve.plot_tempo_curve(f_tempo, t_beat, ax=None, figsize=(8, 2), color='k', logscale=False, xlabel='Time (beats)', ylabel='Temp (BPM)', xlim=None, ylim=None, label='', measure_pos=[])

Plot a tempo curve

Notebook: C3/C3S3_MusicAppTempoCurve.ipynb

Parameters

• f_tempo – Tempo curve

• t_beat – Time axis of tempo curve (given as sampled beat axis)

• ax – Plot either as figure (ax==None) or into axis (ax==True) (Default value = None)

• figsize – Size of figure (Default value = (8, 2))

• color – Color of tempo curve (Default value = ‘k’)

• logscale – Use linear (logscale==False) or logartihmic (logscale==True) tempo axis (Default value = False)

• xlabel – Label for x-axis (Default value = ‘Time (beats)’)

• ylabel – Label for y-axis (Default value = ‘Temp (BPM)’)

• xlim – Limits for x-axis (Default value = None)

• ylim – Limits for x-axis (Default value = None)

• label – Figure labels when plotting into axis (ax==True) (Default value = ‘’)

• measure_pos – Plot measure positions as spefified (Default value = [])

Returns

• fig – figure handle

• ax – axes handle