"""
Module: libfmp.c6.c6s3_beat_tracking
Author: Meinard Müller
License: The MIT license, https://opensource.org/licenses/MIT
This file is part of the FMP Notebooks (https://www.audiolabs-erlangen.de/FMP)
"""
import numpy as np
import librosa
from matplotlib import pyplot as plt
import IPython.display as ipd
import libfmp.b
[docs]def compute_penalty(N, beat_ref):
"""| Compute penalty funtion used for beat tracking [FMP, Section 6.3.2]
| Note: Concatenation of '0' because of Python indexing conventions
Notebook: C6/C6S3_BeatTracking.ipynb
Args:
N (int): Length of vector representing penalty function
beat_ref (int): Reference beat period (given in samples)
Returns:
penalty (np.ndarray): Penalty function
"""
t = np.arange(1, N) / beat_ref
penalty = -np.square(np.log2(t))
t = np.concatenate((np.array([0]), t))
penalty = np.concatenate((np.array([0]), penalty))
return penalty
[docs]def compute_beat_sequence(novelty, beat_ref, penalty=None, factor=1.0, return_all=False):
"""| Compute beat sequence using dynamic programming [FMP, Section 6.3.2]
| Note: Concatenation of '0' because of Python indexing conventions
Notebook: C6/C6S3_BeatTracking.ipynb
Args:
novelty (np.ndarray): Novelty function
beat_ref (int): Reference beat period
penalty (np.ndarray): Penalty function (Default value = None)
factor (float): Weight parameter for adjusting the penalty (Default value = 1.0)
return_all (bool): Return details (Default value = False)
Returns:
B (np.ndarray): Optimal beat sequence
D (np.ndarray): Accumulated score
P (np.ndarray): Maximization information
"""
N = len(novelty)
if penalty is None:
penalty = compute_penalty(N, beat_ref)
penalty = penalty * factor
novelty = np.concatenate((np.array([0]), novelty))
D = np.zeros(N+1)
P = np.zeros(N+1, dtype=int)
D[1] = novelty[1]
P[1] = 0
# forward calculation
for n in range(2, N+1):
m_indices = np.arange(1, n)
scores = D[m_indices] + penalty[n-m_indices]
maxium = np.max(scores)
if maxium <= 0:
D[n] = novelty[n]
P[n] = 0
else:
D[n] = novelty[n] + maxium
P[n] = np.argmax(scores) + 1
# backtracking
B = np.zeros(N, dtype=int)
k = 0
B[k] = np.argmax(D)
while P[B[k]] != 0:
k = k+1
B[k] = P[B[k-1]]
B = B[0:k+1]
B = B[::-1]
B = B - 1
if return_all:
return B, D, P
else:
return B
[docs]def beat_period_to_tempo(beat, Fs):
"""Convert beat period (samples) to tempo (BPM) [FMP, Section 6.3.2]
Notebook: C6/C6S3_BeatTracking.ipynb
Args:
beat (int): Beat period (samples)
Fs (scalar): Sample rate
Returns:
tempo (float): Tempo (BPM)
"""
tempo = 60 / (beat / Fs)
return tempo
[docs]def compute_plot_sonify_beat(x, Fs, nov, Fs_nov, beat_ref, factor, title=None, figsize=(6, 2)):
"""Compute, plot, and sonify beat sequence from novelty function [FMP, Section 6.3.2]
Notebook: C6/C6S3_BeatTracking.ipynb
Args:
x: Novelty function
Fs: Sample rate
nov: Novelty function
Fs_nov: Rate of novelty function
beat_ref: Reference beat period
factor: Weight parameter for adjusting the penalty
title: Title of figure (Default value = None)
figsize: Size of figure (Default value = (6, 2))
"""
B = compute_beat_sequence(nov, beat_ref=beat_ref, factor=factor)
beats = np.zeros(len(nov))
beats[np.array(B, dtype=np.int32)] = 1
if title is None:
tempo = beat_period_to_tempo(beat_ref, Fs_nov)
title = (r'Optimal beat sequence ($\hat{\delta}=%d$, $F_\mathrm{s}=%d$, '
r'$\hat{\tau}=%0.0f$ BPM, $\lambda=%0.2f$)' % (beat_ref, Fs_nov, tempo, factor))
fig, ax, line = libfmp.b.plot_signal(nov, Fs_nov, color='k', title=title, figsize=figsize)
T_coef = np.arange(nov.shape[0]) / Fs_nov
ax.plot(T_coef, beats, ':r', linewidth=1)
plt.show()
beats_sec = T_coef[B]
x_peaks = librosa.clicks(beats_sec, sr=Fs, click_freq=1000, length=len(x))
ipd.display(ipd.Audio(x + x_peaks, rate=Fs))