DEV Community: bfdykstra

Introduction to Music Information Retrieval Pt. 2

bfdykstra — Wed, 04 Dec 2019 00:38:00 +0000

Segmentation and Feature Extraction

(You can view the version with all the audio segments here

In part 1 we learned some basic signal processing terminology, as well as how to load and visualize a song. In this post we will discuss how to break up a signal (segment) and extract various features from it. Then, we'll do some exploratory analysis on the features so that we can get an idea of the interactions among features.

Why do we want to segment a song?

Songs vary a lot over time. By breaking up this heterogenous signal in to small segments that are more homogenous, we can keep the information about how a song changes. For example, say that we have some features such as 'danceability'. Imagine a song that is at first very quiet and low energy, but as the song progresses, it becomes a full blown Rihanna club anthem. If we just examined the average danceability of the entire song, it might be lower than how danceable the song actually is. This context and distinction is important because if we are running a classifier or clustering algorithm on each segment of a song, we could classify that song as the club anthem that it is, not just it's (below) average.

Segment a song using onset detection

An onset in a signal is often described as the beginning of a note or other sound. This is usually found by measuring peaks in energy along a signal. If we find that peak in energy, and then backtrack to a local minimum, we have found an onset, and can use that as a boundary for a segment of a song.

here are some good resources for learning more about onsets and onset detection: https://musicinformationretrieval.com/onset_detection.html, https://en.wikipedia.org/wiki/Onset_(audio), https://www.music-ir.org/mirex/wiki/2018:Audio_Onset_Detection

%matplotlib inline
import librosa
import numpy as np
import IPython.display as ipd
import sklearn
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
plt.rcParams['figure.figsize'] = (16, 7)

By default, onset_detect returns an array of frame indices that correspond to frames in a signal. We actually
want the sample indices so that we can slice and dice our signal neatly with those indices. We'll continue to use our Rock & Roll example from the previous post.

signal, sr = librosa.load('/Users/benjamindykstra/Music/iTunes/Led Zeppelin/Led Zeppelin IV/02 Rock & Roll.m4a')

# use backtrack=True to go back to local minimum
onset_samples = librosa.onset.onset_detect(signal, sr=sr, backtrack=True, units='samples')
print onset_samples.shape

(132,)

We have found 132 segments in the song. Now, lets use the sample indices to split up the song in to subarrays like so:
[[segment 1], [segment 2], ..... [segment n]]. Each segment will be a different length, but when we calculate a feature vector for a segment, the feature vector will be a standard size. The final dimensions of the data after segmenting and calculating features for each segment will be (# segments, # features).

# return np array of audio segments, within each segment is the actual audio data
prev_ndx = 0
segmented = []
for sample_ndx in onset_samples:
    # get the samples from prev_ndx to sample_ndx
    segmented.append(np.array(signal[prev_ndx:sample_ndx]))
    prev_ndx = sample_ndx
segmented.append(np.array(signal[onset_samples[-1]:])) # gets the last segment from the signal
segmented = np.array(segmented)

segmented.shape

(133,)

As a sanity check, if we concatenate all the segments together, it should be equal in shape to the original signal.

print "difference in shapes: {}".format(signal.shape[0] - np.concatenate(segmented).shape[0])

difference in shapes: 0

Listen to a few segments together!

ipd.Audio(np.concatenate(segmented[25:30]), rate=sr)

or just a single short segment

ipd.Audio(segmented[21], rate=sr)

Lets define a more generic segmentation function for use later on

def segment_onset(signal, sr=22050, hop_length=512, backtrack=True):
    """
    Segment a signal using onset detection
    Parameters:
        signal: numpy array of a timeseries of an audio file
        sr: int, sampling rate, default 22050 samples per a second
        hop_length: int, number of samples between successive frames
        backtrack: bool, If True, detected onset events are backtracked to the nearest preceding minimum of energy
    returns:
        dictionary with attributes segemented and shape

    """
    # Compute the sample indices for estimated onsets in a signal
    onset_samples = librosa.onset.onset_detect(signal, sr=sr, hop_length=hop_length, backtrack=backtrack, units='samples')

    # return np array of audio segments, within each segment is the actual audio data
    prev_ndx = 0
    segmented = []
    for sample_ndx in onset_samples:
        segmented.append(np.array(signal[prev_ndx:sample_ndx]))
        prev_ndx = sample_ndx
    segmented.append(np.array(signal[onset_samples[-1]:]))
    return { 'data': np.array(segmented), 'shape': np.array(segmented).shape }

Feature Extraction

Now that we have a way to break up a song, we would like to be able to derive some features from the raw signal. Librosa has a plethora of features to choose from. They fall in to two categories, spectral and rhythmic features. Spectral features are features that have to do with the frequency, pitch and timbre of a signal, where as rhythmic features (you guessed it) give you info about the rhythm of the signal.

The objective with feature extraction is to feed a single function a single segment of a song, that returns an array of the calculated features for that segment

Some of the feature methods return arrays of differing shapes and we need to account for those differences in our implementation. For example, when calculating the Mel-frequency cepstral coefficients for a segment, the return shape is (# of coefficients, # of frames in a segment). Since we're assuming that a segment is a homogenous piece of signal, we should take the average of the coefficients across all the frames so we get a shape of (# of coefficients, 1).

First I will define all the feature functions, and then I will explain what information they add/describe.

def get_feature_vector(segment):
    '''
    Extract features for a given segment
    Parameters:
        segment: numpy array, a time series of audio data
    Returns:
        numpy array
    '''
    if len(segment) != 0:
        feature_tuple = (avg_energy(segment), avg_mfcc(segment), zero_crossing_rate(segment), avg_spectral_centroid(segment), avg_spectral_contrast(segment), bpm(segment))
        all_features = np.concatenate([feat if type(feat) is np.ndarray else np.array([feat]) for feat in feature_tuple])
        n_features = len(all_features)
        return all_features
    return np.zeros((30,)) # length of feature tuple


def avg_energy(segment):
    '''
    Get the average energy of a segment.
    Parameters:
        segment: numpy array, a time series of audio data
    Returns:
        float, the mean energy of the segment
    '''
    if len(segment) != 0:
        energy = librosa.feature.rmse(y=segment)[0]
        # returns (1,t) array, get first element
        return np.array([np.mean(energy)])


def avg_mfcc(segment, sr=22050, n_mfcc=20):
    '''
    Get the average Mel-frequency cepstral coefficients for a segment
    The very first MFCC, the 0th coefficient, does not convey information relevant to the overall shape of the spectrum. 
    It only conveys a constant offset, i.e. adding a constant value to the entire spectrum. We discard it.
    BE SURE TO NORMALIZE

    Parameters:
        segment: numpy array, a time series of audio data
        sr: int, sampling rate, default 22050
        n_mfcc: int, the number of cepstral coefficients to return, default 20.
    Returns:
        numpy array of shape (n_mfcc - 1,)
    '''
    if (len(segment) != 0):
        components = librosa.feature.mfcc(y=segment,sr=sr, n_mfcc=n_mfcc ) # return shape (n_mfcc, # frames)

        return np.mean(components[1:], axis=1)


def zero_crossing_rate(segment):
    '''
    Get average zero crossing rate for a segment. Add a small constant to the signal to negate small amount of noise near silent
    periods.

    Parameters:
        segment: numpy array, a time series of audio data
    Returns:
        float, average zero crossing rate for the given segment
    '''


    rate_vector = librosa.feature.zero_crossing_rate(segment+ 0.0001)[0] # returns array with shape (1,x)
    return np.array([np.mean(rate_vector)])


def avg_spectral_centroid(segment, sr=22050):
    '''
    Indicate at which frequency the energy is centered on. Like a weighted mean, weighting avg frequency by the energy.
    Add small constant to audio signal to discard noise from silence
    Parameters:
        segment: numpy array, a time series of audio data
        sr: int, sampling rate
    Returns:
        float, the average frequency which the energy is centered on.
    '''
    centroid = librosa.feature.spectral_centroid(segment+0.01, sr=sr)[0]
    return np.array([np.mean(centroid)])


def avg_spectral_contrast(segment, sr=22050, n_bands=6):
    '''
    considers the spectral peak, the spectral valley, and their difference in each frequency subband

    columns correspond to a spectral band

    average contrast : np.ndarray [shape=(n_bands + 1)]
    each row of spectral contrast values corresponds to a given
    octave-based frequency, take average across bands

    Parameters:
        segment: numpy array, a time series of audio data
        sr: int, sampling rate
        n_bands: the number of spectral bands to calculate the contrast across.
    Returns:
        numpy array shape (n_bands,)
    '''
    contr = librosa.feature.spectral_contrast(segment, sr=sr, n_bands=n_bands)
    return np.mean(contr, axis=1) # take average across bands

def bpm(segment, sr=22050):
    '''
    Get the beats per a minute of a song
    Parameters:
        segment: numpy array, a time series of audio data,
        sr: int, sampling rate
    Returns:
        int, beats per minute
    '''
    tempo = librosa.beat.tempo(segment) #returns 1d array [bpm]
    return np.array([tempo[0]])

Selected Feature Justification

Energy:

The energy for a segment is important because it gives a feel for tempo and/or mood of a segment. It's actually just the root mean square of a signal.

MFCC:

The Mel-frequency cepstral coefficients relay information about the timbre of a song. Timbre describes the 'quality' of a sound. If you think about how an A note on a trumpet sounds vastly different than that same A on a piano, those differences are due to timbre.

Zero Crossing Rate:

Literally the rate at which a signal crosses the horizontal axis. It often corresponds to events in the signal such as a snare drum or some other percussive event.

Spectral Centroid:

I think the spectral centroid is actually really cool. It's a weighted average of the magnitude of the frequencies in a signal and a 'center of mass'. It's often perceived as a measure of the brightness of a sound.

Spectral Contrast:

An octave based feature, it more directly represents the spectral characteristics of a segment. Coupled with MFCC features, they can provide a lot of information about a signal.

Beats Per Minute:

Provides information about the tempo and (some) percussive elements of a song.

Now what do we actually do with all these features??? We can use them to cluster!!

def extract_features(all_songs):
    '''
    all_songs is a list of dictionaries. Each dictionary contains the attributes song_name and data.
    The data are the segments of the song.
    '''
    all_song_features = []
    song_num = 0
    for song in all_songs:
        print "Processing {} with {} segments".format(song['song_name'], len(song['data']))
        song_name = song['song_name']
        segment_features = []
        for segment in song['data']:
            feature_vector = get_feature_vector(segment)
            segment_features.append(feature_vector)   


        song_feature_vector = np.array(segment_features)
        print "shape of feature vector for entire song: {}".format(song_feature_vector.shape)
        print "shape of segment feature vector: {}".format(song_feature_vector[0].shape)
        n_seg = song_feature_vector.shape[0]
        feature_length = song_feature_vector[0].shape[0]
        song_feature_vector = np.reshape(song_feature_vector, (n_seg, feature_length))

        all_song_features.append(song_feature_vector)
        song_num += 1
    all_feature_vector = np.vstack(all_song_features)
    return all_feature_vector

Visualizing the features of two very different songs

Lets look at the features of Rock 'n Roll by Led Zeppelin, and I Remember by Deadmau5.

rock_n_roll = segment_onset(signal)
rock_n_roll['data'].shape

(133,)

feature_vec_rock = extract_features([{'song_name': '02 Rock & Roll.m4a', 'data': rock_n_roll['data']}])
feature_vec_rock.shape

Processing 02 Rock & Roll.m4a with 133 segments
shape of feature vector for entire song: (133, 30)
shape of segment feature vector: (30,)





(133, 30)

i_remember, sr = librosa.load('/Users/benjamindykstra/Music/iTunes/Deadmau5/Random Album Title/07 I Remember.m4a')
i_remember_segmented = segment_onset(i_remember)
feature_vec_remember = extract_features([{'song_name': '07 I Remember.m4a', 'data': i_remember_segmented['data']}])

Processing 07 I Remember.m4a with 1852 segments
shape of feature vector for entire song: (1852, 30)
shape of segment feature vector: (30,)

These two songs are very different

ipd.Audio(np.concatenate(i_remember_segmented['data'][:30]), rate= sr)

ipd.Audio(np.concatenate(rock_n_roll['data'][:20]), rate= sr)

We need to scale the features to a common range, the feature ranges differ wildly

col_names = ['energy'] + ["mfcc_" + str(i) for i in xrange(19)] + ['zero_crossing_rate', 'spectral_centroid'] + ['spectral_contrast_band_' + str(i) for i in xrange(7)] + ['bpm']
rnr = pd.DataFrame(feature_vec_rock, columns = col_names)
i_remember_df = pd.DataFrame(feature_vec_remember, columns = col_names)

min_max_scaler = sklearn.preprocessing.MinMaxScaler(feature_range=(-1, 1))
rnr_scaled = pd.DataFrame(min_max_scaler.fit_transform(feature_vec_rock), columns = col_names)
i_remember_scaled = pd.DataFrame(min_max_scaler.fit_transform(feature_vec_remember), columns = col_names)

features_scaled = pd.DataFrame(np.vstack((rnr_scaled, i_remember_scaled)), columns = col_names)

rnr_scaled.head()

	energy	mfcc_0	mfcc_1	mfcc_2	mfcc_3	mfcc_4	mfcc_5	mfcc_6	mfcc_7	mfcc_8	...	zero_crossing_rate	spectral_centroid	spectral_contrast_band_0	spectral_contrast_band_1	spectral_contrast_band_2	spectral_contrast_band_3	spectral_contrast_band_4	spectral_contrast_band_5	spectral_contrast_band_6	bpm
0	-1.000000	0.122662	1.000000	-0.516018	1.000000	0.602426	1.000000	1.000000	1.000000	0.540636	...	-1.000000	-1.000000	0.827903	-1.000000	-0.962739	-1.000000	-1.000000	-1.000000	-0.538654	-0.503759
1	-0.242285	-0.648073	0.381125	-0.786313	-0.091164	-0.080649	-0.601857	-0.673427	-0.444300	-0.700703	...	0.604015	0.709584	0.099070	-0.187083	-0.375223	-0.307743	-0.114691	-0.157464	0.903511	-0.250000
2	-0.067639	-0.850079	0.421647	-0.732783	-0.226694	-0.404820	-0.733511	-0.860285	-0.528783	-0.503516	...	0.781001	0.850525	0.758635	-0.287418	-0.213330	-0.196650	-0.161395	0.109064	0.427987	-0.503759
3	-0.433760	-0.969241	0.401582	-0.620675	-0.046703	0.055861	-0.502694	-0.516944	0.055559	-0.147803	...	0.831207	0.924291	0.801980	0.226105	-0.535366	-0.160115	0.185044	0.142246	0.299663	-0.503759
4	-0.282242	-0.686175	0.325491	-0.689084	-0.086337	-0.026573	-0.474765	-0.471087	-0.126714	-0.212362	...	0.496496	0.731278	0.755725	0.126388	0.024655	-0.247040	0.625841	0.129373	0.113422	-0.503759

5 rows × 30 columns

i_remember_scaled.head()

	energy	mfcc_0	mfcc_1	mfcc_2	mfcc_3	mfcc_4	mfcc_5	mfcc_6	mfcc_7	mfcc_8	...	zero_crossing_rate	spectral_centroid	spectral_contrast_band_0	spectral_contrast_band_1	spectral_contrast_band_2	spectral_contrast_band_3	spectral_contrast_band_4	spectral_contrast_band_5	spectral_contrast_band_6	bpm
0	0.058494	-0.647307	0.539341	-0.121477	0.541506	0.437682	-0.272703	0.223814	-0.016933	-0.048312	...	-0.114140	0.491343	0.292913	0.271760	0.192199	-0.773597	-0.625672	-0.067422	0.317499	-0.676113
1	-0.005999	-0.534543	-0.160968	-0.374121	0.505839	0.547995	-0.180246	0.333693	-0.098430	0.029649	...	-0.273774	0.318984	0.266204	-0.026715	-0.369555	-0.694350	-0.538811	-0.494475	0.247816	-0.676113
2	-0.014792	-0.465639	-0.168007	-0.570869	0.449955	0.559676	-0.168267	0.279917	-0.095737	-0.053418	...	-0.327139	0.187170	0.507119	0.038864	-0.463672	-0.728928	-0.451278	-0.269847	0.223124	-0.676113
3	0.127275	-0.410650	0.045166	-0.534741	0.485562	0.635807	-0.145189	0.292996	-0.003948	0.145229	...	-0.474190	0.109161	0.216147	-0.372901	-0.184409	-0.732067	-0.453334	-0.383937	0.103847	-0.676113
4	0.099671	-0.368250	0.332703	-0.435360	0.505144	0.570220	-0.128529	0.181684	-0.068506	-0.032707	...	-0.539708	0.116126	0.209937	-0.236042	-0.115957	-0.544722	-0.496779	-0.378822	-0.068973	-0.676113

5 rows × 30 columns

Lets take a look at the descriptive statistics for the energy

sns.boxplot(x=i_remember_scaled.energy, palette='muted').set_title('I Remember Energy');
plt.show();
print i_remember_scaled.energy.describe()
sns.boxplot(rnr_scaled.energy).set_title('Rock n Roll Energy');
plt.show();
print rnr_scaled.energy.describe()

count    1852.000000
mean        0.184555
std         0.373220
min        -1.000000
25%        -0.009499
50%         0.176136
75%         0.478511
max         1.000000
Name: energy, dtype: float64

count    133.000000
mean       0.354920
std        0.391389
min       -1.000000
25%        0.117536
50%        0.439199
75%        0.639228
max        1.000000
Name: energy, dtype: float64

I remember has a lower mean and median energy, but similar spread to Rock n Roll. I'd say that this fits, as I Remember has almost a melancholic energy, where as Rock n Roll really makes you to get up and move.

What about the zero crossing rate and BPM?

Since zero crossing rate and bpm correlate pretty highly with percussive events, I'd predict that the song with the higher BPM will have a higher zero crossing rate

print 'Rock n Roll average BPM: {}'.format(rnr.bpm.mean())
print 'Rock n Roll average Crossing Rate: {}'.format(rnr.zero_crossing_rate.mean())
print 'I Remember average BPM: {}'.format(i_remember_df.bpm.mean())
print 'I Remember average Crossing Rate: {}'.format(i_remember_df.zero_crossing_rate.mean())

Rock n Roll average BPM: 150.784808472
Rock n Roll average Crossing Rate: 0.14643830815
I Remember average BPM: 136.533708743
I Remember average Crossing Rate: 0.0362864360563

Some scatterplots with spectral centroid on the x axis, and energy on the y axis.

sns.scatterplot(x = rnr.spectral_centroid, y = rnr.energy);
plt.show();
sns.scatterplot(x = i_remember_df.spectral_centroid, y = i_remember_df.energy );
plt.show();

Recall that spectral centroid is where the spectral 'center of mass' is for a given segment. That means that it's picking out the dominant frequency for a segment. I like this because it shows that there's not real relation between the frequency of a segment and its energy. They are both contributing unique information.

build labels and cluster using k means!

song_labels = np.concatenate([np.full((label_len), i) for i, label_len in enumerate([len(rnr), len(i_remember_df)])])

model = sklearn.cluster.KMeans(n_clusters=3)

labels = model.fit_predict(features_scaled)

plt.scatter(features_scaled.zero_crossing_rate[labels==0], features_scaled.energy[labels==0], c='b')
plt.scatter(features_scaled.zero_crossing_rate[labels==1], features_scaled.energy[labels==1], c='r')
plt.scatter(features_scaled.zero_crossing_rate[labels==2], features_scaled.energy[labels==2], c='g')
plt.xlabel('Zero Crossing Rate (scaled)')
plt.ylabel('Energy (scaled)')
plt.legend(('Class 0', 'Class 1', 'Class 2'))

<matplotlib.legend.Legend at 0x13f054f10>


unique_labels, unique_counts = np.unique(model.predict(rnr_scaled), return_counts=True)
print unique_counts
print 'cluster for rock n roll: ', unique_labels[np.argmax(unique_counts)]

[29  9 95]
cluster for rock n roll:  2

unique_labels, unique_counts = np.unique(model.predict(i_remember_scaled), return_counts=True)
print unique_counts
print 'cluster for I remember: ', unique_labels[np.argmax(unique_counts)]

[396 877 579]
cluster for I remember:  1

I suspect that zero crossing rate and energy are not what the determining factors are for a cluster :)

We can actually listen to the segments that were assigned certain labels

Note that these aren't necessarily consecutive segments

First lets look at I remember

i_remember_clusters = model.predict(i_remember_scaled)
label_2_segs = i_remember_segmented['data'][i_remember_clusters==2]
label_1_segs = i_remember_segmented['data'][i_remember_clusters==1]
label_0_segs = i_remember_segmented['data'][i_remember_clusters==0]

Almost all of theses have some of the vocals in them

ipd.Audio(np.concatenate(label_2_segs[:50]), rate = sr)

These are the lighter segments, mostly just synths

ipd.Audio(np.concatenate(label_1_segs[:50]), rate = sr)

0 labels seem to be the heavy bass and percussive parts

ipd.Audio(np.concatenate(label_0_segs[:50]), rate = sr)

Now Rock n Roll

rnr_clusters = model.predict(rnr_scaled)

rock_label_2_segs = rock_n_roll['data'][rnr_clusters==2]
rock_label_1_segs = rock_n_roll['data'][rnr_clusters==1]
rock_label_0_segs = rock_n_roll['data'][rnr_clusters==0]

Again, higher frequencies, vocals included. A lot of consecutive segments included in this class.

ipd.Audio(np.concatenate(rock_label_2_segs[10:40]), rate = sr)

            <audio controls="controls" >
                Your browser does not support the audio element.
            </audio>

Lighter sounds, minor key vocals, part of final drum solo

ipd.Audio(np.concatenate(rock_label_1_segs), rate = sr)

            <audio controls="controls" >
                Your browser does not support the audio element.
            </audio>

All John Bonham here (only drums). Similar to how in I remember label 0 corresponded to the heavy bass

ipd.Audio(np.concatenate(rock_label_0_segs), rate = sr)

            <audio controls="controls" >
                Your browser does not support the audio element.
            </audio>

Conclusion

We've used two very different songs to build vectors of spectral and rhythmic features. We then examined how those features related to each other with boxplots, scatterplots and descriptive statistics. Using those features, we have clustered different segments of the songs together to compare how the different clusters sound together.

More exploration is needed to figure out how to assign a feature value for an entire song that is representative. With more work doing that, it would then be possible to get summary features for entire songs. More on this later!

The rest of this is just some undirected exploring of different features

print rnr_scaled[rnr_clusters==0]['spectral_centroid'].describe()
print rnr_scaled[rnr_clusters==1]['spectral_centroid'].describe()
print rnr_scaled[rnr_clusters==2]['spectral_centroid'].describe()

count    9.000000
mean    -0.360693
std      0.250294
min     -1.000000
25%     -0.370240
50%     -0.280334
75%     -0.260073
max     -0.148394
Name: spectral_centroid, dtype: float64
count    29.000000
mean      0.066453
std       0.363603
min      -0.483240
25%      -0.211871
50%       0.012854
75%       0.156281
max       0.814435
Name: spectral_centroid, dtype: float64
count    95.000000
mean      0.354971
std       0.261067
min      -0.669203
25%       0.218433
50%       0.318239
75%       0.427016
max       1.000000
Name: spectral_centroid, dtype: float64

i_remember_df['mfcc_2'].plot.hist(bins=20, figsize=(14, 5))

<matplotlib.axes._subplots.AxesSubplot at 0x139127290>

sns.boxplot(x=i_remember_scaled.zero_crossing_rate, palette='muted').set_title('I Remember Zero Crossing Rate');
plt.show();
print i_remember_scaled.zero_crossing_rate.describe()
sns.boxplot(rnr_scaled.zero_crossing_rate).set_title('Rock n Roll Zero Crossing Rate');
plt.show();
print rnr_scaled.zero_crossing_rate.describe()

count    1852.000000
mean       -0.683760
std         0.255792
min        -1.000000
25%        -0.880912
50%        -0.762325
75%        -0.535427
max         1.000000
Name: zero_crossing_rate, dtype: float64

count    133.000000
mean      -0.152444
std        0.421004
min       -1.000000
25%       -0.393116
50%       -0.198213
75%       -0.045042
max        1.000000
Name: zero_crossing_rate, dtype: float64

print rnr.zero_crossing_rate.describe()
print i_remember_df.zero_crossing_rate.describe()

count    133.000000
mean       0.146438
std        0.072255
min        0.000977
25%        0.105133
50%        0.138583
75%        0.164871
max        0.344226
Name: zero_crossing_rate, dtype: float64
count    1852.000000
mean        0.036286
std         0.025857
min         0.004319
25%         0.016357
50%         0.028345
75%         0.051281
max         0.206489
Name: zero_crossing_rate, dtype: float64

print rnr.bpm.describe()
print i_remember_df.bpm.describe()

count    133.000000
mean     150.784808
std       31.793864
min       86.132812
25%      135.999178
50%      135.999178
75%      172.265625
max      287.109375
Name: bpm, dtype: float64
count    1852.000000
mean      136.533709
std         7.140772
min       112.347147
25%       135.999178
50%       135.999178
75%       135.999178
max       258.398438
Name: bpm, dtype: float64

# rnr_scaled['energy'].plot();
i_remember_scaled['energy'].plot();
# rnr_scaled['zero_crossing_rate'].plot();
# rnr_scaled['spectral_centroid'].plot();
# rnr_scaled['mfcc_4'].plot();
plt.show()

rnr_scaled[['mfcc_' + str(i) for i in xrange(4)]].plot();
plt.show();

sns.scatterplot(x=rnr_scaled['energy'], y = rnr_scaled['bpm'])

<matplotlib.axes._subplots.AxesSubplot at 0x131c69ad0>

sns.pairplot(rnr_scaled[['mfcc_' + str(i) for i in xrange(5)]], palette='pastel')

<seaborn.axisgrid.PairGrid at 0x15f503c10>

Introduction to Music Information Retrieval Pt. 1

bfdykstra — Mon, 15 Apr 2019 23:56:40 +0000

Introduction to Music Information Retrieval Pt. 1

I love music. I listen to it all day at work, I can play it, and (sometimes) I can make it. I also love analyzing stuff like congressional elections or exploring age of first birth. So, why not try and combine those two things and analyze some of the music in my iTunes library?

Motivation

Spotify knows my music tastes better than I do, and 9 times out of 10, their curated playlists made for me are absolutely spot on. I though that I'd like to try and replicate some of what they have done from a first principles perspective.

I began by looking in to the Spotify API and trying to figure out how to replicate their results in their audio analysis GET route. I was then lead to this reddit discussion where a search keyword given was Subject Audio Classification. My Google Fu then led me to this god send of a site https://musicinformationretrieval.com/. This site has tons of great resources, and also links to the International Society of Music Information Retrieval that has loads of great research papers. A lot of the following examples are based off of the lessons there.

My main objective is to help other people clarify what it is they should be looking for when analyzing music, as well as some tools to help get them started.

What you'll need

I work on macOSX with High Sierra installed. I run an anaconda distribution with python 2.7.13.
For this notebook, all that you'll need are numpy librosa matplotlib and Ipython. If you try to load a song with librosa and it does not work, you may need to install ffmpeg using homebrew.

Getting Started: Initial Challenges and Solutions

Embarking on this project was extremely frustrating for a long time. I did not know what keywords I should google (feature extraction on audio data?) and there were only 1 or 2 tutorials that specifically dealt with voice classification or ambient sound. Furthermore, within the tutiorials, there was no interpretation or explanation of any of the features that were being extracted, and they were very technical terms. Another huge challenge in this space is copyright laws for songs. There aren't huge repositories of pop song mp3s lying around nor are music file formats standardized in any way.

So how did I get around these challenges?

For tools: First thing I found was in the way of Python was Librosa. I immediately started trying to read .m4a files from my iTunes library, but my computer threw a fit. Install ffmpeg using homebrew. Now I could actually read files on my computer.
For songs, I used what I have on my own computer. There are other sources that you can find with raw audio data, although many others leave out the raw files for reasons above.
As far as interpreting features goes, I had to go back to the basics! That's what I'd like to start with :)

Basic Terms and Concepts of Signal Processing

There were a lot of terms that I came across that I was pretty familiar with, but definitely needed some refreshing.

Cycle: Also known as a period. How many radians it takes for a to repeat itself. For sine, this is 2$\pi$ radians. More practically, if a heart beats once every 30 seconds, we say that it's period is 30 seconds

Hertz: (Not the car rental company). 1 hertz = 1 cycle per a second. 440 Hz means 440 cyles/second.

Frequency: How many cycles per a unit of time a signal repeats itself. It is the inverse of cycle. If my heart beats once every half second (cycle), then my frequency is 120 beats per a minute.

Signal: The actual timeseries of frequencies at a point in time from a file. We use librosa.load('filename.mp3', sr=our_sampling_rate) to load a file in to a audio timeseries.

Sampling Rate: The rate at which a sample is taken from an audio file. Defaults to 22050. If you multiply it by some a unit of time, you get the number of samples taken in that time.

Energy: Formally, the area under the squared magnitude of the signal. Roughly corresponds to how load the signal is.

Fourier Transform: This is a hugely important operation in signal processing. While the math of it is pretty dense, what it does is fairly intuitive. It breaks down a signal in to its constituent frequencies that make it up. In other words, it takes something that exists in a time domain (signal) and maps it in to the frequency domain, where the y-axis is magnitude. It's very similar to how a musical chord can be broken down in to the separate notes that make it up, i.e. a C triad chord is composed of the notes C, E, and G.

Now that we have some basic terminology down, let's go through some simple examples that use these terms.

from __future__ import division
import numpy as np
import seaborn as sns
%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (12,8)

import librosa.display
import IPython.display as ipd

Lets create a simple audio wave that is 440 HZ (A note above middle C, specifically A4) that lasts for 3 seconds. This will be a pure tone. We'll use numpy.sin to construct the sine wave. Recall from high school physics/trig that the formula for a sin wave is $A* sin(2*{\pi}*R)$ where R is radians and A is amplitude. We can get the number of radians by multiplying a frequency (radians/time) by the time variable.

sr = 22050 # sample rate
T = 3.0    # seconds
t = np.linspace(0, T, int(T*sr), endpoint=False) # time variable, evenly split by number of samples within timeframe
freq = 440 #hz A4
amplitude = 0.5

# synthesize tone
signal = amplitude * np.sin(2 * np.pi * freq * t) # from our formula above
print "there are " + str(signal.shape[0]/T) + " \
samples of audio per second, if T was 2 seconds, the shape would be " + str(signal.shape[0]*2)

there are 22050.0 samples of audio per second, if T was 2 seconds, the shape would be 132300

Let's play the tone!

ipd.Audio(signal, rate=sr) #

If we plot the entire time series, then it just looks like 1 solid mass because of the high frequency. Instead, let's plot a small sample of the tone so that we can see only two periods.

# sr / freq = # of frames in one period,
two_periods = int(2 * (sr / freq))

librosa.display.waveplot(signal[:two_periods + 1], sr=sr);

Lets make a tone with a much lower frequency

sr = 22050 # sample rate
T = 2.0    # seconds
t = np.linspace(0, T, int(T*sr), endpoint=False) # time variable, 
freq = 110 #hz 
amplitude = 1

# synthesize tone
low_signal = amplitude*np.sin(2*np.pi*freq*t)

You might not be able to play this tone through your computer speakers, try headphones if nothing plays.

ipd.Audio(low_signal, rate=sr) # BASS

Since there are 22050 samples in this signal with a frequency of 110 hz, that means if we divide the number of samples by the frequency, we should be able to see how many frames are in one period of the signal and plot the single period.

# 22050 / 110 = 200.45 => round up to 201
plt.plot(t[:201], low_signal[:201])
plt.show()

To help our own intuition, lets look at how the two signals are related to each other. Since 110 hz is about a quarter the frequency of 440 hz, then 4 periods of higher frequency signal should fit in 1 period of the lower frequency. Green is the higher frequency.

librosa.display.waveplot(low_signal[:4*two_periods], sr=sr);
librosa.display.waveplot(signal[:4*two_periods], sr=sr);

Alright awesome! We've been able to construct our own sound waves and get an idea of the how frequency affects the tone.

Load your own music!

Instead of synthesizing our own tones, lets load a small sample of a song from your music library and take a look at it!

# read 4 seconds of this song, starting at 6 seconds
music_dir = '/Users/benjamindykstra/Music/iTunes/Led Zeppelin/Led Zeppelin IV/02 Rock & Roll.m4a'
data, sr = librosa.load(music_dir, offset=6.0, duration=4.0)

ipd.Audio(data, rate=sr)

librosa.display.waveplot(data, sr=sr); # it just looks loud!

Fourier Transform

Lets see what happens if we take the fast fourier transform of our original signal (440 hz). I expect that there will be a single line at 440 hz mark on the x-axis. After we do our simple case, we can also break down our small song clip.

import scipy

X = scipy.fft(signal)
X_mag = np.absolute(X)
f = np.linspace(0, sr, len(X_mag)) # frequency variable

plt.plot(f[1000:2000], X_mag[1000:2000]); # magnitude spectrum
plt.xlabel('Frequency (Hz)');

Look at that! a single spike right at 440 Hz. Warms me heart :)

Now we can take a quick look at Rock & Roll.

rr_fft = scipy.fft(data)
rr_mag = np.absolute(rr_fft)
f = np.linspace(0, sr, len(rr_mag)) # frequency variable
plt.plot(f, rr_mag) # magnitude spectrum
plt.xlabel('Frequency (Hz)')
plt.title('Rock and Roll Babay');

The reason that the Fourier transform is so important to us, is that it allows us to analyze the occurrence and power of each frequency through a whole song. If we break up a song in to parts, we can start to look at how the distribution changes over the course of the song. We can also use it to separate harmonic and percussive components of a signal. It is a fundamental building block of any algorithms or features that utilize frequency analysis.

Thanks for following along! In my next post we'll get in to the weeds with segmentation and feature extraction with lots of explanations.