Python for Digital Signal Processing

Before starting with this post, you may want to learn the basics of Python. If you’re an experienced programmer and head Python for the first time, you will likely find it very easy to understand. One important thing about Python: Python requires perfect indentation (4 spaces) to validate the code. So if you get an error and you code seems perfect, review if you have indented correctly each line. Python has also a particular way to deal with arrays, more close the the R programming language than to C-like style.

Python’s core functionality is extended with thousands of available free libraries, many of them are incredibly handy. Even if Python is not a compiled language, many of its libraries are written in C, being Python a wrapper for them.

The libraries used on this article are:

  • scipy – Scientific library for Python.
  • numpy – Numeric library for Python.

To load a wav file in Python:

# Loads the scipy.io package for later usage of io.wavefile module.
from scipy import io
#
# Location of the wav file in the file system.
fileName = '/Downloads/Music_Genres/genres/country/country.00053.wav'
#
# Loads sample rate (bps) and signal data (wav). Notice that Python
# functions can return multiple variables at the same time.
sample_rate, data = io.wavfile.read(fileName)
#
# Print in sdout the sample rate, number of items. 
#and duration in seconds of the wav file. 
print "Sample rate:{0}, data size:{1}, duration:{2} seconds" \
    .format(sample_rate,data.shape,len(data)/sample_rate)

The output generated should seem like:

Sample rate:22050, data size:(661794L,), duration:30 seconds

The output shows that the wav file contains in all 661,794 samples (the data variable is an array with 661,794 elements). The sample rate is 22,050 samples per second. Dividing 661,794 elements by 22,050 samples per second, we obtain 30 seconds, the length in seconds of the wav file.

The Fourier Transform

The Fourier transform is the method that we will use to extract the prevalent frequencies from the wav file. Each frequency corresponds to a musical tone; knowing the frequencies from a particular time interval we are able to know which are the most frequent tones within that interval, being possible to infer the key and chords played during that time lapse.

This article is not going to enter into the details of the Fourier transform, only on how to use it to extract information regarding the frequency power from the wav signal analyzed. The video below is an intuitive introduction to the Fourier transform in case the reader is interested on it. It also includes examples of how to implement it algorithmically. It is quite advisable to watch it once now and then come back again to review it after the training in Fourier transform is completed.

Basically, given a signal, a wav file on this post, which is composed by a number n of samples \(x[n]\). We can get the frequency power within the signal with the FFT (Fast Fourier Transform) function. The FFT function is an improvement that optimizes the Fourier transform.

The FFT function receives two arguments, the signar \(x\) and the number of items to retrieve \(k, k\leq n\). The commonly choosen k value is \(\frac{n}{2}\) because the FFT result, \(fft[k]\) is usually symmetric around that length. This means that in order to calculate the FFT, only a half of the total signal length is required to retrieve the different frequencies occurrence. So, in plain words, if the original signal file has 100 samples, only 50 samples are needed to process the complete FFT transform.

In Python language there are two useful functions to calculate and get the Fourier transform from a sample array, like the one where the data variable from the wav file is stored:

  • fftfreq – Returns the frequency corresponding to each \(x_i\) sample from the signal data sample file \(x[n]\) corresponding to the power of the fourier transform. This is the frequency to which each fft element corresponds to.
  • fft – Returns the fourier transform data from the sample file. The position of the elements returned correspond to the position of the fftfreq, so that using both arrays the fft power elements correspond by position to the fftfreq frequencies.

For instance, if the fourier transform function returns fft = {0,0.5,1} and \(\)fftfreq = {100,200,300}\(\), it means that the signal has a power of 0 for frequency 100Hz, a power of 0.5 for 200Hz and a power of 1 within 300Hz; being 300Hz the frequency most frequent.

The following code would extract from a wav file the first 10 second, apply the fourier transform  and the frequencies associated to each item within  the spectral data.

import scipy.io
 
# Package that implements the fast
# fourier transform functions.
from scipy import fftpack
import numpy as np
 
# Loads wav file as array.
fileName = './country.00053.wav'
sample_rate, data = io.wavfile.read(fileName)
 
# Extracting 10 seconds. N is the numbers of samples to
# extract or elements from the array.
seconds_to_extract = 10
N = seconds_to_extract * sample_rate
 
# Knowing N and sample rate, fftfreq gets the frequency
# Associated to each FFT unit.
f = fftpack.fftfreq(N, 1.0/sample_rate)
 
# Extracts fourier transform data from the sample
# returning the spectrum analysis
subdata = data[:N]
F = fftpack.fft(subdata)

F contains the power and f the frequency each item within F is related to. The higher the power, the higher the frequency prevalence across the signal. Filtering the frequencies using the f matrix and extracting the power we could get a graph like the next one:fourier_transform_python

On the y-axis, \(|F|\) is the absolute value of each unit from F and the values of f are the Frequency (Hz) on the x-axis. The green and orange lines can be ignored. To get the subset of frequencies [200-900] displayed on the chart, the next code was used:

# Interval limits
Lower_freq = 200
Upper_freq = 900
 
# f (frequencies) between lower frequency AND
# f (frequencies) upper frequencies.
filter_subset = (f >= Lower_freq) * (f <= Upper_freq)
 
# Extracts filtered items from the frequency list.
f_subset = f[filter_subset]
 
# Extracts filtered items from the Fourier transform power list.
F_subset = F[filter_subset]

Spectral Analysis and Harmony

Chromatic scale tone frequencies

On the previous post, Spectral Analysis and Harmony, it is shown an elementary introduction to harmony and digital signal. We are now going to study the range of tones between A3 an A5. Our central axis is A tone (or A4) which frequency is equal to 440Hz.

The next table shows all the tones and frequencies within the chromatic scale belonging to the range between A3 and A5. The piano key number corresponding to each tone is also displayed.

 Scientific
name
Key
number
Helmholtz
name
Frequency
(Hz)
A3 37 a 220.000
A♯3/B♭3 38 a♯/b♭ 233.082
B3 39 b 246.942
C4 Middle C 40 c′ 1-line octave 261.626
C♯4/D♭4 41 c♯′/d♭′ 277.183
D4 42 d′ 293.665
D♯4/E♭4 43 d♯′/e♭′ 311.127
E4 44 e′ 329.628
F4 45 f′ 349.228
F♯4/G♭4 46 f♯′/g♭′ 369.994
G4 47 g′ 391.995
G♯4/A♭4 48 g♯′/a♭′ 415.305
A4 – A440 49 a′ 440.000
A♯4/B♭4 50 a♯′/b♭′ 466.164
B4 51 b′ 493.883
C5 Tenor C 52 c′′ 2-line octave 523.251
C♯5/D♭5 53 c♯′′/d♭′′ 554.365
D5 54 d′′ 587.330
D♯5/E♭5 55 d♯′′/e♭′′ 622.254
E5 56 e′′ 659.255
F5 57 f′′ 698.456
F♯5/G♭5 58 f♯′′/g♭′′ 739.989
G5 59 g′′ 783.991
G♯5/A♭5 60 g♯′′/a♭′′ 830.609
A5 61 a′′ 880.000

The difference or leap between two tones is called interval. One interesting feature of the chromatic scale is that it is composed by constant intervals. For instance, tone A3 is equal to 220Hz, tone A4 to 440Hz and tone A5 to 880Hz. Each tone frequency is double its analogue tone from the precedent respective octave.

The important idea is that we can analyze tones as numbers and operate with basic arithmetics with them with their frequencies. Who said emotions cannot be explained by Science? Do not be intimidated if you don’t know neither music theory nor Optical Physics; These texts will led you by the hand on a trip at which end you will know how to extract the waves, tones and emotions from digital music even without knowing none of those.

Frequency analysis in a nutshell

In order to analyze the frequencies that compose a piece of music, we take a part from it and extract a subset of frequencies. Like using an equalizer we filter the sound between two specific frequencies or tones. For instance, we could read the first ten seconds of a music mp3 file and generate a table displaying how many times tone A appears within that sequence. Going farther we could analyze how many tones appear and how many times each tone is played within those 10 first seconds.

As seen on the Emotions Within Digital Signals article, those tones can be used to define the chords and keys a piece of music is formed by.

In order to extract the signal frequency occurrences, we can use a frequency spectrum graph. This graph displays how many times a frequency appears on a signal and its power or prevalence other the rest. In this case, the signal is the first 10 seconds of music. Let’s see an example:

frequency_spectrum_graph
Signal and Frequency Spectrum Graphs

From the graph on the right we can see that the most used frequencies, those having higher \(|F|\), are one next to the 200Hz, another between the 300Hz and 400Hz and a third one between the 400Hz and the 500Hz. The x-axis shows the frequency spectrum (or range) we are analyzing, and the y-axis the power of the signal. The higher the line at a certain point on the x-axis, the more the power that signal has over that frequency.

To get an insight of the most used tones, the frequencies that have more power can be extracted, and in this case the dominant frequencies within the signal are in particular 220Hz, 246.942Hz, 329.628Hz and 440Hz. Rounding those frequencies to the nearest integer and comparing them to the ones in the table above, we can extract some of the main tones within the first ten seconds of the song.

Scientific
name
Frequency
(Hz)
A3 220.000
B3 247
E4 330
A4 A440 440

From the data above it can be determined that the dominant key within the first seconds is composed by tones A, B and  E. That key corresponds to chord A2Sus (A 2nd suspended). A chord is how it’s called the sound composed by multiple tones, multiple frequencies. The names of the different chords are not described in this article, since there are many of them.

In terms of music harmony A2Sus, or generically speaking 2nd suspended chords are tones that create a sensation of waiting for something to be resolved. The listener is holding on until the song resolves in something. We could say that the first ten seconds of this song are causing an emotion of expectation.

For more information on music and emotions, search in Google “emotions chords harmony”. For a good introduction to the matter I would recommend the paper Music and Emotions.

This article and the previous one, Emotions Within Digital Signals, set the basis to successfully tackle the problem of extracting emotions from music sequences. I will explain how to perform that task using Python language in the post Python for Digital Signal Processing.