PITCH DETECTION METHODS REVIEW

Introduction

 

The goal of the present paper is to review most of the works done in the field of “Pitch Detection” or “Pitch Tracking”. The topic was chosen because it will help me to guide my current research interests.

 

What is pitch and why are we interested in tracking it? Let’s start trying to define what pitch is. The former definition given by Webster Dictionary states:  “the property of a sound and especially a musical tone that is determined by the frequency of the waves producing it: highness or lowness of sound”. Certainly our psychoacoustic understanding of the term is much more complex than that and involves the model of our hearing system (filter banks) as well as some description of the way we psychologically perceive the frequencies. The ANSI definition of psychoacoustical terminology says that “pitch is that auditory attribute of sound according to which sounds can be ordered on a scale from low to high”[1].

In most of the analysis we are going to describe, a rather simple definition is going to be used: pitch is the fundamental frequency of a harmonic signal, and even in this case the problem seems not to be satisfactory solved yet.

 

Some of the reasons why it is such a complex problem are:  our perception of pitch covers an extremely wide range of frequencies: “ In the width of perception the ear exceedingly transcends the eye; for while the former ranges over eleven octaves, but little more than a single octave is possible to the latter” [2]; our ears sense musical pitch even in the presence of noisy signals; we can follow several pitches simultaneously and also detect slight but expressive pitch deviations (vibrato, melisma, microtonal intervals)

 

There are many possible musical applications for pitch detectors, such as:

 

I am particularly interested in the idea of having a good feedback between the world of “real sounds” and the one of computer generated. Seems to me a very important issue to be able to perform interactive music that might recover some of the (in my opinion) lack of spontaneity in the electronic music.

 

Methods

 

We are going to classify the different algorithms in the following categories:

 

 

Time Domain detection:

 

This seems to be the more straightforward idea, which consists in looking at the input signal as a fluctuating amplitude in time domain and try to find repeating patterns in the waveform that give clues as to its periodicity.

 

 

1.     Zero Crossings: A simple technique that consist of counting the number of times that the signal crosses the 0 level reference. This technique is very simple and inexpensive but is not very accurate. In fact when dealing with highly noisy signals or harmonic signals where the partials are stronger that the fundamental, the method has poor results.

 

 

 

 

 

 

 

 

 

 

 

Figure 1: Harmonic signal in time domain showing that the zero crossing technique could lead to unclear results.

 

                Preprocessing by filters may improve the accuracy of this technique.

 

 

 

2.     Autocorrelation: the goal of the autocorrelation routines is to find the “similarity” between the signal and a shifted version of itself. The mathematical definitions is:

 

 

If the signal is periodic, the autocorrelation function y(n) also will be, and if the signal is harmonic the autocorrelation function will have peaks in multiples of the fundamental frequency. This technique is most efficient at mid to low frequencies. Thus it has been popular in speech recognition applications where the pitch range is limited.

Depending on the frame length, autocorrelation could be extremely expensive involving many multiply-add operations. There is although a cheaper way to calculate it using FFT[4]

 

3.     Maximum Likelihood:  Noll[5]derived an optimal way for pitch detection in the time domain. He went on to actually prove that his method is actually maximum likelihood. To describe his method, consider a length T of a periodic signal r(t) with an unknown period to. Break up the signal into N segments of length t; (1<t≤ T), i.e.

 

       0≤b<t

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2: signal broken into N segments of length t plus one segment of length b.

 

 

Now add these segments (coherently) according to:

 

 

 

 

 

 

 

These segments will add coherently when t  = to. Thus, we can form the objective function:

 

 

and maximize it to obtain to.

However, J(t) is not unimodal but rather has many peaks as is usually the case with ML objective functions. For if J(t) peaks at t= to, then it also peaks at t= mt0 if m divides N, i.e. J(t) will have almost periodic local maxima.

 

4.     Adaptive Filter: Many approaches could be implemented using this technique developed by Prof. Bernard Widrow, such as:

 

a.        Narrow band pass filter: The input signal is sent into a narrow band pass filter. Both the unfiltered and filtered signals are routed to a difference detector circuit.  Then the output is fed back to control the center frequency of the band pass filter, forcing the filter to converge to the frequency of the input signal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3: Pitch detector based on an adaptive filter scheme.

 

 

 

b.       Optimum comb method[6]: This method seeks to find a comb filter[7] that minimizes its input signal. The notches of the comb filter must be adaptively tuned to the dominant frequency of the input.

 

 

c.        Adaptive line enhancer:  I have implemented the structure showed in the following block diagram:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4: Adaptive Line Enhancer for pitch detection.

 

 

It doesn’t show any significant improvement with respect to the time domain methods.

 

 

5.     Super Resolution Pitch Determination[8]: This method uses the idea that the correlation of two adjacent segments is very high when they are spaced apart by a fundamental period or a multiple of it. The method quantifies the degree of similarity between two adjacent and non-overlapping intervals with infinite time resolution by linear interpolation.

 

Frequency Domain detection:

 

The typical analysis consist on breaking the signal into small frames, multiply by a window and get the short time Fourier transform (STFT) of the frame. If the signal is periodic the Fourier transform will show peaks in multiples of the fundamental frequencies, the following algorithms describe ways of finding which peak corresponds to the fundamental frequency.

One problem of this method is the fact that the STFT divides the audio bandwidth into a set of equally spaced frequency channels of bins where each channel is n Hz apart from its neighbors. Since human pitch perception is basically logarithmic, this means that low pitches may be tracked less accurately than high pitches.

 

1.     Harmonic Product Spectrum: The fundamental frequency can be determined by measuring the frequencies of its higher harmonic components[9] and computing the greatest common divisor of these harmonic frequencies[10]. The greatest common divisor can be determined by making an entry to a frequency histogram for each harmonic frequency and at integer divisions of the harmonic frequency. The frequency at the peak of the histogram represents the greatest common divisor, and hence the fundamental frequency.  Some nice features of the method include: inexpensive, immune to additive and multiplicative noise, adjustable to different kind of inputs (changing the number of harmonics to consider or replacing multiplication by adds)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5: Harmonic Product Spectrum implementation

 

 

 

 

2.     Cepstrum: The term “cepstrum “[11] is formed by reversing the first four letters of “spectrum”. The idea is to take the Fourier transform to the log-magnitude Fourier spectrum. Thus, if the original spectrum belongs to a harmonic signal, it is going to be periodic in the frequency representation, and taking the FFT again it will show a peak corresponding to the period in frequency, thus we can isolate the fundamental period.  It can be also interpreted as a de-convolution process. If the input signal is produced by a train of impulses convolved with a filter they are going to be multiplied in the frequency domain, then applying log would transform the multiplication in addition. And applying FFT again would de-convolved the original signal obtaining the fundamental frequency.