CIRCULARITY IN PITCH JUDGEMENT

Introduction

            Pitch can roughly be defined as the aspect of sound which corresponds to frequency. For the basic case, namely sound containing a single frequency (a pure tone), higher pitch  means high frequency.

            For sound containing components with different frequencies, things are much more complicated than just adding up the perceived effect of individual tones.

               When we hear a frequency and its integral multiples (usually called the fundamental and its harmonics) together, the perceived pitch is roughly the pitch of the lowest frequency. This makes the picture more complicated. Even more interesting is that when we here frequencies which are not integral multiple of one of them, the pitch we perceive is the pitch of the highest common factor of these frequencies. For example, if we hear 200Hz and 300Hz together, we hear a pitch corresponding roughly to 100Hz.

            Now we can conclude from these simple examples that our brain roughly assigns the pitch according to the periodicity of the signal. For the case of a fundamental and its harmonics, the overall signal is periodic in 1/f, so its period is f. For the sound including frequencies 3f  and 2f, the period of the overall signal is 1/f , thus the period is f. By the same logic the perceived pitch is 2f for a signal containing 4f and 6f .

 

 

 

 

Risset Scale

            We have seen that our perception of pitch may be misleading in terms of trying to guess the frequency of sound. Are there other ways of playing with pitch, for example in the case of changing frequencies?

            We have dealt with periodic patterns up to now. Let’s listen to a chirp, a signal with exponentially decreasing frequency:

DEMO OF CHIRP

            You heard what is expected, a continuously decreasing frequency. And please listen to this now:

DEMO OF RISSET SCALE

            What have you heard? Probably again a decreasing frequency, but what you have heard was in fact periodic! This acoustic illusion is known as Risset Scale

How Does It Work?

            Start with a frequency f. Then take its first harmonic, and than the first harmonic of the new frequency and so on. You have : f, 2f, 4f, 8f, 16f…. Than we filter the amplitudes by a bell-shaped curve so that central frequencies have high amplitude and lowest and highest frequencies have low amplitude.

           

 

This is the static case. Then let all the frequencies decrease exponentially with the same   

time constant and simultaneously change their amplitudes according to the curve. You

will see the frequencies moving under the constant curve all with same frequency. After a

time, all the frequencies will reach their neighboring lower frequency. At this moment think of the figure again. The only difference from the beginning is the places of the

lowest and highest frequencies, but these have very low amplitude and therefore can be

ignored. At this point, starting the signal from the beginning should not have a

considerable effect and lets try doing this. Here is the paradox, we have an ever-

descending frequency pattern which can be simulated by a periodic pattern. Will our brain choose the periodic case or the ever-descending case for the periodic signal? As you tested a few minutes ago, it perceives this sound as ever-descending.

            This illusion shows the circularity in pitch judgment. It is like walking on a big

circular path and just looking for your next step such that you do not see that the path is circular. In Risset Scale, after a while, the frequency components of the sound are nearly same as the beginning case (like reaching the starting point on a circular path) but all frequency components are still decreasing (we are still walking). Our brain chooses the ever-descending frequencies option (we think that we are still going away from our starting point being unaware of the circularity).

            This type of illusions are named Shepard Scales after R. N. Shepard who invented the first version of it in 1964. Risset Scale was invented by J-C. Risset as a continuous version of the original illusion. There are also other versions described by Burns (1981), Teranishi (1986), Shroeder (1986).

 

Mathematical Recipe

            Although it is easy to understand the general idea behind the Risset Scale, the construction includes many important subtleties. We will start from the very basics and end up with the complete Risset Scale.

            The first thing we need is a pure tone, since our illusion is a collection of these. A pure tone is simply a sine wave with constant frequency. For the unit amplitude case:

            The subscript is merely showing that this is the  harmonic. As we have seen we need a signal with exponentially decreasing frequency. The first idea that comes to one’s mind is usually inserting an exponential expression instead of  in the above equation:

which is wrong. For a sinusoidal of   , frequency is defined as , so we should have the integral of our desired frequency inside the sine in order to have what we want when we differentiate it. The true formula for a tone with changing frequency is:

            We want the frequency to ascend and become its half value at a time, let’s say, , so our frequency is  and our signal is:

 for

            Now, we need an amplitude for this signal. Let’s use a Gaussian for the envelope. As we saw in the introduction, our envelope is a function of the logarithm of the frequency, i.e. for a gaussian:

            Here, f is the instantaneous frequency (dependent on time),  is the frequency with the highest amplitude and  is a parameter determining the width of the curve. Value. Subscript denotes that it is the central frequency. We insert the time dependent expression for the frequency to relate the amplitude directly to time:

           

            As the last step, we sum up all these harmonics:

            Since :

for . This is the part of the Risset Scale from the beginning till the time every frequency reaches the lower one. When we repeat this again and again we have the Risset Scale.

 

 

Generating the Scale by Computer

WE CAN PUT THE MATLAB CODE HERE

Discussion and Conclusion

            In our demo, the values of the parameters are chosen to optimize the effect. Very narrow gaussians allows just one frequency to be heard. Very broad gaussians, on the other hand, does not suppress the newly entering highest harmonic and disappearing lowest harmonic. The perceived sound is periodic for both of these envelopes. For our demo amplitude of the lowest frequency was about of .

            Number of harmonics is important for the strength of the effect. The highest number of harmonics is the best. While doing these, we should not violate the sampling rate. In our demo there were 10 harmonics. The effect is also strong for 4-6 harmonics but lower values it is not as desired.

            The lowest frequency used puts limits on the signal. Human ear is sensitive between 20-20000 Hz which makes a ratio of roughly. This is the limitation on the number of harmonics. To have higher number of harmonics, one should choose the lowest frequency possible. For or demo starting value is 27.5 Hz. One should be careful about the lowest frequency since most of the loudspeakers are not very good in producing low-frequency sound. Our demo is tested on Creative Inspire 5.1 5300 loudspeakers        

            As you might have already realized, a careful observer still can hear periodicity instead of an ever-descending frequency. Moreover, it may become even harder to capture the ever-descending sound if you know the trick already. Risset Scale may not be very strong in these cases but for an unbiased listener it is an astonishing illusion which reveals an interesting aspect of human pitch judgment.