In this next lesson, the second lesson, we're going to talk about pitch and frequency. Again, remember that you always have a pair of parameters here. The objective parameter, which is frequency in hertz that we've talked a little bit about, but we're gonna talk much more about now. And pitch, which is the sense that we have, the response that we have, the subject response that we have to different frequencies. So again, let's go back to the diagram that we already talked about in the relationship between intensity and loudness and then I'll talk about it in the relationship between frequency and hertz, cycles per second, the repetition rate of the sound signal and the sense of pitch that different frequencies elicit. So you remember that the basilar membrane is diagrammed here and I didn't mention what these numbers were for. But these are different positions. One, two, three, four, etc. Positions along the membrane and these positions correspond to the place where a wave is vibrating, changing, mechanically disturbing the hair cells most for whatever that frequency might be. So just to go back a bit and tell you about the nature of the basilar membrane. It's attached at this end, at the end where the over window is, where the sound signal comes in. But it's flapping freely at this opposite end the cochlear apex. And what that means is that the wave that is moving along here is actually a traveling wave in much the same way the wave travels if you take a bullwhip. Imagine taking a bullwhip and cracking it back and forth. You have a travelling wave along the width, it's the same idea here and depending on the intensity the frequency at which you wave the width, the frequency at which the membrane is disturbed. The wave can be at any position along the length of the basilar memory, remember the basilar membrane being about 35 millimeters in length. So because it's attached at this end and therefore much stiffer just like the whip where you're holding it with your hand is bigger and stiffer and moves less this end is responsive to high frequencies and, as you go along the basilar membrane, the responsivity to frequencies declines and this cochlear apex is the part that's responsive to quite low frequencies. So, depending on the place that's maximally disturbed, along the basilar membrane you hear a different frequency because the hair cells that are carried centrally again in the auditory nerve, that we talked about last time, are carrying that information from a particular place that is differentially responsive to frequencies as a function that's placed along the basilar membrane. Now as in the case with the relationship between loudness and intensity, what you hear is not what's there physically, even though there is of course a relationship between them. So one way of thinking about this discrepancy between what's there physically on the basilar membrane and what you hear is what's called the critical bandwidth. If you cycle physically take two sign tones, one let's say at a lower frequency, one at a higher frequency. And you begin to move those two sign tones together. They'll eventually interfere with one another. That is, at first you'll hear them as tone one and tone two, but as they come closer together, they begin to interfere with each other in a disturbing way. And that psychophysical measurement that has been made many many times over the course of the last century or so shows that the distance on the basilar membrane, at which interference occurs, is roughly speaking about a milimeter apart. Whether you're talking about two frequencies that are at the high end of the responsivity of the membrane, or at the low end, the distance at which the frequencies begin to interact and interfere with each other, is on the order of one millimeter. So you can think of this, these one millimeter lengths are called critical bands or critical bandwidths, and you can think of them as being roughly 35 of these along the extent of the inner ear basilar membrane. So what's strange is that when you look, and these were experiments done by Georg von Bekesy, in the 1960s, and 70s, using human cadavers. What he showed, looking directly at the vibration, elicited by some sign tones in the carefully dissected basil membrane of a human cadaver, is that the extend of the membrane that's disturbed by a frequency, whether it's high or low, is many millimeters. So here is the length, roughly speaking, of the membrane, or at least the length that von Bekesy could see with his dissection, going from zero millimeters to about 30, that's the whole extent of the basilar membrane, or most of it. You can see that the disturbance, the physical disturbance, at each of these frequency levels, involves much of the basilar membrane. It's not limited to a little piece. And it's very hard to see how the critical band, a millimeter in length, is related to a much wider resturbance or how it's generated by a much wider disturbance of the basilar membrane en toto. Remember I said that the hair cells are enormously sensitive. They are responsive when you move them as little as something like the diameter of a gold atom. They're just incredibly sensitive to movement. But, these movements that are causing critical bandwidths of approximately one millimeter or disturbances that cover much more of the basilar membrane. Most of it, as you can see in these diagrams, are on the right. So, there's a puzzle here relating the physical movement on the membrane, which is great, and the very limited psycho-physical determination of what you actually hear in response to sign tones, and it's hard to sort of justify how the one is related to the other. That simply remains as a puzzle. Do you remember I told you before that the outer hair cells are changing the compliance of the basilar membrane and sharpening this responsivity, which, because these were cadavers wasn't involved in von Bekesy's experiment. But even that is really not enough to resolve this discrepancy. The next puzzle I wanna tell you a little about that again indicates that what you hear in terms of pitch, pitch being defined as a range of higher or lower, more or less ordinal going from a low pitch to a high pitch of tone. That what you hear is not simply related to what that frequency of repetition is. So, you remember I told you before a little bit about the spectrum of a sound signal. The harmonic series that's generated by an analysis, a spectral analysis of the sound signal and I told you that there was a fundamental frequency, the vibration of the full string, the vibration of the string at half length, third length, quarter length and so on and that whole series was called the harmonic series and represented the modes of vibration on a strung that you plucked or column of air that would be equivalent in a wind instrument. This diagram here is a harmonic series, again just in very simplified form. So here's the fundamental then you have another peak of intensity out of frequency that's twice the fundamental, three times the fundamental, four times the fundamental and so on. That's the harmonic series that we discussed last time and this upper panel. Panel number one is the full harmonic series. Again, in diagrammatic form because you remember that I told that these peaks go down exponentially as the result of the fact that the vibrating string has a lot of power at the full length, half the power at the half length mode. And a third of the power at the third length mode of vibration. So, these should be going down, but simplified here we just have them as bars. This being the fundamental, the first harmonic, second harmonic and so on. Now, what's interesting here is that you hear in response to a vibrating string the fundamental frequency. You don't really hear all these harmonics. They're there, but you don't hear them. Psychophysical experiments show that a really highly trained person in acoustics in music can hear maybe four, five or six harmonics. I can't hear really any of them. But most people can hear a few, but certainly not the full range of harmonics. What you hear in response to a tone is the fundamental frequency. And this diagram shows on the one hand that that's the case. So, in the second panel here, the first fundamental harmonic has been deleted. In the third panel here, the first two harmonics have been deleted, and so on, leaving only a few harmonics in panels four, five, and six. But no matter whether you play this one, this one, this one, this one, this one, this one, you always are hearing the fundamental frequency. It totally corresponds to the fundamental, the zero harmonic, remember this is the fundamental first harmonic and so on. So why is that? Well, we'll have to discuss that later on. But the point here is that, even when you take out the harmonics, these lower harmonics one, two, and three and in particular, the fundamental. So that you have no physical energy at these levels, at the frequencies that would correspond to these pitches. You still hear the pitch of the missing fundamental and the missing harmonics if you can hear two or three harmonics so that's very mysterious. Why is it that despite the absence of physical energy in the sound signal and in the stimulus, the basal memory is not vibrating at those frequencies because there's no disturbance of the atmosphere and no disturbance in the stimulus of the basal membrane of those frequencies at those frequencies. You hear them nonetheless. So, again, that's another dramatic example. It's called hearing the missing fundamentals. You probably have heard of this. It's a common phenomenon in auditory physiology and in considering music as well. And it shows that we don't hear what's expected just on the basis of the sound signal. So, as in the relationship between physical intensity and loudness, so in the relationship between frequency. That is, the actual frequency, the repetition rate of a stimulus. Sound signal in the stimulus. And what you hear, the pitch, the level from high to low on this subjective range of pitch, you're not hearing what's there physically, just as in the case of intensity.
