Most modern concepts of consonance have of course not stuck with Pythagorean philosophy, but have gone on to other ways of thinking about why it is that some music tone combinations are pleasing and others relatively less pleasing. And the person who is historically most important in this respect is Hermann von Helmholtz. So you may well have heard of Helmholtz in other contexts. His major work I think was on vision. He was also a physicist of major proportions in the 19th century, and thermodynamics is attributed to his, in part to his insightful thinking in the second half of the 19th century. But he was very interested in music. A competent musician, as many intellectuals of that era were. A competent keyboardist himself, and very interested in explaining consonance and dissonance in physical terms. What he did was to understand, and he was, this guy was a genius of the first order, not only in physics, but also in thinking about music and vision. Among other things, he invented the modern ophthalmoscope. But his idea was that the reason for consonance and dissonance had to do with the physical nature of what he called beating and roughness that emerge when two tones are played together. And he imagined that, well, he didn't imagine, he demonstrated that beating and roughness arose from tones that, when played together, caused a constructive interference that led to what you see here. So let's consider this in a more specific way. So here is C,and here is C sharp. And the frequencies in the middle range of the piano are 262 Hz for C and 277 Hz, cycles per second, for C sharp. And you can see that when these two tones are played together, and they form a minor second. That they interfere with each other, causing an auditory bumpiness, which we'll hear in a second, that has a frequency of about 30 Hz. So, what Helmholtz surmised, and indeed demonstrated mathematically, is that when two tones interfere with each other in this physical way, generating a tonal combination that has constructive interference that has peaks and valleys at a relatively low frequency. Somewhere in the range of 15 to 30 or a little bit more, in terms of cycles per second, you hear a bumpiness, a roughness, to that. And as the frequency increases, that roughness tends to go away. And based on that, so if you hear a sine tone or a piano tone, any tone, at say a frequency of 262 hertz or 277 hertz, that tone sounds fine. You don't hear any roughness in C or C sharp played in the middle range of a piano. But when you play them together, they combine to form a change in up and down variation in the amplitude that you hear as he described it, rough or bumpy. And his idea was that this roughness or bumpiness, when you heard it, was the definition of dissonance. That, that roughness or bumpiness was annoying, dissonant, displeasing, and for that reason, he defined consonance as the absence of dissonance. Let's listen to two sine tones that present an example of this roughness. Where the destructive and constructive interference of the tones causes this up and down bumpiness that you can hear, and that Helmholtz attributed to the reason that we find certain tone combinations displeasing. And he considers, as I said, the absence of this displeasing quality of bumpiness and roughness to be the definition of consonance. [SOUND] This idea presented by Helmholtz in the latter part of the 19th century. The book that he wrote on this called, On The Sensations of Tones was published in 1862. And as you can see, it's still in press today, indicative of the fact that what he says is eminently sensible, and many people still take this today as the primary explanation of consonance and dissonance. Roughness, dissonant, the absence of roughness, consonant. But there were problems from the outset in accepting this definition of consonance and dissonance. And one of them is indicated here, perhaps the main one, there are others. And in some of the readings that you will have available to you and in the bibliography you can read more about this. But one of the main problems that was recognized, pretty soon in the early 20th century, was that while you present two tones. Let's say two sine tones, to the right ear, and the left ear independently, there's no longer any beating, because the two tones are coming in through two different ears. So they're not being combined physically at the ear. One, the C and the C sharp, for example, are being presented to the right ear and the left ear respectively. So even though they're presented to the two ears, you still hear them as dissonant tone combinations. Even though the physical beating, by virtue of their independent presentation to the right and the left ear, is gone. So that, and several other incidences where consonance and dissonance really didn't make sense as explained just as beating and roughness. Led people to think of other ways of explaining consonance and dissonance, and this is what we're going to turn to next.
