[MUSIC] These early lectures have been laying the groundwork for future lectures dealing with the nuts and bolts of musical materials. Today we're going to talk about acoustics and tuning. I think it's helpful to understand a bit about acoustics and tuning when we talk about scales and keys. Plus, you'll get a little head start on our discussion of intervals next week. If you play guitar, you may have noticed that your perfectly tuned guitar sounds great in one key, but terrible in another key. You may figure that your guitar has simply gone out of tune. Maybe it has, but it's more likely that you tuned it to be acoustically in tune using harmonics or open strings. Believe it or not, tuning open strings to be perfectly in tune can mean that not all keys will be in tune. Why? As Pythagoras showed 2500 years ago, nature isn't in tune. It's impossible to tune a string instrument to be perfectly acoustically in tune for all keys. I'm sure you've seen a piano keyboard and noticed its repeating pattern of white keys and black keys. There are total of 12 notes within each unit of the repeating pattern. The white keys use the seven letters A through G, and the black keys use the sharp or flat symbol with the letter name of an adjacent white key. Sharp raises the pitch, moves to the right on the keyboard and flat lowers it. Notice that each black key can have at least two labels, a sharp one, and a flat one. So this note could be called either D sharp or E flat. We'll talk about the rules for deciding which label to use in future lectures. These notes, both Cs, sound the same except that the one on the right [SOUND] is higher than the one on the left [SOUND]. We say that these two Cs are octaves of each other and the interval [SOUND], the distance between the notes, is called an octave. Octave means eight which is the number of white keys between the two Cs if you count the first and last ones. One, two, three, four, five, six, seven, eight. All octaves must be exactly in tune for the instrument to sound in tune. A note is in tune with another note when there are no interference beats [SOUND] when they are played together. You may think that's a pretty obvious idea. Shouldn't all intervals be in tune? Well, no. If the octaves are going to be in tune on an instrument, other intervals must be out of tune. Let's use my ukulele to discuss this acoustic problem. The C string on my ukulele vibrates between the nut and the bridge. As you can see from the tape measure, the vibrating string is about 13.5 inches long and sounds like this [SOUND]. Here's a picture of the entire vibrating string. This represents the fundamental vibration of the open string on my ukulele. There are other vibrations that aren't being shown here. When I divide the string in half, we get the same note up an octave. So splitting a string in half creates a note that is an octave higher. This is a picture of the vibrating strings split in half, sounding up an octave from the open string. Notice that both sides of the half-way point sound the same note when plucked. This works mathematically too. The C string on my ukulele is vibrating approximately 262 times a second, which we call 262 hertz, which means cycles per second. Here's what 262 hertz sounds like using a pure tone generator. [SOUND] Splitting the string in half makes it vibrate twice as fast. That is we double the frequency to 524 hertz, which sounds like this [SOUND]. 262 hertz is called C4 or middle C, because it's in the middle of the piano keyboard. So an octave higher, 524 hertz is C5. Going back to my ukulele, when we divide the string into thirds by pressing down on the fret and around 4.5 inches, we get a note that is an interval called a perfect fifth above the original. More specifically, we are splitting the string into segments that are one-third and two-thirds the original length. Let's define perfect fit before we go on. In a seven note scale, which the eighth note is the same as the first note up an octave, the interval between the first note and the fifth note is called a fifth. So starting on C, the fifth note up is a G. This fifth between the first note of a scale and the fifth note of a scale is called a perfect fifth. More about that later. Back to our vibrating string. Splitting the string into thirds creates a perfect fifth on the longer segment and a perfect fifth plus an octave on the shorter segments since it's only half the length. The two sides of the subdivision vibrate at different speeds. The shorter the string the faster the vibration and the higher the pitch. So the longer right hand side vibrates an octave lower than the left hand side since the two to one equals an octave. This works mathematically also. Dividing the string into thirds means that each of those thirds will vibrate three times faster than the original or 786 hertz. This is what the short side of this subdivision sounds like. [SOUND] The long side of the string is twice as long so it sounds an octave lower at a frequency 393 hertz, that is half the value of the short side, or three halves the value of the original. Here's what that sounds like. [SOUND] If we do this repeatedly, dividing the string into thirds, going up by perfect fifths, C to G to D to A to E to B to F sharp to C sharp to G sharp to D sharp to A sharp to E sharp, we'll eventually get to B sharp, which is hypothetically the same as C. Mathematically what we are doing is multiplying the frequency each time by three halves until we reach a very high C, which would be C11, which in fact isn't audible, but has a frequency of 33,993.54 hertz. If we lower it by an octave six times, dividing by two, we end up at C5, the octave above middle C, with the frequency of 531.15 hertz, which sounds like this. [SOUND] But if we take our starting C4, 262 hertz, and raise it an octave by doubling its frequency, we get a different number. Octaves are always in tune, so 524 hertz is what an octave above C4 should sound like. Here's what the true octave above middle C is supposed to sound like. [SOUND] Here's what the out of tune C above middle C sounds like. [SOUND] It's a small but noticeable difference. So what? I'll tell you so what. It means that when your piano tuner tunes your piano she needs to tune some of the notes out of tune in order to make the octaves in tune. It means when you tune your guitar using harmonics, which are perfect fifths, you'll end up with strings that are a perfect fifth apart but they won't be exactly an octave apart when you start pressing down on the frets. Tuning is the art of the compromise. Some of the notes on the piano must be out of tune in order for the whole thing to be in tune, as are some of the notes on your guitar. So here's what we covered in this lecture. We looked at the piano keyboard and we learned where the various notes are. We learned about octaves and fifths. We learned how dividing a string in two, doubling the frequency, raises a pitch by an octave. Dividing it into thirds raises it by a fifth. And we saw how a C is not a C if you get to it the long way by repeatedly going up a perfect fifth. [MUSIC]
