July 21, 2006
A Vibrating Guitar String

The last two weeks, we looked at the actual wave forms of some kalimba notes, the middle A and the low A on the ALTO. Now, we are going to look at the wave form of a guitar string vibrating, and we'll understand something of what makes the kalimba so special.

The figure above shows the envelope of the first six seconds of the low A string on the guitar. This note would have lasted like 15 seconds had I not cut it off. (Remember the kalimba's sustain was like 2-4 seconds.) On the far left, there is no sound at first. Then comes a jagged attack, followed by some big amplitude oscillations, and a decay, and THEN after 2 seconds, the guitar starts to get loud again! We will zoom in on three different spots in this wave form.

This figure has zoomed into the first half second of the note such that we can see the individual fundamental vibrations of the low A note. The "Zoom 1" arrow is pointed to the third oscillation peak. The little stuff at the very beginning is probably my thumb nail hitting the guitar string.

Now we have zoomed way into the "Zoom 1" region. This is about 1/3 of a second in past the attack. The vibrations in this note are not at all smooth. Superimposed upon the large scale vibrations of the fundamental A note are a lot of other vibrations at much higher frequency (i.e., they go up and down several times every time the fundamental vibration goes up and down once) -- I call them "hair" on these plots. But a very important point is that the hair is "harmonic" -- i.e., if you see a certain bit of high frequency hair on one of the vibrations, you will see that same bit of hair on the next vibration, and the next.

By the way, if you look at a kalimba note after 1/3 of a second, there is NO hair, just a nearly pure fundemantal vibration with nice smooth curves. On the kalimba, the high frequency hair is either damped out very quickly, or it gets converted into vibrations at the fundamental frequency. So, the question to ask about the guitar note: how long will this hair last?

Another thing we can do is calculate the frequency of the fundamental vibration.

By the way, I gave you some bad information in previous tips of the day: there are 30 frames per second! This explains why my frequencies were coming out low. I just assumed that my kalimba was out of tune! I have since corrected the Friday tips for the last two weeks.

OK: just to the left of the "00:00:00:12" label (12 frames), there is a vibration peak. Start counting peaks as we go right to the "00:00:00:15" label. There are about 10 and 3/4 waves in that 3 frame space. Hence, the fundamental frequency should be 10.75 cycles / (3/30)s = 107.5 Hz. (We could get a more accurate number if we sampled a longer time; A should be 110 Hz.)

This is the "Zoom 2" region, blown up. This is about 1 second after the note's attack, and the hair is dying out a bit, but we don't have a nice smooth note. That is, there is still a LOT of power in the harmonics.

This is the "Zoom 3" region, blown up. This is about 5 seconds after the note's attack, in the region where the note got louder again. Finally, all of that fuzzy hair has gone away, but LOOK AT THIS! The wave is not shaped like a simple SINE WAVE, but the wave's peaks have these noses that stick up and down, and there are little flat shoulders between the up and down noses. What is going on here?

What is going on here is called the first harmonic. As we saw in every single zoomed-in guitar wave form, a guitar string does not vibrate with just one frequency: rather, there are a whole host of frequencies vibrating all at the same time. We call these overtones or harmonics, and they are happening on every single guitar note. If you plug an electric guitar into a distortion box, they are happening even more! What has happened here is that most of the very high frequency harmonics (ie, the hair) have died away, but the first harmonic, which has a frequency exactly twice the fundamental frequency, is still going strong. So, if the frequency of the fundamental is 110 Hz, the frequency of the first harmonic is 220 Hz.

By the way, when we hear a guitar and we know that the sound is different than, say, a kalimba playing the exact same fundamental note, we can tell which is the guitar and which is the kalimba. The difference? Each instrument has its own set of harmonics (high frequency hair), and the different weights of those harmonics will make each instrument sound different to us.

So, a month ago, I put forth a mystery: how can the kalimba, with its tiny tines and little box, play a low note that is almost as low as a guitar?

The answer is: it can't, but it sounds like it can.

When you play a low A on the guitar, you have almost as much power in the first harmonic as you do in the fundamental. Your ears hear all of those, and your brain (at least the TRAINED brain) focuses on the fundamental and says "Oh, you are playing THIS note". Now, bring in the kalimba. The Alto kalimba's low A is actually one octave higher than the guitar's low A. BUT, the kalimba's low A is the same note as the guitar's low A's FIRST HARMONIC. SO, when you play your kalimba and all the harmonics die out very quickly, you have a very pure tone, and it is comparable to lower notes played on the guitar with their extra high harmonics. It is sort of an optical illusion played on you by your ears.