Mathematics Physics High School SMILE Meeting
23 November 1999
Notes Prepared by Porter Johnson

Roy Coleman and Lee Slick [Morgan Park High School]

They supervised our preparation of a "turkey caller" (originally developed by Bob Grimm), for special use just before Thanksgiving. The ingredients were a plastic cup, cotton string, a toothpick, and a rag wet with water. Punch a small hole in the plastic cup, push the string through it, and tie the string around the toothpick. You should break the toothpick in half before tying it. Then, pull the string so that the toothpick is stuck at the bottom of the cup. While holding the cup by the edge, pull the string, using the wet rag as protection for your hand. You should be able to get the most plaintive and distressing sounds out of this device, which is a sound generator. It works through a "slip-stick" mechanism, in which the string alternately slips and sticks as it goes through your hand, producing sounds associated with a driven vibrating string.

Arlyn van Ek [Illiana Christian HS]

Bill Shanks [Music Student at Joliet Junior College; formerly at Joliet Central HS]

The Physics of Music is hard to cover in Physics classes these days, because very few students know much about MUSIC, it seems. What is music, anyway? If a mass is attached to an elastic object [spring or whatever], that mass can be made to vibrate, and such vibrations produce sound. For example, a stretched string or a vibrating column of air may produce sound. One example of such a vibrating object, used both in Physics Labs and by piano tuners, is the tuning fork. The tuning fork is built to oscillate at a specific frequency [along with overtones]. One can amplify the sound by touching the vibrating fork to a flat surface, such as a desk.

A vibrating string with fixed ends held under tension can have standing waves that have an integral number of half wavelengths on the string. If the string is of length L, then the wavelengths will be ln, where L = n ln/2; or ln = 2 L/ n. The resonance frequencies are multiples of the fundamental frequency n0; i.e., n0; 2 n0; 3 n0; 4 n0; ... In a plucked string, several harmonics are excited, depending upon how and where you pluck it. The difference in between various stringed or wind instruments is determined by the excitations of different distributions of harmonics. You can achieve "vibrato" in stringed instrument by changing the tension in the strings by moving your fingers.

How do you tuned musical instruments, such as pianos? Well, you can set one string frequency absolutely with a tuning fork, by listening to the beat of the tuning fork with the string. Also, take into account that an octave is a factor of two in frequency, and that the standard note is "A", corresponding to a frequency of 440 Hz. There are 12 half tone notes on a piano in an octave, and according to the generally accepted chromatic scale or equal temperament scale, the half tone notes are taken a factor of 21/12 = 1.059463094 apart in frequency.

Here are the ratios of frequencies for notes that are n half-tones apart, for various n.

[I have given them all, so that you can see the approximate correspondences upon which primitive western harmony is based -PJ]

Factor			  n

1.05946309		  1
1.12246295		  2
1.18920712	*	  3
1.25992105	**	  4
1.33483985	***	  5
1.41421356		  6
1.49830708      ****      7
1.58740105		  8
1.68179283		  9
1.78179744               10
1.88774862               11
2.00000000               12
Note that, for n = 3, the Factor is fairly close to * 6/5, for n = 4 it is fairly close to ** 5/4, for n = 5 it is fairly close to *** 4/3, and for n = 7 it is fairly close to **** 3/2.

Alternate scales exist, such as the Pythagorean scale, the Mean Tone 1/4 scale, or the Just Scale, in which some of these ratios are set exactly at the "harmonic values" for some notes, but you cannot set up all the notes in that fashion, since that would be equivalent to making powers of the twelfth root of two to be rational numbers, and they aren't.. The chromatic scale is the one to use, in spite of the fact that the harmonics are approximate, but not precise, in it, because the ear is a forgiving detector!