## Math, Music, and Pentatonic Scales

No one knows how long humans have been creating music, but the earliest attempts at understanding the theory seem have been made by that strange society called the Pythagoreans. Its founder, Pythagoras, lived on the Greek island of Ionia, sometime between 600 and 500 B.C.E. He is famous mostly for the statement of the *Pythagorean Theorem* — a fact known to the Babyloneans for at least a millenium before him. There are lots of interesting legends — some of which may actually be facts — about the Pythagoreans, but I don’t have time to go into them now. As Casey Stengel used to say “You could look it up.”

Anyway, the Pythagoreans seemed to have experimented with the sounds produced by strings and hollow flute-like tubes of differing lengths. They noticed that (all other things being equal, like tension for strings or diameter for tubes) that the most pleasant intervals between notes are produced when the lengths were in simple ratios. For example, if the length of one string is double the length of another, then the shorter one produces a tone that, in our current terminology, is called *one octave* higher than the longer one. This is the interval between, say, a C on the piano, and the next C above it. It’s the interval from Do to Do in the scale Do Re Mi Fa Sol La Ti Do.

The next most interesting ratio is 3 to 2. If one string is 3 units long, and the other 2 units, the shorter string will be a *perfect fifth* above the lower one. This is the distance from C to G on the piano, or from Do to Sol, as in Do Re Mi Fa Sol. Other ratios are: 4 to 3: a perfect fourth (C to F or Do to Fa) and 81 to 64: a major third (C to E or Do to Mi). The third was considered a dissonance in early music.

We now know that sound is simply a vibration of the air; the faster the vibration, the “higher” the pitch. A modern “middle C” is 256 cycles or vibrations per second. The vibrations per unit time — say second — is called the *frequency*. If two similar strings or tubes have lengths in a certain ratio, then the *frequencies* they produce will be in the opposite ratio. In other words, the shorter the length, the higher the frequency. If C has a frequency of 256, then G should have a frequency of 256 times 3/2 or 384.

It turns out that the fifth, or 3:2 ratio, can be used to get all the other “named” notes in the scale that we use. Starting with C, say, we keep going up by fifths. This gives C – G – D – A – E – B – F# – C# etc. You eventually get back to C, so this is called the “circle (or cycle) of fifths.” If you take the first five of these, C, G, D, A, E and arrange them in order: C, D, E, G, A, you get what’s sometimes called the “major pentatonic scale”. *Pentatonic* means, simply, “five-toned.” There are other forms of pentatonic scales, but this is what is usually meant by the term. Play these on the piano or whatever instrument you can find, or go to

to hear them on your computer.

It turns out that there are serious flaws in trying to create all of the notes of the modern “12 tone” scale using the circle of fifths: you don’t exactly get back to where you started, and some of the intermediate notes which should be, say, a fifth apart themselves, aren’t! This created problems for centuries in playing music in various keys and on different instruments. It wasn’t solved until about the (early) 18 century with the development of the “well-tempered” musical tuning. The idea in well-tempering, basically, is to divide the ratio 2 to 1 (the octave) into 12 equal parts or ratios; 12 because there are 12 “half tones” in an octave: C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C. The frequency ratios between successive notes in the “chromatic” scale (all 12 tones) are set equal. This means that the product of all 12 of these ratios must equal the ratio 2:1 ( = 2 numerically) of the octave. Thus, each ratio is the twelfth root of 2. Now the twelfth root of 2 (written 2^{1/12}) is about 1.059463094 (by my calculator). Thus, to get from C to G, a space of seven of these half tones, requires multiplying the frequency of C by 2^{1/12} seven times. Mathematically this is 2^{7/12} which is approximately 1.4983: very close to 3/2 ( = 1.5), but not exactly. By making this ever so slight mathematical adjustment, we can build and tune instruments which sound beautiful together in all keys and in all combinations.

Thus we use math in our daily musical lives.

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