I am going to explain the strange math in nature that requires us to “temper” our system of pitches in the chromatic scale. There have been many different ways of tempering our tuning system going back to the Renaissance, including no tempering at all, which required one fifth in the circle of fifths to be smaller than the rest. Those of you familiar with the music of Bach may be surprised to learn that “well tempered” is not necessarily the same thing as “equal tempered”.
The octave is the most basic part of our system of pitches. You get an octave by doubling the frequency of whatever is vibrating (a string, a wind column, a bell…). Conversely, you get an octave below the fundamental by halving its frequency. In all the tuning systems this ratio of 2 to 1 has been kept exact. We never tweak the octave.
You can double the frequency of, for instance, a string, by dividing the string in half. Doubling the frequency is the same thing as halving the wavelength of the sound wave. (frequency = 1/ wavelength, taking into account the speed of sound).
The next simplest frequency ratio after 2/1 is 3/1. Tripling the frequency (dividing a guitar string in thirds) gives us what we call “a perfect fifth”. This is the fifth above the first octave (of the fundamental frequency; the one we started with). The octave of the fundamental and the fifth above that octave are the first two overtones in our overtone series. (In well behaved pitched Western instruments, an overtone is the same thing as a harmonic, except harmonics include the fundamental and overtones start with the octave of the fundamental).
Since the perfect fifth above the octave of the fundamental is 3x the frequency of the fundamental, dividing that frequency in half gives us an octave lower, the fifth between the fundamental, F, and its octave. Therefore, F x 3/2 = perfect fifth, and F x 2 = octave.
To get through the circle of fifths on a piano you need to start on the lowest C and move up in fifths to the highest C (the highest note on the piano). This is 12 fifths. To get from the lowest C to the highest C via octaves takes seven octaves.
So if a perfect fifth were usable, that would mean that doubling the fundamental frequency seven times should get us to the same frequency (the highest C) as multiplying the fundamental by 3/2 twelve times.
Therefore, F x 2 x 2 x 2 x 2 x 2 x 2 x 2
should = F x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2 x 3/2.
We can simplify this with exponents and say:
F x 212 = F x (3/2)7
Using a simple fundamental frequency such as 100 Hz, we get:
100 x 27 = 100 x (3/2)12
12.8 kHz = 12.9746 kHz
Those pitches are off enough for us to hear them as out-of-tune.
Tempering means adjusting all the fifths by a tiny amount to make the circle consistent, whether you go by fifths or octaves. They must all be made a little bit smaller. Equal temperament means dividing that discrepancy by 12 and reducing the size of all the fifths by that amount. Well temperament can mean adjusting all the fifths by reducing the size of all the fifths by different amounts, so that all fifths are not the same but the math still works out and the music sounds “in tune” in every key. The sum of all the adjustments should still equal the discrepancy noted above.
There is so much more to all this and many implications that can be examined in much more detail. In the 16th century they would use perfect fifths for 11/12 of the circle and have one fifth be too narrow and sound horrible. This was called “the Howl”. Such a system would not work in today’s highly chromatic and modulating music. One drawback of using today’s most common tuning system, equal temperament, instead of another well tempered system is that our keys do not have as distinct of flavors or personalities as they did when each key might contain some slightly different intervals than the last.
You can find out lots of more details and implications by Googling “circle of fifths”, “equal temperament”, “the Howl”, etc. I also highly recommend a few books:
– How Equal Temperament Ruined Harmony (and why you should care) by Ross Duffin
– The Math Behind the Music by Leon Harkleroad
– The Science of Musical Sound by John Pierce
– On the Sensations of Tone by Herman Helmholtz – chapter XVI – The System of Keys