[This is mainly for those of you who (like me) are interested in the mathematical qualities of music, but it's good information for anyone.]

Producers of electronic music deal quite a bit with frequencies; cutting them out, shifting them over time, boosting them, changing the proportions between them, and so on. "Sound waves" are our way of conceptualizing the motion of air molecules that occurs when some mechanical or electrical event (which we refer to as "striking a note") happens, and frequency is a measure how fast these waves of molecules move up and down. Frequency is specified in "Hertz (Hz)," which means "cycles per second."

Low frequency wave (top) and high frequency wave (bottom):



In terms of human hearing, lower and higher frequencies mean lower and higher "pitches," and musicians and musical theorists in the past made organized systems of pitches to designate what frequencies their instruments should play; they based these systems of pitches both on what was pleasing to the human ear and on what was mathematically "beautiful," and we still use these systems today.

What a tuning system specifies is how the physical frequencies designated by musical "notes" relate mathematically to one another. In Western music, the "reference frequency" used to tune most instruments is called "A 440" or "A4." That is, "A4" on a keyboard or other instrument is defined as the key which, when pressed, will cause the instrument to release a sound wave with a frequency of 440 Hz. When a note (call it "X") is an octave above or below another note (call it "Y"), that means that note X has a frequency either twice that or half that of note Y, respectively. A half-step is the distance between one note and the next (ratio between the two frequencies) in a given Western tuning system.

All the stuff mentioned in the last paragraph is universal in traditional Western music and in all non-experimental modern pop music, which includes most electronic music. Where things get complicated is in how specific tuning systems derive the frequencies of individual notes from octave to octave. The most popular tuning system in modern Western music is called equal temperament tuning (more specifically, twelve-tone equal temperament tuning). It gradually became the most popular because it contains no "wolf intervals" (very harsh-sounding frequency ratios which make it unfeasible to use an acoustic instrument to play in multiple keys without retuning it). Here is how equal temperament tuning works:

Equal Temperament Tuning

Twelve-tone equal temperament tuning, or 12-TET for short, specifies the frequency difference from one note to the next by using a constant ratio. This ratio is derived in the following manner:

Remember from above that two notes being an octave apart means that the pitch of one is twice that of another. Consider two notes, A4 and A5, which are an octave apart. We can represent their frequencies in the following way:

Frequency of A4 = f(A4)

Frequency of A5 = f(A5)

Since A5 is twice the frequency of A4, the ratio of f(A5) to f(A4) is 2 to 1 (2:1).

Now remember that there are twelve tones in our desired tuning system, and we want the tones to increase or decrease using a constant ratio from frequency to frequency. We will designate this ratio "R." If our system is going to have twelve tones, then it must be the case that R is a number that when multipled by itself twelve times will equal two: otherwise we could not use it to specify the twelve intervals that go to make up an octave in a Western tuning system. Another way of saying "R multiplied by itself twelve times" is just R^12 ("R to the twelfth power").

So this means that:

Ratio of f(A5) to f(A4) = 2:1 = R^12.

How do we remove the exponent from R^12? Easy: we take the twelfth-root of both terms in the following equation:

2/1 = R^12

...so...

Twelfth Root of 2 = R

The twelfth root of 2 is approximately 1.05946309. So this is the number we will use to specify the ratio of the frequency of one note to the frequency of the note below it. Consider A4; its frequency is 440 Hz. To derive Bb4, the next highest note in the tuning system, we would multiply 440 by 1.05946309. This gives us 466 Hz, which will be the frequency of Bb4 in our system. To get the next highest note, B, we would multiply 466 by 1.05946309 (or 440 by 1.05946309 times 1.05946309, which amounts to the same thing). This will give us 494 Hz, the frequency of B.

By now you can see the pattern: to get a note that is X half-steps up from the "root" note, we must multiply the frequency of the root note by R^X. If you have any trouble understanding this process, the following chart should help clear things up:



How would you derive the frequencies of notes below A4? Easy: instead of multiplying in order to go up, you would divide to go down, using the exact same pattern -- X semi-tones down from A4 = 440 / (R^X).

Other Tuning Systems

For most of Western history, tuning systems other than 12-TET were used. These include meantone temperament, Pythagorean tuning, and just intonation. It would take many more pages to explain all of them and their advantages and disadvantages, so I have made the terms into Wikipedia links for those of you who are interested.

Hope you enjoyed reading this.