Equal Temperament: Why Thirds Are Out of Tune

In the lesson on harmonics, you saw that every note contains a series of neatly lined-up frequencies: two, three, four, five times the fundamental. These multiples are the source of the intervals that sound most pleasant to the ear. Yet your DAW doesn't use them as they are. It divides the octave into twelve perfectly equal semitones. This lesson explains why, and what you gain as well as what you lose.

Natural intervals

When two notes have frequencies in a simple ratio, their harmonics meet exactly, and the interval sounds calm, with no beating. A ratio of 2 to 1 gives the octave, for example 110 and 220 hertz. A ratio of 3 to 2 gives the perfect fifth, for example 220 and 330 hertz. A ratio of 4 to 3 gives the perfect fourth, 5 to 4 the major third and 6 to 5 the minor third. These are called pure, or natural, intervals.

Take the major third above A3, which vibrates at 220 hertz. The pure ratio puts it at 275 hertz. The fifth harmonic of the A, at 1,100 hertz, then lands exactly on the fourth harmonic of the third, also at 1,100 hertz. Nothing beats, and the third sounds perfectly settled.

The fifths don't close the circle

If pure intervals sound so good, why not tune the whole keyboard with them? Let's try. Start from C and go up by pure fifth after pure fifth: G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, E♯, and the twelfth fifth should bring you back to B♯, that is, to a C, seven octaves higher. The math says otherwise.

Twelve pure fifths multiply the frequency by 1.5 twelve times in a row, which gives about 129.75. Seven octaves multiply it by 2 seven times in a row, which gives exactly 128. There's a small surplus of about 23.5 cents left over, called the Pythagorean comma. In other words, pure fifths never land exactly on the octave: somewhere, you have to cheat.

For centuries, musicians tried different ways of spreading out this cheat, called temperaments. Some kept very pure thirds in the common keys, at the cost of a few keys that were almost unplayable. The solution that won out for keyboards and for most Western music today is the most even-handed one: equal temperament.

Twelve perfectly equal semitones

Equal temperament divides the octave into twelve identical semitones. Each semitone multiplies the frequency by the twelfth root of 2, about 1.0595, and measures exactly 100 cents. The comma is shared out equally among the twelve fifths: each one is shortened by about 2 cents, and the circle closes exactly. All the keys then sound the same; only the pitch changes.

That's what makes your piano roll so convenient. You can select a loop and move it up three rows without anything changing color. A C major chord has exactly the same shape and the same sound as an F sharp major chord, just higher. Scales, Roman numerals, transposing a sample: everything you've learned in this course rests on this regular division.

What the ear loses

The price to pay is that, apart from the octave, no interval is exactly pure. The table compares each pure interval with its version in equal temperament. The fifths and fourths are almost perfect. The thirds and sixths, on the other hand, drift clearly away from their natural ratio.

So your DAW's major third is about 14 cents too wide. On A3, the pure third would vibrate at 275 hertz, whereas C♯4 vibrates at about 277.2 hertz. The harmonics that used to meet exactly now brush past each other a few hertz apart, and they beat, about eight to nine times per second. Listen to the difference with a sound rich in harmonics.

The tempered third has a small restlessness, a slight shimmer that the pure third doesn't have. For the fifth, the difference is almost imperceptible: its harmonics beat less than once per second. Don't let that worry you, though: our ears are used to equal temperament, and most listeners don't notice a thing.

In the last lesson of this part, we move from the time of vibrations to the time of the track, translating tempo into milliseconds to set your delays, reverbs and LFOs.