Musical scales are determined by a musical interval called the perfect 5th. Pythagoras discovered that if one tuned a string to any given note and then shortened that string by a third but keeping the same tension caused that string to play a perfect 5th higher. Likewise lengthening the string by a third caused it to play a perfect 5th lower. The perfect 5th is the basis of all harmony and all music. It is actually a scale in its own right, a 2-note scale. To add more notes to the scale, we simply pile on the perfect 5ths.

Western musicologists have always insisted that the full octave be perfect. A string halved exactly plays an octave higher. Conversely, when the string length is doubled, it plays an octave lower. The ear can pick out even tiny variations from the 1:2 ratio and so it is never violated. This octave was divided into 12 parts or half-steps, which are the smallest division of the octave. Two half-steps equaled a whole step the value of which is 9/8. The problem that Pythagoras discovered was that other intervals as perfect 5ths do not fit inside the octave exactly.

For example, the major 3rd interval spans three notes in 4 half-steps. Therefore there are three major 3rd intervals in a 12-half-step octave. The value of a major third is closest to 5/4 so we raise 5/4 by the power of 3 which is 1.953125. Very close to the octave value of 2/1 but not quite and the ear can hear that variation. So the major thirds must be stretched a little to fill the octave exactly. Since there are 6 whole steps in an octave then 9/8 raised to the power of 6 should equal exactly 2. It does not. The value is 2.02728653.

Again, the difference is very small but very noticeable. The reason this happens is that fundamental building blocks of the octave—the half-steps—are not exact. They have an arbitrary value of 16/15 but actually vary slightly from one note to the next. So we must stretch or shrink our other intervals to make them fit exactly within an octave. This process is called tempering.