Tuning Systems

Throughout history, many different tuning systems were developed and used by musicians. The very first tuning system emerged from Ancient Greece, but it was not very effective: certain notes "clash" under this system. Much later, systems such as just intonation and equal temperament were developed, in attempts to fix the tuning system created by the Greeks. Below, we will outline two different tuning systems. (Warning: math and music theory ahead!)

Pythagorean Tuning

The system of Pythagorean tuning was developed in Ancient Greece. Although it is named after Pythagoras, the famous Greek mathematician did not invent this system. According to legend, Pythagoras heard consonant sounds produced by hammers striking a forge, and discovered that the hammers' masses were related to each other by whole number ratios; this led him to develop the tuning system. In reality, Pythagorean tuning is much older than its namesake.

Pythagorean tuning relies on whole number ratios between note frequencies. Given a note with frequency f, we can create a new note by multiplying f by a certain ratio. For simplicity, let us assume that f corresponds to the note C. The most basic interval we can create is an octave, and it is obtained by multiplying f by 2. In other words, the notes with frequencies f and 2f correspond to the notes C and C', with the latter being an octave higher than the former. Next, if we take the note with frequency (3/2)f, we obtain the note G, which is a perfect fifth above C. Similarly, we can find F, which is a perfect fifth below C, by dividing f by 3/2 and then multiplying it by 2 (multiplying a frequency by 2 will result in the same note except an octave higher, as seen above). This gives us the frequency (4/3)f.

We now have the following note frequencies: C is f, F is (4/3)f, G is (3/2)f, and C' is 2f. The interval between F and G is a whole tone, and is represented by the ratio ((3/2)f )/((4/3)f ) = 9/8. Multiplying C by 9/8 we obtain D, which has the frequency (9/8)f. Likewise, we obtain the frequency of E by multiplying the frequency of D by 9/8, and so on. In this manner, we can "fill out" all the notes in the octave. The notes in the C major scale are therefore as follows:

Now, the interval between E and F is a semitone. The ratio representing the semitone is therefore ((4/3)f )/((81/64)f ) = 256/243. Say we want to find the frequency of C#. At this point, we run into a problem. Going "up" from C, we get (f )*(256/243) = (256/243)f. But going "down" from D, we get ((9/8)f )/(256/243) = (2187/2048)f. Clearly, these are not the same frequency! This reveals the flaw in Pythagorean tuning.

The further you go from the original note, the more apparent this kind of problem becomes. This becomes a major issue when modulating to distant keys; the instrument will sound very out of tune.

Equal Temperament

Several attempts were made throughout history to "fix" Pythagorean tuning. For instance, the Italian music theorist Giuseppe Zarlino (1517 - 1590) created just intonation, a system which uses even simpler frequency ratios than Pythagorean tuning. Although it works well in a single key, just intonation becomes problematic when modulating to a different key. Another tuning system was proposed by Vincenso Galilei (ca. 1520 - 1591), the father of the famous Galileo Galilei. He proposed that the ratio between any two note frequencies be 18/17, if the notes are separated by semitone. Although this fixes the problem of "clashing" frequencies, perfect fifths and octaves will always sound flat.

Finally, the equal temperament tuning system invented by Simon Stevin (1548 - 1620) fixed all the problems of Pythagorean tuning. Under equal temperament, if f is the frequency of the initial note, and F is the frequency of the note n semitones higher than f, then F is given by the equation F = 2^(n/12)*f. The resulting frequencies approximate Pythagorean tuning very closely, and allow the musician to play in any key. The German composer J.S. Bach (1685 - 1750) extensively took advantage of equal temperament in The Well-Tempered Clavier, where he wrote two preludes and fugues for every single key (see Baroque Keyboard Music for more information).