The Pythagorean Scale

The proportions of musical tones can be found in many different manifestations of nature, from the microcosm to the macrocosm. The Pythagoreans believed that music has hidden within it the harmony of the universe. They claimed that what we essentially hear when listening to music are the sacred proportions of nature.

Pythagoras was the first to observe the harmonic vibrations and formulate their relationship. In other words, a string vibrates not only as a whole but also in its various segments, which are integer divisions of its original length.

By knowing that the length of a string is inversely proportional to the pitch (frequency) of the sound it produces, Pythagoras was able to determine the pitches of the sounds generated by these harmonic vibrations. Using only two intervals, he defined all the tones of a musical scale. These two intervals are the Octave (dividing the string in half) and the Fifth (dividing the string into two-thirds).

A more detailed explanation is as follows: If a string producing the note C is divided in half, it produces a harmonic note that is also a C but one octave higher. This is known as the Octave Rule.

If the string is divided into 2/3, the resulting pitch is at a ratio of 3/2 of the fundamental pitch, and the musical interval is a fifth. This is where the note G is defined, and this is known as the Fifth Rule.

If we further divide the G note by 2/3, the resulting note is D, but one octave higher. In other words, the Interval of a 9th is created. To fit within the original octave, Pythagoras introduces a new rule called the Octave Complement. What it essentially does is double the length of the string, resulting in a ratio of (4/9) * 2, which is 8/9.

So we have:

G = 2/3 of C

D = 2/3 of G = (2/3) * (2/3) of C = 4/9 (a fifth of a fifth). But: 4/9 < 1/2
Therefore: D = (4/9) * 2 = 8/9 (octave complement)

A = 2/3 of D –> 2/3 * 8/9 = 16/27

E = 2/3 of A –> 2/3 * 16/27 = 32/81 However: 32/81 < 1/2.
Therefore: E = (32/81) * 2 = 64/81 (octave complement)

B = 2/3 of E –> 2/3 · 64/81 = 128/243

This is the “mechanism” through which are defined the notes.

From C, he goes to G. From G, it goes to D, where it is applied the octave complement. From D, again using the rule of the fifth, we finds A. Then we perform the octave complement and continue with the same reasoning. Thus, with the rule of the fifth, which unlocks everything, and the octave complement, which is tautophony (unison), we have all the notes.

Below are all the notes of the Pythagorean scale and their corresponding numerical ratios.