The famous Greek mathematician, Pythagoras, and his followers studied numbers and their relationship to musical harmony. The Pythagoreans believed that harmony consisted of numerical ratios. The concept that ratio between two numbers is an entity is essential in music theory. For example, we can apply ratios to

• Two tones, C# / C = 1.059463094… or
• Two frequencies in an octave, A₄ / A₃ = 440 Hz /220 Hz = 2 / 1.

Pythagoras conducted experiments with a musical instrument called a monochord, a stretched string with a movable bridge as shown in the diagram below.

He found that the shorter the string, the higher the pitch (frequency).

Pythagoras moved the bridge to shorten the string to half of its original length. What is the ratio of the higher pitch (new length) to the lower pitch (original length.)

frequency ratio = f _new / f _old = 2 / 1

string-length ratio = l _new / l _old = (1/2) / 1 = 1 / 2

These two ratios are reciprocals of one another. Also, octave is obtained by shortening the string to one half its original length.

Pythagoras also found that in addition to the octave, the fifth and fourth notes sound good (called consonant.)  What length should you shorten the string to obtain these notes?

Estimate the answer:

The ratios of the new length to the original length are.

Fourth note 3 to 2 ; fifth note 4 to 3
Fourth note 3 to 1 ; fifth note 4 to 1
Fourth note 4 to 1 ; fifth note 5 to 1