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
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
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
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