To understand modern musical harmony, we must have an understanding of its discovery. Pythagoras is one individual credited with helping discover musical harmony some 2,500 years ago. At this point in time, music was not associated with mathematics. Rather, music was seen as a manifestation of the spirit of God: a theological or philosophical discussion, rather than a mathematical one (Pythagoras and Harmony). Since Pythagoras was one of the first to merge philosophy with mathematics, he found interest in turning seemingly unmathematical issues into mathematical ones. The story goes that one day, Pythagoras was walking past a blacksmith’s shop when he heard a pleasing harmony resonating from the hammers and anvils within the shop. He entered and was able to conclude that the relative notes produced were a result of the weights of the hammers. The following are the results he came upon:
By doing this, Pythagoras discovered something interesting when plucking a string, something we now call overtones (Ashton). By touching the string at certain ratioed positions and then plucking the shorter side, he was able to create the pitches described in the following diagram:
As is made obvious by these illustrations, Pythagoras’ fascination of harmony came down to one simple mathematical concept: proportions. These proportions and recurring numbers became so prevalent, that Pythagoras came to the conclusion there must be a metaphysical link to nature through harmony as well. His followers, enthralled by these mystical findings, created the following shape called a tetractys (Pythagoras and Harmony):
Pythagoreans became so obsessed with this figure that it grew to have religious significance for them. They would often swear oaths on the image, growing it to become a sacred symbol that represented celestial harmony. This led to an extension of applying proportional principles to the planets and the galaxy. They concluded that the distances between planets and other celestial bodies in proportion to each other produced celestial music. This led to further applications to the human body and self-harmony, even influencing Plato’s eventual famous philosophy on dualism (Robinson).