Pythagoras (570-495 B.C.E ) was an ancient Greek philosopher, mathematician, musician, and founder of Pythagorean-ism. Pythagorean teachings revolve around the idea that numbers are the source of all things:
Arithmetic = Number in itself
Geometry = Number in space
Astronomy = Number in space & time
Music = Number in time
Pythagoras believed that everything in the universe was directly related to numbers and mathematics and that each number carried its own specific vibration (This is still believed by some modern numerologists).
This idea was associated with the ancient philosophical concept of the Music of the spheres - the music made by the planets and the sun as they orbit the Earth. This “music” was considered more of a mathematical (relying on ratios) or religious concept than an audible sound. Due to the religious and scientific nature of this concept, Pythagoras believed that music was a holy science which should be studied but never be used for entertainment.
Pythagoras also studied the relationship between musical pitch and the length of string needed to produce it. He documented these ratios and developed the first scale based on mathematics. This scale divides an octave into intervals of tones and semitones. Although the ratio of frequencies of two notes an octave apart is 2:1, Pythagoras assigned the octave a ratio of 12:6 - allowing the remaining intervals to be calculated using only positive integers.
Pythagoras described the first four overtones which have become the building blocks of musical harmony: The octave (1:1 or 2:1), the perfect fifth (3:2), the perfect fourth (4:3), and the major third (5:4). He discovered that whole number ratios always produce harmonious intervals and Non-whole number ratios produce dissonant intervals. He used this idea of harmonious ratios to develop a medicinal practice for healing diseases of the body, the emotions and the soul. He “aligned souls to their divine nature” and performed what he called, “Soul Adjustments”.
A new educational scheme (or curriculum) was born from the teachings of Pythagoras. The quadrivium was the intellectual discipline of classical education consisting of Arithmetic, Geometry, Astronomy and Music.
Morris Kline (Professor of Mathematics and writer on the history, philosophy, and teaching of mathematics) classified the four elements of the quadrivium slightly differently:
Arithmetic = Pure
Geometry = Stationary
Astronomy = Moving
Music = Applied
It is interesting to note that music was considered a subject of the quadrivium (maths and science) rather than the three-pronged trivium (consisting of grammar logic and rhetoric). For Pythagoras, music was a way of understanding the structures of nature - a scientific endeavor which could lead to the creation of a perfect society.