The Pythagoreans were ridiculed on stage as superstitious, filthy vegetarians, yet never as mathematicians. The Pythagoreans exercised ascetic, monastic living, vegetarianism and common-ownership of belongings. They believed the elevation of the soul to God was to be attainable by means of mathematics. God is unity, and all on earth is plurality, consisting of contrasting elements. Harmony is divine, as it is a ratio of numbers.
Musical harmonies are numerical ratios.
A string or flute shortened to half of its original length produces a tone which is one octave higher. All of these are of which the Pythagoreans had knowledge. They classified numbers in to categories of odd, even, prime, composite, perfect and amicable numbers. They used stones or pebbles, in groups to form different patterns of which they classified figurate, triangular and square numbers. Aside, the Pythagoreans used the symbol of heath to distinguish them from others.