music and math
Music is the harmonization of opposites, the unification of disparate things, and the conciliation of warring elements...Music is the basis of agreement among things in nature and of the best government in the universe. As a rule it assumes the guise of harmony in the universe, of lawful government in a state, and of a sensible way of life in the home. It brings together and unites. - The Pythagoreans. Every school student will recognize his name as the originator of that theorem which offers many cheerful facts about the square on the hypotenuse. Many European philosophers will call him the father of philosophy. Many scientists will call him the father of science. To musicians, nonetheless, Pythagoras is the father of music. According to Johnston, it was a much told story that one day the young Pythagoras was passing a blacksmith’s shop and his ear was caught by the regular intervals of sounds from the anvil. When he discovered that the hammers were of different weights, it occurred to him that the intervals might be related to those weights. Pythagoras was correct. Pythagorean philosophy maintained that all things are numbers. Based on the belief that numbers were the building blocks of everything, Pythagoras began linking n
umbers and music. Revolutionizing music, Pythagoras’ findings generated theorems and standards for musical scales, relationships, instruments and creative formation. Musical scales became defined and taught. Instrument makers began a precision approach to device construction. Composers developed new attitudes of composition that encompassed a foundation of numeric value in addition to melody. All three of these approaches were based on Pythagorean philosophy. Thus, Pythagoras’ relationship between numbers and music had a profound influence on future musical education, instrumentation, and composition. The intrinsic discovery made by Pythagoras was the potential order to the chaos of music. Pythagoras began subdividing different intervals and pitches into distinct notes. Mathematically he divided intervals into wholes, thirds, and halves. Four distinct musical ratios were discovered: the tone; its fourth; its fifth; and its octave. (Johnston, 1989). From these ratios the Pythagorean scale was introduced. This scale soon revolutionized music. Pythagorean relationships of ratios held true for any initial pitch. This discovery, in turn, reformed musical education. With the standardization of music, musical creativity could be recorded, taught, and reproduced. (Rowell, 1983). Modern day finger exercises, such as the Hanons, are neither based on melody or creativity. They are simply based on the Pythagorean scale, and are executed from various initial pitches. Creating a foundation for musical representation, works became recordable. From the Pythagorean scale and simple mathematical calculations, different scales or modes were developed. The Dorian, Lydian, Locrian, and Ecclesiastical modes were all developed from the foundation of Pythagoras. (Johnston, 1989). The basic foundations of musical education are based on the various modes of scalar relationships. (Ferrara, 1991). Pythagoras’ discoveries created a starting point for structured music. From this, diverse educational schemes were created
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