# 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 blacksmiths 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 numbers 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 upon basic themes.

Pythagoras and his mathematics created the foundation for musical education as it is now known. According to Rowell, Pythagoras began his experiments demonstrating the tones of bells of different sizes. “Bells of variant size produce different harmonic ratios. ” (Ferrara, 1991). Analyzing the different ratios, Pythagoras began defining different musical pitches based on bell diameter and density. “Based on Pythagorean harmonic relationships and Pythagorean geometry, bell-makers began constructing bells with the principal pitch prime tone, and hum tones consisting of a fourth, a fifth, and the octave. (Johnston, 1989). Ironically or coincidentally, these tones were all members of the Pythagorean scale.

In addition, Pythagoras initiated comparable experimentation with pipes of different lengths. Through this method of study he unearthed two astonishing inferences; When pipes of different lengths were hammered, they emitted different pitches, and when air was passed through these pipes respectively, alike results were attained. This sparked a revolution in the construction of melodic percussive instruments, as well as the wind instruments.

Similarly, Pythagoras studied strings of different thickness stretched over altered lengths, and found another instance of numeric, musical correspondence. He discovered the initial length generated the strings primary tone, while dissecting the string in half yielded an octave, thirds produced a fifth, quarters produced a fourth, and fifths produced a third. “The circumstances around Pythagoras discovery in relation to strings and their resonance is astounding, and these catalyzed the production of stringed instruments. ” (Benade, 1976).

In a way, music is lucky that Pythagoras attitude to experimentation was as it was. His insight was indeed correct, and therefore the realms of instrumentation would never be the same again. Furthermore, many composers adapted a mathematical model for music. For example, Rowell Schillinger, (a famous composer and musical teacher of Gershwin) suggested an array of procedures for deriving new scales, rhythms, and structures by applying various mathematical transformations and permutations. His approach was enormously popular, and widely respected. “The influence comes from a Pythagoreanism.

Wherever this system has been successfully used, it has been by composers who were already well trained enough to distinguish the musical results. ” In 1804, Ludwig van Beethoven began growing deaf. He had begun composing at age seven and would compose another twenty-five years after his impairment took to full effect. Creating music in a state of inaudibility, Beethoven had to rely on the relationships between pitches to produce his music. “Composers, such as Beethoven, could rely on the structured musical relationships that instructed their creativity. (Ferrara, 1991). Without Pythagorean musical structure, Beethoven could not have created many of his astounding compositions, and would have failed to establish himself as one of the greatest musicians of all time. In speaking of the great musicians, perhaps another name comes to mind; Wolfgang Amadeus Mozart. “Mozart is clearly the greatest musician who ever lived. ” (Ferrara, 1991). Mozart composed within the arena of his own mind. When he spoke to musicians in his orchestra, he spoke in relationship terms of thirds, fourths and fifths, and many others.

Within deep analysis of Mozarts music, musical scholars have discovered distinct similarities within his composition technique. According to Rowell, initially within a Mozart composition, Mozart introduces a primary melodic theme. He then reproduces that melody in a different pitch using mathematical transposition. After this, a second melodic theme is created. Returning to the initial theme, Mozart spirals the melody through a number of pitch changes, and returns the listener to the original pitch that began their journey. “Mozarts comprehension of mathematics and melody is inequitable to other composers.

This is clearly evident in one of his most famous works, his symphony number forty in G-minor. ” (Ferrara, 1991) Without the structure of musical relationship these aforementioned musicians could not have achieved their musical aspirations. Pythagorean theories created the basis for their musical endeavors. Mathematical music would not have been produced without these theories. Without audibility, consequently, music has no value, unless the relationship between written and performed music is so clearly defined, that it achieves a new sense of mental audibility to the Pythagorean skilled listener.

As clearly stated above, Pythagoras correlation between music and numbers influenced musical members in every aspect of musical creation. His conceptualization and experimentation molded modern musical practices, instruments, and music itself into what it is today. What Pythagoras found so wonderful was that his elegant, abstract train of thought produced something that people everywhere already knew to be aesthetically pleasing. Ultimately music is how our brains interpret arithmetic, sounds, and nerve impulses.

Leading from this, how each brain might absorb this interpretation matches what the performers, instrument makers, and composers thought they were doing during their respective creation. Pythagoras simply mathematized a foundation for these occurrences. “He had discovered a connection between arithmetic and aesthetics, between the natural world and the human soul. Perhaps the same unifying principle could be applied elsewhere; and where better to try then with the puzzle of the heavens themselves. ” (Ferrara, 1983).

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