FREE ESSAY ON MUSIC AND MATH

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 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 Mozart's 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.
Mozart's 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).

Bibliography
Bibliography 
Benade, Arthur H. (1976). Fundamentals of Musical Acoustics. New York: Dover
Publications. 
Ferrara, Lawrence. (1991). Philosophy and the Analysis of Music. New York: Greenwood
Press. 
Johnston, Ian. (1989). Measured Tones. New York: IOP Publishing. 
Rowell, Lewis. (1983). Thinking About Music. Amhurst: The University of Massachusetts
Press.