Pythagorean Influence On Musical Instrumentation Essay, Research Paper

Pythagorean Philosophy and its influence on Musical Instrumentation

and Composition

by Michael Anderson

Philosophy 101

“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 occured 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 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

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 the realms of instrumentation would never be the same again.

Furthermore, many composers adapted a mathematical model

for music. According to 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 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 two greatest musicians of all

time. Speaking of the greatest musicians of all time, 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 endeavours.

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 Pathagoras 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 intrepret the arithmetic, or

the sounds, or the nerve impulses and how our interpretation

matches what the performers, instrument makers, and

composers thought they were doing during their respective

creation. Pythagoras simply mathematized a foundation for these

occurances. “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

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.