Pythagorean Essay, Research Paper

Pythagorean Philosophy and its influence on Musical Instrumentation and


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


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).

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.

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