Pythagorean Essay, Research Paper
Pythagorean Philosophy and its influence on Musical Instrumentation and
Composition
“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).
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.