Essay, Research Paper
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. "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).