l on the belief that numbers were
l Instrumentationand Composition
by Michael Anderson
“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.
Mathematicallyhe 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
Rowell, Lewis (1983). Thinking About Music. Amhurst: The
University of Massachusetts Press.