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“Music is a moral law. It gives
soul to the universe, wings to the mind, flight to the imagination, and charm
and gaiety to life and to everything.” The philosopher Plato perfectly
summarized how music affects the individual. He also states that music is a
law, which is definite and unquestionable in science, which tells us that music
is strictly bonded to the field of mathematics and science. Many people
overlook the mathematics of music. However, understanding the symbiotic
relationship between the two is fundamental to enjoy it further. From chords to
scales, learning the theory behind music is filled with ratios, sequences, and
more. There are many people who analyzed and displayed the mathematics of
music, but most memorable of these people are the philosopher Pythagoras, and
the renowned musician J.S. Bach. These people led to breakthroughs in music,
not only letting people use their research for beautiful compositions, but
gifting people with an understanding of an art which is still taken for granted
to this day.

listening to music, there is one thing anyone can realize immediately: that no
instrument sounds the same. Though you can play the same note, the smooth
vibrations of a saxophone differ greatly from the reedy humming of a violin, or
the plucking sound of a piano. To understand why this happens, you must realize
that each instrument vibrates at a different frequency, which is the rate at
which something occurs over a period of time. Each instrument has a unique
frequency, which makes an orchestra sound in harmony rather than simply louder.
For example, in “Ava Maria” by Charles Gounod superimposed over “Prelude No. 1
in C Major, BWV 846” by Johann Sebastian Bach, the cello and the piano are
played together, which gives a charming effect rather than a boost in volume.
This charming effect is a blend in vibration from both instruments, which
matches the amplitude of the sound wave. This combination leading to such a
romantic sound is known as interference, or two sound waves interacting in a
positive or negative manner in order to create a new sound wave. This can lead
to a different wave frequency, or amplitude. As dictated by physics, the sound wave is always shown as a
sine or cosine curve, where the amplitude is marked by the number that is
multiplied to the sine expression, and the frequency is shown through the
number inside the sine function. This lets us graph sine and cosine curves for
musical notes, as well as analyze the differentiation between the curves by
juxtaposing them on the same plane. Using mathematics, we can find definite
values for musical notes, beats and more.

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harmonics and frequencies are important to music, nothing is more relevant to
the creation of music than the fractions of music. When writing sheet music,
the first thing you must do is decide how many parts of the whole are to be in
a measure. More commonly artists pick to use a 4 beat measure. In each measure,
notes must be distributed to match the mental metronome made by the four part
beat. Thus to record music on paper for a measure, notes were divided into
fractions. From whole notes, half notes, quarter notes, all the way to notes
that are one part of 64, rhythm is made by adding fractions to make a whole.
These notes are not limited to having the same denominator, which leaves a
multitude of combinations in a measure alone. Sheet music has a vast amount of
measures depending on the song, which creates many more possibilities in

addition to frequencies, there are various other components of music that are
deeply rooted in mathematics. Music would be nothing without intervals, or
ratios of pitch. To differentiate and create new sounds out of instruments,
octaves, fifths and thirds were formed. Intervals are used for scales, which is
a set of notes that are ordered by the frequency of the note. Using the ratios,
you can start with a note and figure out a scale from the frequencies of the
note. Scales are used for many things, like instrumental solos. Musicians are
drilled into knowing scales in different keys in order to improvise. For a more
common instance of these intervals, we must realize that vocal harmonizing is a
great method in which these musical ratios are used. In order to blend vocals
together when doing a duet, one member of the pair must deviate an octave
higher or lower than the other member. Octaves are musical intervals displayed
through the ratio 2:1, which means that from 100 Hertz to 200 Hertz in
frequency is an octave. As we understand about ratios, this means that no
matter what the original number is, when factoring the numbers to their
smallest form, we will always get the ratio of 2:1. A fifth is recorded as the
ratio 3:2, or 300 Hertz to 200 Hertz. As for the other three main intervals used,
a fourth is shown as 4:3, a Major Third is 5:4, and a Minor Third is 6:5. These
intervals are very helpful with creating music. The Circle of Fifths, shows a
relationship between notes in the chromatic scale. This can be used for many
things, such as transposing songs into a different key and writing music, which
both are used frequently in the world of music.

has been analyzed in a mathematical manner for centuries. One of the pioneers
of this was the philosopher Pythagoras, who contributed many ideas to the field
of mathematics, such as the Pythagorean Theorem. For music, Pythagoras analyzed
the first differentiation in frequencies through a blacksmith. As the story is
told, Pythagoras visited a blacksmith who was hammering metal with different
sized hammers. He realized that the different-sized hammers on the same metal
would create variations in pitch. He then created a wooden device that he
attached multiple, equal sized and weighted cords with different weights
attached to them. This gave him knowledge about the diatonic scale. He then
took the ratios of the strings, and noticed that certain strings together would
harmonize with each other. This led to him noticing octaves, fifths, and more.
He compared the weights with each other in order to make these ratios, and he
then created the Pythagorean tetractys, which is a diagram for all the ratios
and scales. Pythagoras treated music as a sub-division of mathematics, and knew
that the harmonies were dictated by proportions. To him, these harmonies became
synonymous with numbers and fractions.

on to another pioneer in the mathematics of music, we take a look at Johann
Sebastian Bach, whose music displays how mathematical this area can be. Bach is
lauded by many as the man whose history books invented “musical grammar,” or
the proper way to approach music. His collection of Preludes and Fugues, called
the Well-Tempered Clavier BVW 846-893 showed a precise placement of
notes in a mathematically accurate manner. Most of these are juxtaposed over
different instrument compositions in order to create a blend and give a feeling
that the song is completed. This is again shown in “Ave Maria” in which
displays the smooth cello of Gounod placed over one of Bach’s Preludes in C
major. Numerous artists have done this method, since Bach’s music was like a
formula. It was easy to compute, and even today’s artists keep it in mind at a
sub-conscious level. To many, J.S. Bach’s music analyzes why all these methods
work and why things we consider to be laws in music can be proven. Bach treated
all parts of music with a cold analytical glance, and was able to piece each
sound together with corresponding sounds on a scale or progression. Whether it
was composing, or even dancing, Bach realized that all of these are interwoven
together in order to create an individualistic world inside each song. This
explains why his preludes and fugues were used as a base for many other
instrument players.

essence, music is defined by the mathematics it is related to. Without math,
music would be less profound, less defined, and less interesting. There are
many things that taking math away from music would destroy. A concerto would be
more complicated to perform, as blending instrumental sounds would become more
difficult. A rock venue would feel forced and bland, as the musicians would
never variate from the song they created, which would end improvisation. Rap
and R&B music would fall flat as syllable counts and beats would lose
timing and rhythm. All the works of Johann Sebastian Bach would never have been
paired with any music, just leaving us with his book “Well Tempered Clavier”
with no understanding of the possibilities it held. When Pythagoras began to
analyze the mathematics of music, what he did not realize was how he would
shape the music world for years to come. He did not hypothesize that his
findings would lead to people deconstructing the sounds that some of us take
for granted, nor did he know that resonance and interference would allow for
amazing concertos, duets, and more. Taking away the mathematics of music would
be the equivalent of making music no longer an art form.

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