“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.

When

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.

While

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

writing.

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.

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.

Moving

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.

In

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.