Mathematic Exploration: The Relationship

Between Math and Music

Destre

Tashjian

International

Baccalaureate Mathematics 4.1

December

20th, 2017

Table of Contents

INTRODUCTION

3

EXPLORATION

5

Fibonacci sequence

5

Golden RATIO

6

Connection to Math………………………………………………….…………………………..6

Conclusion………………………………………………………………………………………..9

INTRODUCTION

When you stop to

think about it, there’s a remarkable connection between math and music. At the most basic level, a musician must

master fractions and decimals. Because

every measure has the same number of beats, the musician must perform the same

mental gyrations as a mathematician does when calculating percentages. Educators claim that students who play musical instruments

seem to perform better in mathematics. I have a love for music, and through this analysis I

hope to get a deeper appreciation of both topics.

The

math goes much deeper than simply rhythm and meter. Chord progressions develop predictable

harmonic patterns. Notes are frequencies

that are measured mathematically in hertz.

Tuners are calibrated mathematically.

“Most

amazingly, the most beautiful chords can be defined mathematically. The Fibonacci sequence is a series of numbers

where a number is found by adding up the two numbers before it. Starting with 0

and 1, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth. Written

as a rule, the expression is xn =

xn-1 + xn-2.”1 This was called the golden ratio. It has been observed in everything from

nautilus shells, to sunflowers and even to galaxies. The Fibonacci sequence in musical

scales is also a series golden ratios.

Even

the famed mathematician Pythagoras has something to teach us about music. “Pythagoras (6th century BC) observed that

when the blacksmith struck his anvil, different notes were produced according

to the weight of the hammer.”2 He further went on to document that music

notes had proportional relationships that could be measured and predicted

mathematically.

In

this essay, I will use frequency graphs to demonstrate the golden ratio in math

and music.

“Named after Fibonacci, also known as

Leonardo of Pisa or Leonardo Pisano, Fibonacci numbers were first introduced in

his Liber abaci in 1202.” Fibonacci was the son of a

Pisan merchant who traveled and traded often and extensively. From his youth,

Fibonacci loved numbers. This helped him

as a trader because math was a vital part of the trading industry.

Growing up in North Africa, Fibonacci

studied in the Hindu-Arabic arithmetic system: what is said to be the first of

number systems. Thus, he wrote several books ranging in the topics of geometry

and commercial, arithmetic, and irrational numbers and took part in the

development of the concept of zero.

To put it simply, the Fibonacci

sequence is the foundation for art, beauty, and life.

It has showed itself in the oceans through nautilus shells, in fields through

sunflowers, and ultimately, in our own and other galaxies. Likewise, is shows

itself in all kinds of music.

Fibonacci Sequence

It

is incredible how such complex features in our lives originate from a basic

sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …The next number is found by adding

up the two numbers before it. The 2

is found by adding 1+1, the 3 is

1+2, the 5 is (2+3), and so on. When

squares are made with those widths, a spiral is formed.

Like

all math, this must be written as a rule. When written out, the terms are

numbered from zero onward like this:

For example, term number 7 would be called X7,

equaling 13. When written as a rule, it is:

xn =

xn-1 + xn-2

Golden Ratio

With a grasp on

what the Fibonacci sequence is, one may now delve into the idea of the Golden

Ratio. When any two successive Fibonacci Numbers are taken, their ratio is very

close to the Golden Ratio:

Golden Ratio = ?

? 1.618034…

Any number in the Fibonacci

sequence can be calculated using the Golden Ratio:

The answer

always comes out as a whole number, exactly equal to the addition

of the previous two terms.3

Connection

to Music

So, how

does this comprehensively connect to music? The answer to this question is

simultaneously simple and complex. There

are several connections uniting the Fibonacci sequence and a piano. Many argue

that math is in the keys themselves and the music they make.

For example, when we look at a piano, we see:

1

2,

3

5

8

13

Octave

Number

in each set of black keys

Number

of black notes

Number

of white notes

Total

number of notes

This clearly displays the first few numbers of the Fibonacci

sequence. Some do argue that this theory is a stretch, but

Let’s talk music.

“There are 13

notes in the span of any note through its octave. A scale is composed of 8 notes,

of which the 5th and 3rd notes create the basic foundation

of all chords, and are based on a tone which

are combination of 2 steps and 1 step from the root tone, that is the 1st note of the scale.”3

If

you divide an octave by a perfect fifth, (13/20), you get the golden ratio. If

you divide a perfect fifth by an octave, (8/13), you get the golden ratio. If

you divide a perfect fourth by a major sixth, (6/10), you get the golden ratio.

And if you divide a major third by a perfect fifth, (5/8), you get the golden

ratio.

“Notes

in the scale of western music are based on natural harmonics that are created

by ratios of frequencies. Ratios found in the first seven numbers of the

Fibonacci series (0, 1, 1, 2, 3, 5, 8 ) are related to key frequencies of

musical notes.”4

The

Pentatonic scale has five notes, the Diatonic scale has eight notes, and the

Chromatic scale has thirteen notes. Along with this, the 1st, 3rd, and 5th

notes in any scale create the basic foundation of chords.5

We

even find the sequence in the composition of songs and instrument design,

demonstrating exactly how much of an impact the sequence has in music industry

development, as well as music itself.

Conclusion

Understanding

the depth behind music and harmonic scales has widened my appreciation for the

mathematicians that came before me. I think it many times, we take things for

granted, especially when we are ignorant of the thought processes and planning

that went behind it. The Fibonacci sequence is around us at all times; and

whether that is comforting or terrifying is anyone’s guess.

We see this sequence in

atoms, sea shells, wires, flowers, galaxies, and of course, music. Something so

simple: the addition between 0 and 1, then 1 and one, then 1 and 2, and so on

and so on until super computers can’t even comprehend the foundation of our

world. I think this idea is endearing; understanding that the basic building

blocks of life on Earth are just a few, simple numbers.

After we wrap our heads

around this idea, calculus doesn’t seem so hard anymore, we develop an

appreciation for nature and the universe, and perhaps most importantly, it

brings us together. Despite our differences, we live on the same earth with the

same sea shells and the same flowers and the same galaxies. Let the Fibonacci

sequence allow us to appreciate each other and of course, the beautiful music

made possible by the sequence.

Works

Cited:

Goldennumber.net,

www.goldennumber.net/music/.

“Harmony

and Proportion by John Boyd-Brent.” Harmony and Proportion: Pythagoras: Music and

Space, www.aboutscotland.com/harmony/prop.html.

Hom,

Elaine J. “What is the Fibonacci Sequence?” LiveScience,

Purch, 14 June 2013, www.livescience.com/37470-fibonacci-sequence.html.

Fibonacci Sequence,

www.mathsisfun.com/numbers/fibonacci-sequence.html.

“Music

and the Fibonacci Sequence w/ Rory PQ | Dubspot.” Dubspot Blog,

1 June 2016, blog.dubspot.com/fibonacci-sequence-in-music/.

1 http://www.livescience.com/37470-fibonacci-sequence.html

2 http://www.aboutscotland.com/harmony/prop.html

3 https://www.mathsisfun.com/numbers/fibonacci-sequence.html

4

5 http://blog.dubspot.com/fibonacci-sequence-in-music/