Mathematic Destre Tashjian International Baccalaureate Mathematics 4.1 December

 

 

 

 

 

 

 

 

 

Mathematic Exploration: The Relationship
Between Math and Music

 

Destre
Tashjian

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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

Music and the Fibonacci Sequence and Phi

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