Transcript The Mathematics of Music

The Mathematics of Music
By: Phil Boyer, Mark Forman, Brian
Glenn, and Arvind Kanna
What is a harmonic?
• Any of a series of musical tones whose
frequencies are integral multiples of the
frequency of a fundamental tone
• A wave whose frequency is a whole number
multiple of that of another
f
r
e
q
u
e
Harmonic Series
• This series Diverges.
• For each whole number a
harmonic exists.
• In music, only the odd
harmonics sound pleasing to
the ear.

1

n
n 1
E
A
R
Structure of Human Ear
• Sounds travel through the ear by way of the
cochlea and basilar membrane
Why Sine Waves?
• Sine waves most accurately model
how a sound particle travels through the
ear.
• The equation that best represents how
musical sound waves travel through the
ear is the equation for simple harmonic
motion
Simple Harmonic Motion
F   ky
F  ma
2
d y
a 2
dt
d 2 y ky

0
2
m
dt
ky
y
0
m
k
k
y  A cos(
t )  B sin(
t)
m
m
Vibrating strings
Vibrating Strings Equation





k
k
  Bn sin  n

y    An cos n





mt
mt
n 1 





•n is the value of the harmonic being
played.
Musical Scale
• The Western Musical Scale consists of
twelve notes forming an octave.
• There are 7 natural notes and 5 sharp/flat
notes.
• # means “sharp”, b means “flat”
• A A# B C C# D D# E F F# G G#
• A Bb B C Cb D Eb E F Gb G Ab
Musical Harmonies
• There are many musical harmonies that can
be played on guitar or any instrument for
that matter.
• Harmonies can be played between different
types of instruments.
• Most popular artists use harmonies in their
songs. Even samples and electronic beats
use them.
Why Do Mathematical People
Excel in Music?
• 1. generally this is so because of the mind set and
thought, which is similar between both subjects.
• 2. math as a subject is generally about thinking,
problem solving, seeing patters.
• 3. math is also about working with what you know
to find a method of exploring and understanding
new concepts.
• 4. Music is very similar to math in this sense.
• 5.
Arranging a melody on an instrument
and finding fingerings that correspond to
certain sequences of notes(like piano for
example) is definitely a type of math
problem.
• 6. Playing the same melody on different
instruments is math, as is playing a stringed
instrument and changing the tuning.
• 7. Playing harmonies, playing the same song in
different keys--these things all involve recognizing
the structure of a piece of music.
• 8. Good musicians can often listen to a song,
observe the musical structure, and play along with
it, without really knowing it or rehearsing it,
because they recognize familiar patterns.
• 9. this type of pattern recognition and mind set of
math oriented individuals helps them excel in
music.
Bibliography
• Dave, Benson. "Music : A Mathematical Offering."
Mathematics and Music. 14 May 2006. Cambridge
University. 25 May 2006
<http://www.maths.abdn.ac.uk/~bensondj/html/mathsmusic.html>.
• Harrison, David M. "Standing Waves." 24 Mar. 2003.
University of Toronto. 25 May 2006
<http://www.upscale.utoronto.ca/GeneralInterest/Harrison/
Vibrations/Vibrations.html>.
• Reid, Harvey. "On Mathematics and Music." Woodpecker
Records. Nov. 1995. Woodpecker Records. 25 May 2006
<http://www.woodpecker.com/writing/essays/math+music.
html>.