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Fourier analysis in musicology
Junlin Hou
Huangyan Pan
Yifan Li
Jie Liu
contents
 Mathematics and Music
 The explanation of Fourier analysis in musicology
 The application of the theory
 Summary
Mathermatics and Music
Music is the arithmetic exercises of the soul
——Leibniz
The world can be made up of the music notes ,also the
mathematical formula
——Einstein
 Algorithmic Music
 《S+/10-1.080262》
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Music for Strings,
Percussion &
Celesta
Fibonacci
sequence
explanation
 Q:
What characteristic makes the different sounds of
different instruments?
Musical quality
Sine wave
 y=Asin(ωx+α）
 A：loudness
 ω：tone
 Musical quality???
The superposition of sine waves
 sinx+(sin3x)/3+(sin5x)/5+……+(sinnx)/n
n=199
Square wave
n=9
The superposition of sine waves
 sinx+(sin2x)/2+(sin3x)/3+……+(sinnx)/n
Sawtooth wave
 Fourier
 Periodic function → the superposition of sine waves
 Musical quality
Wave form
The Basic Process
 1. A simple example
 2. Mathematic ideas
 3. The process( more difficult)
A simple example
 two–dimensional space
 three–dimensional space
 multi–dimensional space
Mathematic ideas
 S: v1，v2，v3，……vn
 vp=c1·v1 +c2·v2+c3·v3+…+cn·vn
 (v-vp ) ·vj = 0
v·vj =vp ·vj (j=1,2,…,n)
 system of linear equation:
The process
 Basic vectors：
v : f (x)
vp : Tn(x)
Same?
summary
A common phenomenon:
A person who do well in mathematics may not be proficient in music ,and
a musician who also have achievement in mathematics is rare.
A close connection:
a sound synthesizer 、the Fourier series in the piano black and white keys 、
the different musical sounds because of different chord lengths and so
on …
A fantastic world of mathematics:
Even Fourier series, not only concludes the factor of music ,but also
concludes the factor of painting.
The
end!
Thanks for your
time!