Math 236  •  9 May 2001

Middle A with harmonics (overtones): synthesizing  periodic functions

Introduction

Middle A is a pure tone with a frequency of 440 Hz.  The functions
	sin(((2  π)/440) t), cos(((2  π)/440) t), and a cos(((2  π)/440) 
              t)  +  b sin(((2  π)/440) 
              t) 
for constants a and b all describe such a pure tone.

λ  =  (1/440);

We look at and listen to such functions as well as functions derived from them by adding some harmonics (overtones) whose frequencies are integral multiples of the fundamental frequency of 440 Hz.

Graphs

Plot[
        3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]   ,  {t,  0,  
          5  λ},  PlotRange  ->  {-7,  7}];

Shift upwards, then successively add overtones of 880 Hz and 1760 Hz:

Plot[
        0.5  +  3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]   ,  {t,  0,  
          5  λ},  PlotRange  ->  {-7,  7}];
Plot[
        0.5  +  3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]   +   
-0.75 Sin[((4  π)/λ) t]  +  
          1.68 Cos[((4  π)/λ) t] ,  {t,  0,  
          5  λ},  PlotRange  ->  {-7,  7}];
Plot[
        0.5  +  3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]   +   
-0.75 Sin[((4  π)/λ) t]  +  
          1.68 Cos[((4  π)/λ) t]  + 
0.1  Sin[((6  π)/λ) t]  -  
          0.5 Cos[((6  π)/λ) t] ,  {t,  0,  
          5 λ},  
        PlotRange  ->  {-7,  7}];

Sounds

Sound of the fundamental:

Play[
        3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]  ,  {t,  0,  1},  
        PlayRange  ->  {-7,  7}];

Sounds of the upward-shifted fundamental and then with overtones of frequencies 880 Hz and 1760 Hz:

Play[
        0.5  +  3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]  ,  {t,  0,  1},  
        PlayRange  ->  {-7,  7}];
Play[
        0.5  +  3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]   +   
-0.75 Sin[((4  π)/λ) t]  +  
          1.68 Cos[((4  π)/λ) t]  ,  {t,  0,  1},  
        PlayRange  ->  {-7,  7}];
Play[
        0.5  +  3 Sin[((2  π)/λ) t]  +  
          4 Cos[((2  π)/λ) t]   +   
-0.75 Sin[((4  π)/λ) t]  +  
          1.68 Cos[((4  π)/λ) t]  + 
0.1  Sin[((6  π)/λ) t]  -  
          0.5 Cos[((6  π)/λ) t] ,  {t,  0,  1},  
        PlayRange  ->  {-7,  7}];