NERCOMP • Amherst College • 22 Feb 2000

Middle A with harmonics: synthesizing  periodic functions

Middle A

The fundamental is Middle A— is a pure tone with a frequency of 440 Hz.  The functions
	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);

Look at and listen to such a function.
Add harmonics with frequencies being integral multiples of the fundamental frequency 440 Hz.

Graphs

The fundamental

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

Fundamental plus harmonics

Add harmonics of 880 Hz and 1760 Hz:

Plot[
        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[
        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

The fundamental

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

Fundamental plus harmonics

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

Play[
        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[
        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}];

Main notebook