Harvard University,FAS
Fall 2004

Mathematics Math21b
Fall 2004

Linear Algebra and Differential Equations


Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu

Soundmanipulation and Linear Algebra

Analoguously to image manipulation, we can also do sound manipulation with Mathematics and Mathematica.

A piece of music is just a function f(t). Alternatively, given a function f, you can play it. Mathematica allows to do that with the command "Play". 

f[x_]:=Sin[3000 2Pi x]
Play[f[x],{x,0,10}]


for example, plays a 3000 Herz sound for a few seconds.



A Scherzo of Chopin: a function f
Mixture of the two pieces: the function (f+g)/2.
Accordeon Musik. A function g.




The mixture of the two sound pieces was obtained with the following Mathmematica commands: (The Mathematica code is a bit more complicated then anticipated because by building the song vector (a list of sampled amplitudes), the left and right sound channel are built differently). 
a=Import["scherzo.wav"]; b=Import["wirke.wav"];
type=a[[1,2]]; n=Length[a[[1,1,1]]]; m=Length[b[[1,1,1]]]; nm=Min[n,m];

song={
   Table[(a[[1,1,1,k]]+b[[1,1,1,k]])/2,{k,nm}],
   Table[(a[[1,1,2,k]]+b[[1,1,2,k]])/2,{k,nm}]
};

c=Sound[SampledSoundList[song,type]];
Export["mix.wav",c,"WAV"]
Here are the files: you first have to convert the mp3 into .wav files. We did not include the .wav files due to size. You can convert it easily. in linux for example, type at the commandline 
mpg123 -w scherzo.wav scherzo.mp3 
mpg123 -w wirke.wav wirke.mp3
The conversion from .wav to .mp3 is done in linux with 
lame mix.wav
For Macs or PC's you can do the conversion using standard sound manipulation programs.


Music inversion



An other example: we take a function f(t) and apply the linear transformation T(f)(t) = f(-t). Lets do that with the Scherzo: 
a=Import["scherzo.wav"]; 
type=a[[1,2]]; n=Length[a[[1,1,1]]]; 

song={
   Table[a[[1,1,1,n-k]]/2,{k,0,n-1}],
   Table[a[[1,1,2,n-k]]/2,{k,0,n-1}]
};

c=Sound[SampledSoundList[song,type]];
Export["scherzo_reversed.wav",c,"WAV"]
Here are the files:


Speed manipulation



An other example: we take a function f(t) and apply a linear transformation T(f)(t) = f(g(t)). Lets do that with an Enya song "adiemus" 
a=Import["adiemus.wav"]; 
type=a[[1,2]]; n=Length[a[[1,1,1]]]; 

song={
   Table[a[[1,1,1,Floor[k+(n/10) Sin[Pi*k/n]] ]]/2,{k,0,n-1}],
   Table[a[[1,1,2,Floor[k+(n/10) Sin[Pi*k/n]] ]]/2,{k,0,n-1}]
};

c=Sound[SampledSoundList[song,type]];
Export["rap.wav",c,"WAV"]
Here are the files:


A simple synthesizer



( More details in the Fourier exhibit). 
PlaySong[hull_,tune_,name_,ground_]:=Module[{},
  u=ToCharacterCode[name]; 
  soundfilename=FromCharacterCode[Join[u,{46,119,97,118}]];
  scale[n_]:=ground*2^(n/12); beatlength=1/5;
  songlength=Length[tune]*beatlength;
  frequency[x_]:=tune[[1+Floor[x/beatlength]]];
  song[t_]:=hull[scale[frequency[t]]*t];
  P=Play[song[t],{t,0,songlength}];
  Export[soundfilename,P,"WAV"];
]

t1={0,0,4,4,7,7,5,4,2,2,0,4,7,4,0,0,0,0,0,2,0};
f1[x_]:=Sin[x]; n1="sin"; g1=2000;
PlaySong[f1,t1,n1,g1]


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