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This is the sound of violin:

snd = ExampleData[{"Sound", "Violin"}]

These are data behind the sound file (it'll be the same for imported .WAV file):

dat = snd[[1, 1, 1]];

This is the sampling rate

sr = snd[[1, 2]]

22050

This is direct Fourier transform of data:

f = Fourier[dat];

Now my gut tells me that if we take less Fourier harmonics we need to reduce the sampling rate proportionally:

Manipulate[
  Sound[SampledSoundList[{Re@InverseFourier[
  Take[f, {1, Round[k Length[f]]}]]}, Round[k sr]]], 
{k, .001, 1, Appearance -> "Labeled", ImageSize -> Small},
FrameMargins -> 0]

k means fraction of harmonics taken to reconstruct the sound. I am taking real part of reconstructed signal. This all maybe a bit naive but works nicely to instructively demonstrate something to non-dedicated crowd. I actually love how it sounds. This is animation of how you are adding harmonics.

Now a very neat stereo effect (perhaps quite artificial) is when we put real and imaginary parts into different channels. At some settings a very cool stereo feel can be heard - like hear:

Manipulate[
 Sound[SampledSoundList[{Re[#], Im[#]} &[
    InverseFourier[Take[f, {1, Round[k Length[f]]}]]], 
   Round[k sr]]], {{k, .1865}, .001, 1, Appearance -> "Labeled", 
  ImageSize -> Small}, FrameMargins -> 0]

The imaginary part will be zero when sound is fully reconstructed.

You can use direct Fourier and then feed just part of the result to InverseFourier. This is the sound of violin:

snd = ExampleData[{"Sound", "Violin"}]

These are data behind the sound file (it'll be the same for imported .WAV file):

dat = snd[[1, 1, 1]];

This is the sampling rate

sr = snd[[1, 2]]

22050

This is direct Fourier transform of data:

f = Fourier[dat];

To keep duration of sound the same, when we take less Fourier harmonics we need to reduce the sampling rate proportionally:

Manipulate[
  Sound[SampledSoundList[{Re@InverseFourier[
  Take[f, {1, Round[k Length[f]]}]]}, Round[k sr]]], 
{k, .001, 1, Appearance -> "Labeled", ImageSize -> Small},
FrameMargins -> 0]

k means fraction of harmonics taken to reconstruct the sound. I am taking real part of reconstructed signal. I actually like how it sounds. This is animation of how you are adding harmonics.

Now a very neat stereo effect (perhaps quite artificial) is when we put real and imaginary parts into different channels. At some settings a very cool stereo feel can be heard - like hear:

Manipulate[
 Sound[SampledSoundList[{Re[#], Im[#]} &[
    InverseFourier[Take[f, {1, Round[k Length[f]]}]]], 
   Round[k sr]]], {{k, .1865}, .001, 1, Appearance -> "Labeled", 
  ImageSize -> Small}, FrameMargins -> 0]

The imaginary part will be zero when sound is fully reconstructed at k=1.

This is the sound of violin:

snd = ExampleData[{"Sound", "Violin"}]

These are data behind the sound file (it'll be the same for imported .WAV file):

dat = snd[[1, 1, 1]];

This is the sampling rate

sr = snd[[1, 2]]

22050

This is direct Fourier transform of data:

f = Fourier[dat];

Now my gut tells me that if we take less Fourier harmonics we need to reduce the sampling rate proportionally:

Manipulate[
  Sound[SampledSoundList[{Re@InverseFourier[
  Take[f, {1, Round[k Length[f]]}]]}, Round[k 22050]]], 
{k, .001, 1, Appearance -> "Labeled", ImageSize -> Small},
FrameMargins -> 0FrameMargins -> 0]

k means fraction of harmonics taken to reconstruct the sound. I am taking real part of reconstructed signal. This all maybe a bit naive but works nicely to instructively demonstrate something to non-dedicated crowd. I actually love how it sounds.

This is the sound of violin:

snd = ExampleData[{"Sound", "Violin"}]

These are data behind the sound file (it'll be the same for imported .WAV file):

dat = snd[[1, 1, 1]];

This is the sampling rate

sr = snd[[1, 2]]

22050

This is direct Fourier transform of data:

f = Fourier[dat];

Now my gut tells me that if we take less Fourier harmonics we need to reduce the sampling rate proportionally:

Manipulate[
  Sound[SampledSoundList[{Re@InverseFourier[
  Take[f, {1, Round[k Length[f]]}]]}, Round[k sr]]], 
{k, .001, 1, Appearance -> "Labeled", ImageSize -> Small},
FrameMargins -> 0]

k means fraction of harmonics taken to reconstruct the sound. I am taking real part of reconstructed signal. This all maybe a bit naive but works nicely to instructively demonstrate something to non-dedicated crowd. I actually love how it sounds. This is animation of how you are adding harmonics.

Now a very neat stereo effect (perhaps quite artificial) is when we put real and imaginary parts into different channels. At some settings a very cool stereo feel can be heard - like hear:

Manipulate[
 Sound[SampledSoundList[{Re[#], Im[#]} &[
    InverseFourier[Take[f, {1, Round[k Length[f]]}]]], 
   Round[k sr]]], {{k, .1865}, .001, 1, Appearance -> "Labeled", 
  ImageSize -> Small}, FrameMargins -> 0]

The imaginary part will be zero when sound is fully reconstructed.

This is the sound of violin:

snd = ExampleData[{"Sound", "Violin"}]

These are data behind the sound file (it'll be the same for imported .WAV file):

dat = snd[[1, 1, 1]];

This the sampling rate

sr = snd[[1, 2]]

22050

This is direct Fourier transform of data:

f = Fourier[dat];

Now my gut tells me that if we take less Fourier harmonics we need to reduce the sampling rate proportionally:

Manipulate[
  Sound[SampledSoundList[{Re@InverseFourier[
  Take[f, {1, Round[k Length[f]]}]]}, Round[k 22050]]], 
{k, .001, 1, Appearance -> "Labeled", ImageSize -> Small},
FrameMargins -> 0FrameMargins -> 0]

k means fraction of harmonics taken to reconstruct the sound. I am taking real part of reconstructed signal. This all maybe a bit naive but works nicely to instructively demonstrate something to non-dedicated crowd. I actually love how it sounds.

This is the sound of violin:

snd = ExampleData[{"Sound", "Violin"}]

These are data behind the sound file (it'll be the same for imported .WAV file):

dat = snd[[1, 1, 1]];

This is the sampling rate

sr = snd[[1, 2]]

22050

This is direct Fourier transform of data:

f = Fourier[dat];

Now my gut tells me that if we take less Fourier harmonics we need to reduce the sampling rate proportionally:

Manipulate[
  Sound[SampledSoundList[{Re@InverseFourier[
  Take[f, {1, Round[k Length[f]]}]]}, Round[k 22050]]], 
{k, .001, 1, Appearance -> "Labeled", ImageSize -> Small},
FrameMargins -> 0FrameMargins -> 0]

k means fraction of harmonics taken to reconstruct the sound. I am taking real part of reconstructed signal. This all maybe a bit naive but works nicely to instructively demonstrate something to non-dedicated crowd. I actually love how it sounds.