Derivative of a Song: Finding a Connection in Music & Calculus

[DirtyBurge33]

Does anyone have any ideas of what would happen if you somehow graphed a piece of music and found its derivative? Would it sound like anything? And how would you do this? This is something I have been mulling over for a while now after learning about derivatives in Calculus. I wouldn't be surprised to see a connection of some sort. Any thoughts?

 

[ice109]

the same thing that would happen if you took the integral of an apple.

 

[DaveC426913]

Maybe it would sound http://strangemaps.files.wordpress.com/2008/02/world-beat-music6.jpeg".

 

[Cincinnatus]

Lets say you define the function f(t)=pitch at time t

Then the derivative of this function with respect to time would be the "instantaneous rate of change of the pitch" at each point. So when the pitch is changing quickly, like when you have a lot of fast notes at different pitches you'll have a high rate of change. When you have a slow song where the pitch changes more slowly you'll have a lower rate of change.

So, if you then identify rate of change with pitch and play the "derivative song" you'll get something high pitched if the original song was fast and something low pitched if the original song was slow.

A more interesting operation would be to take the Fourier Transform...

 

[K.J.Healey]

Well if you express any wavepacket as a function where its just a sum of sine's, the second derivative would just be its negative times two frequency.

But a single derivative, its equivalent to flipping its symmetry from symetric to anti, or anti to symmetric (sin->cos, cos->sin). Which is equal to:

Music' = (Music - pi/2 phase change)*frequency

So its distribution would stretch and it would get higher frequency, and would be phase shifted.

Right?

 

[ice109]

Well if you express any wavepacket as a function where its just a sum of sine's, the second derivative would just be its negative times two frequency.

But a single derivative, its equivalent to flipping its symmetry from symetric to anti, or anti to symmetric (sin->cos, cos->sin). Which is equal to:

Music' = (Music - pi/2 phase change)*frequency

So its distribution would stretch and it would get higher frequency, and would be phase shifted.

Right?

wave packet of what? only cincinnatus has stated what we're plotting here as a function of time and it wouldn't work anyway because I'm sure his function would have discontinuities everywhere.

 

[John Creighto]

I think someone mentioned the answer above. A derivative is a high pass filter where the gain increases linearly with frequency. The following link may help the reader better understand the concept:

http://en.wikipedia.org/wiki/Bode_plot

 

[jostpuur]

My guess is that higher frequencies get louder, and lower frequencies more quiet, and that the song itself would remain the same but sounds would get different.

Different Fourier modes get phase shifted by different amounts, but since the frequencies are quite high anyway compared to the speed of music, I don't think the phase shiftings would affect the song much. It's just that the sounds probably get absurd, and different instruments probably are left unrecognizable.

 

[Cincinnatus]

only cincinnatus has stated what we're plotting here as a function of time and it wouldn't work anyway because I'm sure his function would have discontinuities everywhere.

If you are playing a violin it is certainly possible to "slide" from one note to another, playing all the pitches in between.

However, it is true that for many instruments (like the piano) this isn't possible. You still take something like the derivative though. You just need to set a sampling rate and draw secant lines between sampled points. If the sampling rate is high then this is effectively the same as the derivative which corresponds to doing this with an infinite sampling rate. This is the only way to really calculate derivatives of time series data like this anyway.

Some of the other responders are speaking in terms of frequency. I'd like to point out that high frequency is the same as high rate of change. If you are approximating a function by some sinusoid (linear combination of sines and cosines) then it takes a high frequency component to follow a quick change in value.