# Finding the strongest beat in a sound wave using autocorrelation

1. Nov 26, 2012

### wildetudor

Hi everyone,

I have a sound wave representing a piano piece played at a steady tempo, and would like to get a graph of the saliency of each beat (essentially, a probability distribution for how strong each possible tempo is). I understand that this is done by plotting the autocorrelation function, however I don't quite understand why a graph of r coefficients against each possible lag value (which is, as far as I understand, the deffinition of an autocorrelogram) would have anything to do with beats.

The following Matlab code produces a graph that doesn't in any way suggest anything to do with the actual steady beat of the piece (60 BPM):

Code (Text):
[r,lags]=xcorr(y,'coeff');
plot(lags,r)
Clearly I'm understanding autocorrelation wrongly. For instance, in this very simple example, the frequency of the sine hidden in noise is nowhere visible from the autocorrelation graph - or is it? Furthermore, that frequency would actually be the pitch of the sound, and not any rhythm-related measure!

The MIR toolbox for Matlab has a function specifically for finding the tempo of a waveform - however what I'm after now is understanding these things at a theoretical level.

Anticipated thanks for any clarifications!!

2. Nov 26, 2012

### Staff: Mentor

You can see a clear sine wave in the autocorrelation function, which corresponds to a "hidden" sine wave in the original function with a period of ~10.

Can you find the time steps where the frequency in the signal changes? An autocorrelation of that should help.
Alternatively, search for steps in the autocorrelation function, this could help as well.

3. Nov 26, 2012

### sophiecentaur

Problem with music is that it can be very subtle. However, to find an 'obvious beat', why not find the short term RMS values (Energy with time) of a long passage and FFT that to find low frequency (sub Hz) energy variations. Some pre-filtering could identify components like cymbal strikes or bass drum beats. (Sound to light 'disco' boxes do this quite well)

4. Nov 26, 2012

### wildetudor

Thanks for your replies! I'm afraid I still don't fully understand, though, the link between a time signal's (waveform's) periodicity and its autocorrelation function; and, I could also add, the link between the signal's autocorrelation function and its Fourier transform (the latter of which I understand how it relates to pitch but not to rhythm)

5. Nov 26, 2012

### sophiecentaur

The rythm would correspond to peaks in level (rectified and integrated?) and the rate at which they repeat. That's why I included the idea of "sub Hz".

I am not sure that the ACF would necessarily reveal the rythm unless there were rhythmic repeats of the same instrument producing a note that was somehow phase consistent? It could be worth while trying with some actual examples of music.

6. Nov 26, 2012

### wildetudor

Thanks, that's quite helpful, I'll play around with it in Matlab on my own a bit to facilitate intuition :)

7. Nov 26, 2012

### AlephZero

The autocorrelation for the audio of a complete piano piece probably won't show anything useful.

If you calculate autocorrelations for short "windows" of the data, you might be able to see the CHANGES in frequency content at the start of each new note. But even a single note on a piano is far from being a simple sine wave, and most piano music has several notes at different pitches sounding at the same time, so this isn't going to be easy to see.

There are a few commercial computer programs that claim to be able to convert an audio recording into music notation, for example http://www.neuratron.com/audioscore.htm, but they don't work very well for anything except very simple audio, like a single instrument that can only play one note at a time.

But it is possible ... http://www.ted.com/talks/john_walker_re_creates_great_performances.html

8. Nov 26, 2012

### sophiecentaur

If your task is to find a reliable method using DSP and not a brain, it will fall over for some examples, I'm sure. It would depend on how good it needs to be. Good luck and have fun.

9. Nov 27, 2012

### rbj

better look up "beat detection" which is a similar problem to pitch detection. i would suggest googling jean larouche beat detection without quotes and reading any number of documents this guy wrote.

it ain't an easy problem.

10. Nov 28, 2012