Comment by anigbrowl | Hacker News Reader

Comment by anigbrowl | Hacker News Reader

This is to say, there was roughly three delayed notes per beat, or as Tim Darling points out, it's roughly 3/16 tempo (though really I think he meant 6/16 time or 3/8 time, where 3/8 = 0.375, which is a close approximation to 0.36788).

No, I think he meant 3/16, especially since he explains the derivation of that value. It's a fixture in reggae music and dub because it provides instant syncopation, and later found its way into a lot of electronic dance music for the same reason. Get started with Dub at http://en.wikipedia.org/wiki/Dub_music and then study the early studio history of Lee Perry, who pioneered a great many audio production tricks by necessity. This 1978 track is a seminal work: http://www.youtube.com/watch?v=hs9Z2TEqSZo All the sound effects going on are done with a mixer and 2 delays, using tricks like splitting the output of a channel back into 2 inputs and inverting the polarity on one.

The 'number of delayed notes per beat' comes in around 3 because the delay unit is feeding back on itself, and a 4th repeat (being equal to 12/16ths) is likely to fall exactly on a beat: 1/16, 5/15 and 12/16 are the strongest beats in rock and dance music. If you hit the 9/16 beat it's straight rock or dance, if you delay it by half a beat you get the basic rhythm of hip hop. You can of course turn the feedback up higher but above a certain level it tends to run away and make a horrible noise, independently of the delay time.

Edited to add: I hope that explanation didn't sound blithely dismissive of the mathematical investigations. The 1/e hypothesis is compelling, but has the air of being 'so beautiful, it must be true' - be careful of this! I have several notebooks' worth of similar explorations of geometry, golden ratio and so forth as applied to music. It's wonderfully inspiring, but it's easy to find yourself trying to square the circle or retrieve the Lost Chord.

Per your comment I think you've got a good handle on Music and Math. I'm totally naive but, briefly, what background should one have if he intends to start working on extracting a melody from a song? I will appreciate your reply (didn't find your email in your profile so posting it here, thanks).

Can you be more specific - do you mean so you can learn to play the tune yourself, or extract it via software?

Latter i.e. extract the melody via software?

OK, then you want to get into the world of Digital Signal Programming, or DSP. Before you do so, be aware that this is a Hard Problem if you want to achieve more than the most basic results. The basic tool of DSP is the Fourier Transform, which allows you to convert a 1-dimensional signal in the time domain (such as an audio file) to a 2-dimensional signal in the frequency domain (such as a spectrogram aka graphic equalizer display). Many problems that look knotty or impossible in the Time domain are soluble with simple math in the Frequency domain. So you do an FFT, modify or analyze your signal, and then do another FFT if you want to convert it back to an audio stream.

This is a really excellent starter book that you can also download for free: http://www.dspguide.com/ It's far better written than most other books on the field and will help you to develop an intuitive understanding of the fundamental math. Many books just say 'here's the math,' without discussing why it works or why you would want to do it one way rather than another. Many more cover DSP from the point of view of radio or wireless communication - although the same principles apply here as for audio, it's somewhat confusing. This book is very audio-friendly.

The state of the art in pitch extraction from usic recordings is Celemony's Melodyne: http://www.celemony.com/cms/ The company was started in the mid-90s by a German audio geek named Peter Neubäcker with his wife and a programmer. He says in interviews that he's using a different approach based on the shape of sounds, but has never published his methods. I've met him a couple of times at conferences and trade shows, but he knows how to keep a secret! However, you'd be well advised to try out the demo version of his software. How he does is it is a mystery, but he's way, way ahead of any commercial or academic methods.

If you like Matlab, this is the best academic work on the task so far: http://isophonics.net/content/reverse-engineering-mix and you should also grab a copy of Sonic Visualizer, which is a slow-but-flexible analysis tool: http://isophonics.net/SonicVisualiser Be sure to follow up the links on the Isophonics site, which will lead you to a rich variety of libraries and tools for audio programming.

Fascinating, can you recommend any good resources that you stumbled across in your explorations of music theory and math?

Offhand not really; discussions about math and music from the math end tend to either be very dry and technical or quickly tilt into pseudoscience, and good musicians (of which I am not one) tend to look at the math as an interesting curiosity, but only for as long as it takes to get a handle on some new compositional technique.

http://electro-music.com/forum/index.php is the best place I know for serious-minded discussion of the interface between math and music, though it's (surprise) heavily geared towards users/designers of synths and sequencers.

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