Harmonic Overtones in binaural beats

I'm probably going to be very uneven maintaining this blog, but if an idea pops into my head that's worth writing about, I'll add it here.

One thing I was playing around with over the Summer of 2017 was maintaining stable harmonic intervals within paired binaural beats.

I Harmonic Intervals

A Harmonic Interval is basically a chord, but with just two notes represented. (Whereas a chord would have three or more, typically.)

So for instance, a C note and a G note on the piano would form a perfect fifth. This is the 'music' name for the harmonic interval the two tones create when played together.

In terms of numbers, (in its purest form*) it basically says the two notes have a 3 to 2 ratio (Written 3:2). So, if the first frequency is 200 HZ, the second one would be 300 HZ**.


* Western Music 'corrupts' the harmonic intervals just a tiny bit - it's a compromise in the name of creating a system where you can play the same scale starting at any key on a piano. The absolute most perfect, perfect fifth is 3:2. (They call that a 'just perfect fifth', 'just' meaning it relies on the simplest form of the harmonic interval, typically with the tiniest whole numbers in the ratio.)

** If you divide the smaller number of the harmonic ratio into the larger number, it will tell you what you need to multiply the smaller frequency by to get the larger frequency. So, 3 divided 2 = 1.5. 200 HZ times 1.5 equals 300 HZ.

II Creating (fairly) Stable Harmonic Intervals In Binaural Pairs

So, if you want to create a nice clean harmonic internal between two tones, each containing harmonic beats, how would you do that?

The first disclaimer - don't feel it's necessary to do this. I find the minor fluctuation created in harmonics as a result of the binaurals being introduced is actually quite nice. It's going to be more apparent the lower the carrier frequencies of the binaural beat are, since small changes in the frequency make a bigger difference in the harmonic relations in the bassier range, since the octaves are packed closer together. (Higher carrier frequencies 'absorb' the harmonic impact of the binaural better.)

With that disclaimer out of the way, let's assume you want to have a stable harmonic interval between two tones, each containing a binaural beat. Let's say we want it to be a perfect fifth (second tone 1.5x the other).

The trick to doing this :

1) Make sure one set of the tones (those going to the left ear or the right ear) contain the 1.5 harmonic interval. So, if I have a 200 HZ tone in the left ear for the first binaural, I'm going to want to use a 300 HZ tone in the left ear of the second binaural.

2) Next, if you make the binaural beat value themselves a 1.5 harmonic interval of each other, the carrier frequencies in the right channel will also fall in line with the 1.5 harmonic interval in the process. So, 8 HZ (alpha-theta range) and 12 HZ (beta range) have a 3 to 2 ratio with each other. If we add the 8 HZ to the 200 HZ (left ear) value, that leads to a 208 HZ audio frequency for the right ear of the first binaural. Conversely, adding 12 HZ to the 300 HZ (left ear) value of the second binaural will lead to a 312 HZ value for the right ear.

If you do this, it will preserve the 3:2 harmonic interval, removing any harmonic artifacts. The downside is if you're gunning for a purely alpha-theta binaural, it is introducing a higher binaural value to the mix (12 HZ). You might be able to combat this somewhat by dropping the volume on that higher 12 HZ binaural.

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III Fun With Tables

Just a visual representation of what I'm describing above :






One thing I am catching now - even this way, you probably are introducing at least some 'harmonic compromise'. Just as you create accidental cross-binaurals through the left side of Binaural 1 interacting with the right side of Binaural 2 (and vice-versa), you'd also create accidental harmonics in the process.

• The 200 HZ left tone from Binaural 1 will interact with the 312 HZ right tone of Binaural 2 to form a greater harmonic interval than the 1.5 value we were gunning for. You get a 1.56, which is pretty damn close the next note up on the piano keyboard.

• The 208 HZ right tone from Binaural 1 similar interacts with the 300 HZ left tone of Binaural 2 to form a 1.44(and some change) harmonic interval. This is close to the augmented fourth or Tritone, considered to be one of the least harmonic intervals on the keyboard.***

*** I actually think Tritones get a bad rap. Played on an instrument that sounds quickly and abruptly halts, they can be a little jarring, but on a synthesizer or something with a longer sustain, they actually are quite nice.
 
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Enough typing for now -- I'm writing this out as much for myself, since I have all this math written out on the backs of CVS receipts strewn about my apartment, and while they do make for great ROH-style streamers, I do need to recycle them at some point.