U2's Natural Logarhythm: Exponential Decay in the Delay of The Edge's Guitar
Ever wonder what it is that makes The Edge's guitar playing so special? In particular, how does he achieve his signature delay sound? Even before he started using a delay pedal, like on Boy, The Edge had a distinctive way of "lagging" his notes that didn't correspond to the natural linear rhythm of the song (a good example is "Twilight"). Another good example of where he hits these beautifully off-rhythm notes without a delay pedal is in "Like a Song" on War. It's almost like he's finding gaps to fill, blank spots on the musical canvas, that don't fall in the normal places you'd normally look at 1/2, 1/4, 1/8 or 1/16 of the beat, analogous to the way he often hits harmonic notes that don't fall on the traditional scale. You can spend a lifetime analyzing pitch intervals (and it is interesting to note that binary or base-2 logarithms are often used to analyze pitch intervals, and the Western chromatic scale is based on 2 to the 12th power, or binary logarithmic, pitch intervals). But what I'm specifically talking about here is the temporal delay--that off-beat, modulated, harmonic rhythm--that The Edge uses in his guitar playing.
On War, The Edge started using delays (generated through a delay or echo effects pedal) that hinted at this signature rhythm. If it wasn't apparent by then, he hammered you over the head with them on the opening notes of Joshua Tree. I remember where I was when I first heard Joshua Tree, driving with my brother on the Pasadena freeway on the way back from Tower Records. When you first hear an album, it takes some getting used to to know if you're "hooked" by the melody, and I remember listening to the opening chord progression and thinking it was overly simple. But it was the rhythm of the guitar that just made your neck bristle. The Edge had tapped into something that defied description. As Lee Krasner tells Pollock (in the movie anyway), he'd "blown it wide open."
It's hard to quantify how much I listened to that album, probably at least once a day through the late 80s and into the early 90s. As a guitar player at the time, of course I was, like many others, wondering, how the hell does he do that? Back then there was no internet. Guitar or music magazines would talk about his use of delay but never specifically what he did to get that sound. Recently I tried searching around on the internet to see if anyone else had discovered what I had discovered back then, and while I found some interesting articles (in particular this one by Tim Darling), nobody seems to have happened on the discovery that I made. Darling does a thorough and at times technical analysis of the equipment and delay settings that The Edge uses to get his sound, his conclusion being that his delay is set at roughly 3/16 tempo, or for example on "Where The Streets Have No Name," his pedal was set at 340-350 ms. But this 345 ms setting is not relevant unless we talk about beats per minute.
Any guitar player that has fooled around with a delay pedal (and preferably a drum machine or metronome at the same time), knows that that there is a certain point when you are cranking the setting that this "Edge" sound kicks in. Of course there's the obvious point where it matches the beat or half-beat. For example, if The Edge was using delay on "I Will Follow" (which I don't think he is anywhere on Boy) then it would be set like this, so the delay would coincide exactly with the next note, creating doubled up chords. But if you keep cranking it (tighter, so the notes get more bunched together), you get to a point where it just hits a groove. It's like riding a horse where you are trotting really fast and then suddenly the horse breaks into a gallop. It's a modulating rhythym that is self-sustaining, self-propagating. There's a tension there, like it's offbeat, on the edge of being chaotic, but then it resolves itself with each beat. But back to the question, what is this delay interval?
Around the time of Joshua Tree, not only was I fascinated by music, but also mathematics. I was studying both, but opted for degrees in math, and then physics, thinking it was the more practical choice. But also because the more I "studied" music, the more I didn't want to know what made it tick. The natural thing to do when studying this interval, mathematically, was to find this ratio between the two rhythms (the rhythm of the drum and the rhythm of the guitar). Listening to the music, it's easy to count the number of beats per minute, but counting the number of delayed notes (the guitar rhythm) per minute is difficult. But if you have a digital delay and a drum machine, this can easily be quantified. In fact, I had implemented this sound in my own music , some old songs of which I have recently put on a MySpace page. In particular, on "Shadow to the Wind," I am not even playing the rhythm guitar part, but had programmed it according to this "Edge" delay.
When I'd count and take a ratio of (beats per minute)/(delayed notes per minute), I'd always get a number close to 0.36788. 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). But I wasn't interested in rough approximations. I wanted to know the significance of that number. When you invert the number it is obvious. (1/0.3678794) = 2.7182821 or e, Euler's number, probably the single most important number in mathematics that is used to explain a lot of natural phenomena such as exponential decay. That is, if you divide the (delayed notes per minute)/(beats per minute) you get e! Moreover, the function of the inverse of e, or 1/e, is what is referred to as a natural logarithm, that is used to explain all sorts of natural multiplicative phenomena.
This is what it would look like on a timescale if you plucked a note (black dot) followed by another note a half beat away. The smaller gray notes following are the delayed notes.
The delayed note would come again right before (0.13212 of a beat) before the next note. A further study could probably be made on the various successive intervals generated between the notes and the significance of them. But the important correlation is that the ratio of the beat interval to the delay interval is e.
If we go back to Tim Darling's analysis, he says the secret setting for "Where The Streets Have No Name" is 340-350 ms. I went back and counted the number of beats per minute in the song, and its actually like 64 beats per minute, so each beat is about 0.9375 seconds. So now if we divide the beat interval of 0.9375 by this setting of 345 ms or 0.345 seconds, we get (0.9375)/(0.345) = 2.7173913, which is pretty damn close to e (2.718281...), which explains why this particular setting works for this particular song. So, more precisely, if you have a song that is exactly 60 beats per minutes, or a beat interval of exactly one second, then to get the "Edge" delay, you'd set it to 367.88 ms. For any other beat interval, all you have to do is divide by e to figure out what to set your delay on.
This is obviously nothing The Edge needs to "know" to play guitar like he does, at least not consciously, just like a sunflower or pine tree doesn't need to "know" what a Fibonacci number or Golden Ratio is to generate the whorling patterns in it's flowers and cones (the subject of my math thesis). And the question still remains as to "why e?" In the case of the plants, the golden ratio allows them to optimally space leaves on an axis for maximum sun exposure. There is a physical reason. But why the delay ratio of e sounds most appealing, still remains to be determined. I am not sure I want to know.
(c) 2007 Derek White