The devil's in the details, they say, and god is in the gaps. We organize the world into thoughts, clauses and words, even though the real conversation happens in the transitions and in the spaces in between them. True meaning requires context and nuance, and true dialogue requires some understanding beyond what the words are saying, some faith of intent, some invisible working models of yourself and your partner (and of reality itself) which are always the hidden participants in any exchange, interacting above the robotic flow of data. Inflection matters. What's not said matters. It's why we go nuts over irony. Or poetry. Or fiction. It's why online conversations can be difficult sometimes. And it's what, in music, gives life to what is otherwise a bunch of notes.
Music is just one more thing that we humans--or maybe the universe itself--has made an annoying effort to discretize. And I'm not just talking about rhythm here. In many ways, pitch, to be arranged artfully into music, has to get stacked in set intervals in frequency space. The urge to do this may well be biological. The pleasure of hearing harmonics--frequencies that are integer (or simple integer fraction) multiples of each other--may well be a necessary artifact of being blood-and-guts machines. Deep in our inner ears, the actual business of sound detection occurs via arrays of tiny electromechanical resonators, and while the transduction of sound into nerve signals is complex and interesting, the vibrations themselves still must be supported on a bunch of actual, physical things, in order to get the sensation funneled into our brain. The fact that we have, fundamental to our hearing, all these bunches of guitar-string cells, each tuned to different notes, means that the same receptors will be sensitive to harmonics, and when we register "consonance," we're thrumming groups of hair cells that naturally buzz together, massaging them in a similar way, in somewhat different proportions, and it can't help but feel good, like scratching each other's itch.
To step it back for the non-nerds, we can envision sound harmonics as waves on a string. When the string is plucked, it will want to vibrate back and forth, keeping the ends anchored. It can move in lots of ways under these conditions, but after a very short time, only a few stable vibrations (standing wave modes--the picture shows a few of them, taking a few snapshots in time) will emerge, corresponding to how many wavelengths can fit neatly on the string's length. The second mode has half the wavelength, and, as though a string was fretted at that node point, it will move back and forth twice as fast as the first mode, and so on. Likewise, when a string encounters an external source of motion that pushes it at one of its natural frequencies, then it will start to buzz there on its own (what it means when a mode is "excited"). Some wiseass is going to say, "Hey Keith, sound waves are longitudinal, man, those vibrations breathe in and out instead of bend up and down," but the modal analysis is the same--a solid sustaining a vibration at some frequency (possibly in a complex way) will radiate longitudinal sound waves into the air. Maybe less obviously, the modes are evenly spaced here only because we're imagining the string as a one-dimensional system, where the disturbance at points along the straight line is the only one that matters. That's a fine approximation when it comes to your hair cells, which are rod-shaped, and for almost all musical instruments, which excite strings like in the picture or the air inside a pipe. (I think the brass family is the only group of instruments that is designed to excite higher order modes when it's played, however. Usually, you just change the length of the cavity. Imagine how fast you'd have to blow across a flute to get the n=2 stuff out! You can do it on a stringed instrument, if you hit it just right, tap it halfway down its length, but there's not a whole lot of point to that.)
You might argue that once you choose a frequency to tune the band on, you're pretty well stuck with the corresponding set of modes. And it's true! There are entire volumes of musical conversation that are ruled out once you define your key. But fortunately, not all is lost. We can add in a lot of obscure modes to give a given instrument a unique timbre, and there are an important class of instruments that don't work this way (and more on this later). Even all the ones that do, they allow varying access to this space, letting the skilled player suggest all the drama that lies just off resonance. Also, we are given to work with the fact that Western tuning is inherently fucked.
Personally, I find the intersection of musicianship and acoustics to be vexing. It's irritating to me that a "third" interval has nothing to do with the third harmonic up there, that a "fifth" is a half, and so on. (The problem is really the English language: the ordinal number and the fractional number should never have been the same word.) In the usual convention, we break up every doubling of the frequency (every octave) into twelve equal divisions. (Equal on a logarithmic scale, that is. It's twelve even ratios, really, and, well, people seem to have a biological imperative to divide things by twelve too.) But let's note that in terms of harmonics, this makes sense only going from the fundamental to the second one. Let's also note that any in-between harmonics that might please our ears--however many 1/3s and 1/2s it'd take to add together to sound consonant to our ears (as though we were hearing the double and triple of some lower frequency)--don't get represented very well in this scaling.
It's a lot easier to see that when it's written out. I put some notes and frequencies in a table here, starting with the reference A at 440 Hz and proceeded up the "equal temperament" chromatic scale. It gets pretty close to true with the fourth (1/3) and the fifth (1/2) which is presumably the primary point of it. The fact that our music hammers so hard on the IV and the V chord no doubt has a lot to do with tuning, but these are important harmonic intervals, and we probably would have stressed them anyway. The major and minor thirds (1/4 and 1/5, grrrrr...) carry a lot of musical weight too, and now we're getting a little degraded. Every other step is struggling to get any real harmonic significance, and I swear they're there only to make more use of the frequency space, to give some means for the musician to move on and off of the scaffold imposed by the more basic harmonics.
[I mean, okay, the point of equal temperament is to universalize twelve keys too. I understand that they used modulation before they began to tune instruments this way, but did standardizing the space make it easier to wing chord progressions around? I don't really know, but that is my suspicion. Because for anything with fixed frets or pipes, tuning with a just (as in, "just the harmonics, ma'am") temperament is going to be pretty damn limiting. The question of tuning is an old one though. When you got instruments that could efficiently hit multiple strings, the wizards of the day quickly realized that tuning could be easily designed to support some harmonics or other ones, as well as more universal temperaments that fudged them. I know even less about music history than I do about music, and I realize that the tuning argument had been circulating for a couple hundred before his time, but from this angle, I'm convinced that the only reason J. S. Bach didn't solve every remaining technical question in the field was because no one had invented the pianoforte yet. I almost wish he ended up on team leftbrain instead.]
I understand a lot of the musical conversation as dissonance that resolves to a more pleasing consonance. (I don't mean to suggest that I can carry on such a conversation, mind you--which ones are the blue notes again?-- but you don't necessarily need to be able to build a house to have a feeling for how a hammer works.) And the scale notes are in there to lead us toward or away from the (more) harmonic chords or intervals, which is great and all, but it's still holding us back. We're better off when we can weave in and out of consonance on a smaller level. You goddamn well need to have slurs and slides and hammer-ons, vibratos and bends. Honestly, it's what makes playing with feeling worth a damn. A minority of instruments are really flexible in this regard, either by using a fretless fingerboard, or else they have a sensitive enough action that you can change the pitch continuously within some range. On a fiddle or an electric guitar--or with the human voice--we can pinpoint the harmonics precisely (choirs often do it by sheer instinct), or we can wheel nauseatingly away from them, and do it all on purpose. I imagine that this is about 2/3 of the reason violins came to rule the orchestra, and why Les Paul came to rule the rock band. [Why not the trombone, slide whistle, dobro, or theremin? Because those instruments just aren't as cool. Sorry.] Man, when you're playing them right, nothing sounds more lugubrious than a violin or an electric guitar. And that's because they can avoid or accept the tyranny of harmonics at will.
I mentioned timbre. Even though instruments are producing the same pitch, or almost the same pitch, they all have different sounds. Any given instrument is going to have enough subtleties in its structure or action so that a number of minor vibrations can get through into the output as well. The one-dimensional assumption is great, but it only goes so far. Shaped cavities and resonators are the usual way to do change the voice, increasing or decreasing the quality factor (a measure of how "wide" in frequency the modes are, how much impure sound can sneak in at the edges) to soften or strengthen the tone, or which can set the instrument's modes just off even so as to avoid producing the overtones. Materials of construction can be more or less lossy, and even odds and ends such as handles and bolts might resonate at some unexpected frequencies and contribute to the sound as well. The nature of the input matters too. A vibrating reed is going to produce a different spectrum of frequencies at the front end, which may or may not be supported in the tube, than buzzing lips or a bowed string will (and when you strike things, the input is really different). If you took an oscilloscope and measured the waveform, you'd see it was periodic at the expected frequency, which is where the primary pitch comes from, but the shape of the pressure wave would be pretty funny-looking, with a bunch of little spikes and blobs in it. It's as if a whole bunch of clean sine waves that might have "fit" in that frequency were added to the primary one. Which would be exactly how to think of it mathematically.
The math would suggest that any periodic function can be correctly represented as combination of a lot of simpler periodic functions. That the physics suggests that it really does contain smaller waves is something I have occasionally found a little spooky, but it's not really any different than breaking things out in terms of other orthogonal components, just like on a rectangular grid. The idea is that any periodic function can be represented exactly by a sufficient number of sines and cosines. When you have a wave that can be imagined to persist for so many cycles that it doesn't matter where it begins or ends, then adding simple waves together is easy to imagine, but it really does get a little more mystical when you mix it in with the time domain. A quick pulse is fundamentally different than a blown or a bowed note.
If you think of the latter as a wave extending, for all practical purposes, infinitely, then that's what a sine function looks like. A pulse, although it has a discernible duration, is different because it disappears at the ends. Maybe you get something close to this from a really stiff drum, just one big and more or less uniform burst of pressure coming out its ass end. For a moment, let's ignore what happens far from the ends and just imagine a square pulse--on suddenly, and then off suddenly--as if it's inside one period. (For math purposes, we have to think for now that it repeats after this period.) One big bubble of a wave is obviously what you need to get the basic size of it right, and from there on, you can keep adding waves of different integer frequencies, each with smaller and smaller contributions (which you can calculate), to make the ends and the top get flat. That's Fourier analysis, and it's cool stuff. To make an exact square wave, you'd need an infinite many small sine waves. But even here, you could rightly argue that we're still using lots of little harmonics to finally address the non-harmonic space. Well, we did have to assume that the pulse repeated...
To describe just one pulse, you need to imagine frequency as a continuous thing, and the contribution of frequencies as a smooth curve too. (This is the Fourier transform now, and not just the coefficients of an added-up Fourier series.) It's got a straightforward formula, that we can at last apply it a single pulse, which is what has been done in the right side of the figure below. The wavy curve represents how much any given frequency contributes to the simple beat. (If we were, by contrast, to take the Fourier transform of a continuous sine wave, it would only have one value, and not be curvy. A sine wave is only defined by just one frequency.) This has some interesting implications. First, a pulse has an infinite number of frequencies associated with it, even though they get really small far away from the center line. If the pulse is shorter, the frequency curve will widen, and the strongest frequency component will be higher-pitched. If the pulse is stronger, then the Fourier transform of it has higher amplitude too, and the fading end bumps are more likely to be noticed.
There is only one part of the band that produces pulses of sound, and I had never quite managed to cross the line and think of drums as musical instruments before. But last week, talking to my friend switters, I had an epiphany, and music hasn't looked the same since. (I'm going to steal a little from that conversation here.)
[Plucked strings might count as a middle ground, although most of the time you let them ring, so that the secret frequencies fall right out. Redneck bands use the mandolin for rhythm usually, and there you try to "chop" it, arguably working it like a drum--whack the strings and then immediately deaden them. Done right, it gives a pop that only suggests the chord. Good rhythm players will have a little conversation of their own, opening the chops and closing them back up.]
Do drums really produce a pure pulse? Well, a taut drumhead does have a set of vibrational modes of its own (2-dimensional now, so hitting the surface in different locations can excite different ones, and the modes are no longer consonant), and there is a very short resonator section. And you can, to an extent, control the duration of the strike (expand that envelope of frequencies or narrow it) by softening the head or the mallets or by hitting it harder. So obviously, different drums and drummers have different voices. Some orchestral ones are built to sound more tuned (evidently for depth), and that's great, but I'll take a deader head to fuzz out the harmonics, hit hard, with a deep primal voice. Get some of that big voodoo to come out the other end.
I think that the primal sound is difficult to get with a single drummer, and that modern popular music too often fails to get all that it can from the shadowy acoustical depths. (Although I believe that this is exactly why the standard rock drum break is a couple of quick hard-pounded phrases down along the heavier toms. Whacka whacka, thudda thudda, boom.) Two ways you can try and get it with the one-man kit, either with an overwhelming quantity of accurately placed and voiced notes (color, as switters calls it) or to just beat the living fuck out of the things in order to get a denser (not just louder) piece of sound to come out. On the drum kit, for normal human beings, this seems like it must involve a serious tradeoff. If you hit the skins hard, it's going to be more difficult to hit them with speed and precision. I suspect that a bomber type needs a very good time sense (but what drummer doesn't?) if he's going to get by with fewer whacks.
We can all name some of the bombers--Keith Moon, Jon Bonham and Lars Ulrich immediately came up. I'm going to add this clip though, because for me, the itch started here. I never much thought of Bad Religion until I got this CD, and the difference is entirely this young guy with the sticks. This track might make an especially good example, because the lyrics are pretty sub-par for these guys (a shitty pun, a cheerleader chant, and no rhyming big words) and the musical flavor is pretty damn unsubtle (talk about hammering on that fifth interval) without much contribution from of the usual vocal harmonies. I'd call it a phoned-in effort, but sweet holy fuck, those drums. They fill the lower register with ominous power. If you have to hear it first on this youtube clip, at least make sure you put on some headphones to get any of the effect. When that phrase at about 2:13 comes in, you can feel from the first roll that the whole thing is going to explode in a couple of bars. Well, too bad the rest of the song wasn't better.
I've had that BR itch for a year or two now. But of all places, the revelation came while watching and discussing the Battlestar Galactica remake, which has a surprisingly excellent score. And while I have never considered myself a real audiophile, I have come to absolutely love this home theater setup I sprung for; it gets a lot of the primal sound fidelity that I never knew I was missing. To go with the clever orchestration, the show incorporates some serious drum rhythms (more than one player, I think, judging by all the darkness they're getting out of them) fit in with some complexity and nuance, pulling you every which way, and for a drama that slugs you in the guts so much, it's just a brilliant touch.
That conversation went to a lot of interesting places, and at this point I have a lot of things I need to be off listening to. That's all for now!