It's 3 -- ah, 3:30 -- argh, 4:00 In The Morning And I Need A Strobelight
(Or: Physics Goes 'Twang')
I should be asleep. I'm not. Go figure. And I'm puzzled.
Pluck an open string, and you get the fundamental frequency of that string (i.e., that combination of length + mass + elasticity + tension). There are overtones present, but to a large extent what you've got is the vibration of one segment with nodes at the nut and the bridge. From what I remember of plugging a guitar into a sillyscope[1] years ago, the wave produced is overwhelmingly the fundamental with much smaller amplitude overtones crawling slowly along it (which means they're not exactly integer multiples of the fundamental, I guess -- izzat predicted by the model or a side effect of making strings out of real-world materials?), pretty much as the elemtary-school physics example suggests. And pretty much as we expect to hear, unless we pluck the string way down at one end (or select a pickup, or combination of pickups, placed where certain overtones are easier to detect) to add "twang" or "bite". For now let's only consider the "plucking in a normal place" situation.
And natural harmonics do the obvious thing for the very straightforward reasons one ought to expect. Lightly touch the string at its midpoint while plucking with the other hand, and you force a node at that point, which remains a node after your finger is withdrawn, resulting in what looks like a pair of standing waves, each half the length of the open string, one on each side of that midpoint node (it's one standing wave with a wavelength half the length of the open string, with a third node at the midpoint). Half the length, twice the frequency, and we hear the octave. All well and good. (Why the octave harmonic sounds "more pure" and more bell-like than the same note fretted, I'm not certain. And I don't recall whether that looked the same on the sillyscope as I'd expect from its sound.)
Here's what I'm having trouble picturing, and what I think I need want a strobelight (and maybe a fast motion picture camera (and maybe a sillyscope too, as long as I'm wishing)) for:
Pick up a bass guitar (my regular guitar is downstairs right now and I'm in bed, or I'd verify that this works on that as well; for now I'll limit myself to the electric bass because it's the current bed-instrument[2]) and play the octave harmonic on the G string. Then hammer on at the second fret. What do you hear?
What the simplistic model of plucked strings suggests I should hear is either the natural A (because hammering on disrupted the two-division standing wave, and the energy of the hammer-on got added to the kinetic energy already in the string but as though you'd just plucked it stopped at the second fret -- as is what happens if you hammer on at, say, the fourth fret) ... or I should hear the octave A (because hammering on didn't disrupt the node structure, just moved the nut-end node to the second fret, raising the pitch -- like what happens if you play an artificial harmonic and then slide). But neither of those is what I hear.
I hear both notes: the A that I would hear playing normally at the second fret, and the A that I woud hear at the fourteenth fret. It sounds very much like two strings an octave apart playing together, but (hold on while I repeat the experiment with my thumb muting the A string to be sure I'm not just hearing a sympathetic vibration) but it's all coming from the one string.
Okay, thinking numerically I can understand this as a fundamental and an overtone of approximately equal strength, and expect that on the sillyscope I would see a shape very close to f(x)=sin(x)+sin(2x) if I ignore the wee ripples of higher order overtones. Which is also what I'd expect to see (but messier) if it were two strings played together. And thinking logically, it makes sense that if overtones can exist at all -- and I've seen 'em on the sillyscope so I know that's the math I'm hearing when I hear them -- then having a really loud overtone is merely a difference in magnitude, not a fundamentally[3] different phenomenon.
But what's got my three-in-the-morning brain (it took me a while to type this) confusled[4] is that I want to think visually here, so I keep trying to picture this as a standing wave, and I can't figure out what f'ed up shape that wave would have to be to work.
Am I just not seeing the right shape, or am I approaching the problem completely wrong in the first place? Is it a standing wave, or just a travelling wave (a*sin(b*x)+a'*sin(2*b*x)) bouncing back and forth like the jumprope tied to a fencepost in the elementary-school demonstration?
And that's why I want a strobelight. And maybe a high speed motion picture camera. At a quarter to four in the morning. I, ah, don't suppose anyone reading this knows the answer off the top of your head or knows what search terms to fling at Google or Wikipedia to zoom in on this narrow subtopic without wading through three or four reams of stuff I already know plus ten times as much background I don't know with lots of math to chew on to get to it? What the hell is happening in my G string?[5]
I'm not sure what happens if there are no frets. I'll check that tomorrow.[6]
[1] Oscilloscope. Not to be confused with an osculascope, which I've never heard of but am having a great deal of fun trying to imagine.[7]
[2] Well, at the moment I've only got two[8] bed instruments (sometimes I have just one), but when I've been doing a lot of composing and haven't gotten around to putting any of my toys away, I occasionally wind up with two instruments beside the bed and three or four more in bed with me. Anyhow, I gotta have at least one instrument -- usually a solid-body electric guitar or bass -- close at hand in case a) a nifty tune idea pops into my head, b) I just get a "must play guitar now" craving, c) I can't sleep and want to play myself a lullaby (or distract myself from the frustration of not being able to fall asleep), or d) wake up not feeling well enough to wander downstairs for an instrument, but feeling that I should practice. Oh, or e) I bump into sheet music while surfing the web and want to hear how it sounds. And yes, yes, I've occasionally woken up curled around my guitar as though it were a teddy bear, and yes, I've already been teased about it.
[3] Sorry[9]. Couldn't resist.
[4] Not entirely certain how that wants to be spelled, but that's the spelling that makes the most sense to me. Pronounced "con-f(y)ooz-'ld" or "con-fooz-əld". Hey, anyone know the etymology of that? Is it something conscious like a portmaneau of "confused" and "puzzled", or nothing more than a deliberately too-cutesy-by-half version of "confused" that just happens to be really fun to say?
[5] Y'all were really hoping I'd get around to leaving you an opening like that, weren't you? C'mon, admit it. And yeah, I was tempted to make that the subject header for this entry, but that would've made it too easy.
[6] The double bass is too big for the bed. It doesn't get a turn as a bed instrument. (The mandolin, on the other hand, gets extra time on the bed because it takes up so little room even though I almost never play it on stage.)
[7] Not the first time I've entertained myself with such musing/imagining over the years. It's just a fun word to contemplate possible meanings/implementations of. Though I expect that inventing such a device would be even more up madbodger's alley.
[8] The other is this double-whistle thang, with three holes for the left hand and four for the right, that I should probably put in the woodwinds rifle-case and take to 3LF sometime )though I can't do much with it yet).
[9] But not quite sorry enough to go back and edit it, obviously. Deal.