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Other than fixing myself a larger-than-I-meant-it-to-be dinner (I got carried away with the number of other ingredients and had to add another egg to compensate) and some migraine-abatement tea (lavender+basil), I didn't really accomplish anything since my last entry -- never got that second wind after all. Oh well. I'll try to get to the store after I get up again, and I think I can do a temporary fill where the acryllic is growing out that'll hold for the 3LF gif this weekend and hit the nail salon next week -- I'll only be playing a little bit of guitar on Sunday, mostly recorder and double bass. Anyhow, since I'm feeling a bit more alert than early this evening, I'll toss out an incomplete bit that I'd been meaning to post before...
About three weeks ago, I posted about a phenomenon I was curious about, in which I get two distinct notes out of one string at the same time. (They're harmonically related -- an octave apart -- so you can think of the higher frequency as "merely an overtone", but those two frequencies are approximately equally loud and swamp any other overtones present, so the acoustical (or psychoacoustic?) effect is of two distinct notes.)
Well, I still don't have a good picture in my mind of what the
string is physically doing (though ![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif){kind=small}
juuro's comment
that, "From what I remember [...] there is almost always a
traveling-wave component as well [...]," (and the mention of
helical modes) does answer the most basic question I asked
then. (Thanks, by the way.) But what I do have (and will show
y'all after I get all my screencaps and diagrams and notation sorted
out and uploaded somewhere when I'm a little more awake than now and
have less in the 'urgent to-do' category than I expect to tomorrow)
is an oscilloscope-like trace and some math. The scope trace is
actually a plot from a .wav file captured on a friend's digital
recording rig in his studio, and I found it rather interesting.
It clearly shows that yes, both frequencies are present (it's
not some audio illusion), but I was fascinated by the asymmetry
of it. The first section we looked at looked like a series of
capital 'M's -- the positive portion of the wave was not a
reflection of the negative portion, and the bottom went farther
from zero than the top did.
I suspected (and I'm sure a bunch of other math geeks and vaguely-mathish geeks here already guessed while crossing the space between the preceeding paragraph and this one) that this unusual shape resulted from a particular phase relationship between the fundamental and the octave. This week, while failing to fall asleep one night, I finally got around to investigating that -- I threw a few lines of BASIC at my PDA to tell it to plot y = a1 × sin(θ) + a2 × sin(2×θ + φ) for different phase offsets φ, and it turns out that if the octave is π/2 radians (90°) out of phase with the fundamental, the 'M' wave results. So plug either of these into your graphing calculator, favourite math app, or the programming language of your choice:
y = sin(θ) + sin(2×θ+π/2)
y = sin(θ) + cos(2×θ)
and you'll see the waveform I'm talking about.
Yeah, I'd have gotten around to this much sooner and with less effort if I had a pair of sine wave generators and an oscilliscope lying around, but I did eventually get to it. And yes, mathematically this is really, really trivial stuff, but I still think it's nifty trivial stuff, that such cool, asymmetrical waveforms emerge from something so very simple as the sum of a sine wave and its first harmonic, just by playing with the phase relationship.
I'll post that trace later.
I still want stroboscopic footage to see just how that
string is dancing around. Stereostroboscopic video so I
can clearly see what it's doing in 3-D! And a pony!