Ideal Electric Guitar--Up and Down Pulses

[H S]

Thought this might be interesting to the group.  This comes out of a lot of work trying to make guitar-like sounds with an additive synth.  The bottom line is that the pure, ideal, Platonic Form of an electric guitar (but not necessarily best-sounding!) would make an "up-and-down pulse" waveform like Fig. 3.

Here's how it goes:  According to Fletcher and Rossing, The Physics of Musical Instruments (great book!), when you pluck a string you start by pulling it, thus making a “corner.”  When you release the string that corner travels around the string in both directions and back again, like Fig. 1:



So what’s happening over the pickup?  The string position over the PU is like the line in Fig. 2.  But the PU responds to string velocity, not position, which is like the line in Fig. 3.  This is the waveform you’d get from an electric guitar in an ideal world--if there was no friction, no string stiffness, no stray capacitance, and if you had a pickup that responded to only one point on the string.  (The pulse width and the distance between the pulses depend on the PU position and the pluck position.)

So, two things:  First, a fuzz isn’t just distortion, it can be more like an “ideal” guitar than the clean channel is.  And second, a fuzz with up and down pulses instead of just one direction would be even more “ideal.”  (Not that being mathematically ideal is necessarily the best sound--but sometimes it’s the most extreme you can get.)

P.S.

Z.VEX â€" I met you in Twin Town Guitars onceâ€"I was buying a SHO.  I tried to explain this stuff then, but I made a hash of it.  I hope this makes more sense!

[A.S.P.]

pic-show-problems... book-link won`t work, too

[R.G.]

I'm confused. Doesn't the string velocity go to zero just as the corner passes over the pickup and the string changes directions? It seems like the string's velocity is fairly constant between corners and goes to zero as it changes directions at corners.

[H S]

I'm confused. Doesn't the string velocity go to zero just as the corner passes over the pickup and the string changes directions? It seems like the string's velocity is fairly constant between corners and goes to zero as it changes directions at corners.

After the string is released, the point over the PU stays in the up position, where it was when the string was pulled, until a corner passes.  It doesn't "know" that the string was released until the corner passes.  Then it moves downward, until the corner passes by again (after reflecting off the bridge.)  Then it's frozen in the down position until a corner passes again and starts it moving upward, back to the up position.  (The string movement part is from Fletcher and Rossing, I just added the pickup interaction part.)

The lines in Fig. 1 are strings, stretched between the nut and bridge, at different points in time.  The line in Fig. 2 is the position of one point on the string (at the PU) over time.  Even though something's always moving, that one point can spend a lot of time standing still.  It's like when "the wave" is passing around a stadium--most of the time you're sitting still.  The line in Fig. 3 is the velocity of that point over time, and thus the voltage across the PU.

[R.G.]

OK.

After the string is released, the point over the PU stays in the up position, where it was when the string was pulled, until a corner passes.

I'm confused. The corner is a displacement of the string. When the string is released, the corner propagates away from the point where it's plucked. The corner cannot propagate infinitely, as the amplitude must be zero at the end of the string. So as the corner moves, its amplitude changes. Because the ends of the string are mechanically clamped to zero motion, the slope of the string on either side of the corner must change for the corner to propagate up and down the string. For the slope to change, there must be motion of the string at each point except the two ends. The motion cannot be only at a corner, as the constantly changing slopes on each side of the corner amount to motion to/from the pickup all the time. A corner passage is a point where the motion changes direction.

After the string is released, the point over the PU stays in the up position, where it was when the string was pulled, until a corner passes. It doesn't "know" that the string was released until the corner passes.  Then it moves downward, until the corner passes by again (after reflecting off the bridge.)  Then it's frozen in the down position until a corner passes again and starts it moving upward, back to the up position.

Maybe I am just being thick headed, but from my reasoning, that sounds physically impossible. The constantly changing slope beside the corners must and will make the string have motion. It cannot stay suspended in one position waiting for a corner.

(The string movement part is from Fletcher and Rossing, I just added the pickup interaction part.)

I wonder if Fletcher and Rossing are confusing acceleration and velocity. The acceleration does have pulses, being the derivative of velocity.

[H S]

After the string is released, the point over the PU stays in the up position, where it was when the string was pulled, until a corner passes. It doesn't "know" that the string was released until the corner passes.  Then it moves downward, until the corner passes by again (after reflecting off the bridge.)  Then it's frozen in the down position until a corner passes again and starts it moving upward, back to the up position.

Maybe I am just being thick headed, but from my reasoning, that sounds physically impossible. The constantly changing slope beside the corners must and will make the string have motion. It cannot stay suspended in one position waiting for a corner.

Well, we're talking about things happening in a fraction of a cycle--a millisecond or less--while these corners are moving around the string at the speed of sound.  Maybe "frozen" is an overstatement!  When the string is released, the news moves at the speed of sound--that is, the corner moves at the speed of sound--but the point over the PU can't move until the news gets there.  In the fraction of a millisecond between the release of the string and the moment the corner can get to that point, that point is still "frozen" in its original position.  That same "frozen" moment is repeated in successive cycles.

Anyway, that's my best understanding.  Fletcher & Rossing discuss plucked strings at pp. 40-44.  Fig. 1 here is based on Fig. 2.8 on p. 43.

[Squeal]

We're gonna need an animation.

[R.G.]

Well, we're talking about things happening in a fraction of a cycle--a millisecond or less--while these corners are moving around the string at the speed of sound.  Maybe "frozen" is an overstatement!  When the string is released, the news moves at the speed of sound--that is, the corner moves at the speed of sound--but the point over the PU can't move until the news gets there.  In the fraction of a millisecond between the release of the string and the moment the corner can get to that point, that point is still "frozen" in its original position.  That same "frozen" moment is repeated in successive cycles.

Hmmmm... the speed of sound in steel is significantly faster than the speed of sound in air. In air, it's about a millisecond per foot; I think it's five to ten times as fast in steel.

Here's the problem: I've seen stroboscopic movies of string motion. It's not corners. It's simple harmonic motion in modes. The main belly of the string moves in a simple sine motion in two axes, perpendicular to and parallel to the pickup. Depending on where you picked the string, you excite some of the other modes - half the string length, 1/3 the string length, etc.

Those other modes of oscillation are why pinched harmonics work; the full length of the string is damped by touching the string at a node, which remains still, and the only string motion is the 1/2, 1/3, 1/4, etc lengths of the open string.

Come to think of it, how does the corner theory account for pinched harmonics? the point where your touch the string remains (relatively) motionless while the modes oscillate. Where do the corners go? When you remove your finger from the node, the node position is no longer held in place mechanically, so the corner must be free to move through there, but you can see on a stroboscope that it doesn't.

I'm not trying to be difficult. It's just that I can't understand what Fletcher and Rossing were thinking. I can well understand transmission line propagation of pulses (that's what they're trying to say; it happens in electrical lines all the time) but I think the "corner" is so small that it has no bearing on what the pickup picks up.

OK, there's another thought. The corner theory implies that the string motion that we hear is composed of corners passing a pickup. That necessarily means that the  corners passing have to occur at the frequency of the string vibraiton if it's correct. The fundamental frequency of the string corresponds well to modelling the string mass, tension, and damping. If you measure the string mass, stiffness, and tension, then compute what the fundamental frequency is, you get a number that corresponds to what we pick up and hear. However, the corners move with the speed of sound in the material. That speed is different for each material, and its' independent of the tension on the string. So if corners move at the speed of sound in the material, it can't be related to how tight the string is and we'd get the same  string response regardless of the string tension because the speed of sound in a material is mostly independent of external stresses on it, being a property of the bulk material.

[H S]

I think I used the term "speed of sound" too carelessly--the point is that change happens quickly but can't happen instantaneously all over the string.  I'm not sure what that speed is, though.

F&R talk about "time analysis" and "frequency analysis."  The time analysis is the moving corners thing, the frequency analysis is the description of the harmonic partials, which is closer to what you're describing.  The thing is, when the partials are in their original phase relationship, they add up to make the string shape described by the time analysis.  In the real world, they quickly get out of phase, but the harmonic content is still essentially the same.  The same should be true for the guitar output--the shape changes as the partials get out of phase, but the harmonic content is still the same.  On the oscilliscope I can sometimes see a waveform that looks like a soft-corner version of Fig. 3, like a heartbeat.

For playing the octave harmonic, I imagine the time analysis would describe the two corners making a figure 8 between the two halves of the string.  When you release the node you add two more corners.

Good discussion.

[R.G.]

It is a good discussion.

I'll go dig through their stuff. Maybe they were making some other point.

[George Giblet]

Believe it or not you are both talking about parts of the same thing.

Take the case of 1st harmonic.  At an instant in time imagine the string is in one of the extreme positions with a half sinusoidal shape.  That position is treated as two waves, one have sine-wave "A" going to the left and another "B" going to the right; the magnitude is half the original  displacement.   When each half moves part of the wave appear to "goes" of the end, what happens here is the part that *would have* gone off the end gets a reverses in direction of travel and also the displacement is flipped - this is a result of the pinned boundary.

It will take a certain time for the peak of the wave to make it to the fixed boundary, this is the time to travel 1/2 the string length.  At this point in time the wave A will have two parts one travelling left and it's flipped part which is travelling right.  The two parts cancel out.  The exact same thing happens to wave "B".  This condition corresponds to the string in the central position of it's motion.

Now after this point most of "A" is moving right and most of "B" is moving left, they are moving towards each other and are lipped in displacement.   From the point in the last paragraph, the time it takes for the waves to travel another 1/2 string length the peaks of the two parts of the wave co-incide and the result is the full displacement in the flipped direction.

In between these times it's possible to show the shape of the string is a sinusoid with lower displacement than the initial position - this is all in agreement with the usual string moving up and down theory.

Now imagine the initial position is a narrow triangular pulse located at 3/4 the length of the string (ie a pip to the right) and in the down position.  As before the pulse is spit into two pulses.  The time it takes to travel 1/4 the length of the string the "A" pulse will be at the centre and the "B" pulse will be at the boundary, there is no "B" displacement at that point.  The time it takes to travel 3/4 the length of the string the "A" pulse will be at the left boundary and the "B" pulse will be flipped in displacement and would have travelled to the half way position. The time it takes to travel one string length the "A" pulse will be flipped in position and will be located and the 1/4 string length postition from the left, the "B" pulse will have made it to the same position and will add to the "A" pulse.  The string shape produces will be the same as the original shape except it has been flipped left/right and up/down.

Now imagine the pulse and the half sine being super imposed at the same time.  The parts of the wave travel at the same speed (called non-dispursive) and they will both line up such to produce this flipped left/right and up/down replica of the original string shape.

In reality when you pluck a string you don't put a local triangular pulse on the string, you actually pull the string from the fixed end points and there is a kink at the pick point.  The impact that the fixed pivot points have is to limit the frequencies on the string to harmonics of the fundamental.   You can still have a kinked shape on the string travelling up and down but the shape will be made up of harmonic components.  The shape and position determines the harmonic distribution - plucking near the bridge sounds brighter than plucking near the neck.

So both the sinusoid and the kink can be described by the same mechanism - ie. both views are the same.

The last thing to note is the string is a linear system which means all the frequency components can be separated an added together ie. they don't interfere with each other and superpostion applies.

Here's some links which might convince people further:
http://www.mhhe.com/physsci/physical/jones/graphics/jones2001phys_s/ch15/others/15-1/index.html
http://www.phys.unsw.edu.au/~jw/strings.html

On the simulation I recommend right clicking on the graph to move the pluck position to the right so you an see how the shape of the string  "toggles".

[Squeal]

Wave speed in an ideal string: square root of(Tension/(mass per unit length))

[H S]

Thanks for the explanations.  The first animation link really connects the two perspectives, especially if you start with one harmonic and then add one at a time.

I dug up more of my old additive synth stuff.  In the electric guitar output, the amplitude of the nth harmonic should be:

A(n)=(1/n) x sin(Pi·n·X%) x sin(Pi·n·Y%)

where X% and Y% are the distance of the plucking position and the PU position from the bridge, as a percent of the total string length.

Harmonics that have a node at the plucking or PU position get zeroed out.