# Re: Calculating harmonics of a vibrating guitar string

"Squier" <squier@xxxxxxxxxx> wrote in message news:241020111115517392%squier@xxxxxxxxxxxxx
RichL <rpleavitt@xxxxxxxxx> wrote:

guitar-string harmonics:

http://mysite.verizon.net/vzex6vrt/sitebuildercontent/sitebuilderfiles/harmonics.xls

What the first worksheet (called Triangle) does is assume an initial string
shape which is a triangle whose apex corresponds to the point along the
string at which the string is plucked, measured from the bridge. Ignore the
other two worksheets for now.

You can enter different values into the boxes highlighted in green.

The first box is scale length, e.g., 24.75" for a Les Paul, 25.5" for a
Strat, etc.

The next box represents how far away from the bridge you pluck the string.
The value that's currently entered, 3.5", is roughly the mid-point between
the two pickups on a Les Paul.

Next you choose a string: this is a drop-down box, and you pick 1-6. Next
to the drop down box, the spreadsheet will display the string name (e.g., G,
A) and the string's fundamental frequency.

Finally, you choose from another drop-down box where you fret the string: 0
represents an open string, 5 the fifth fret, etc.

On the right is a graph displaying the amplitudes (in dB) of the various
harmonics relative to the fundamental which is chosen as 0 dB, all the way
up to harmonic # 21. Also, on the left is a list containing the fundamental
and harmonic frequencies, amplitudes, squared amplitudes, and amplitudes
(relative to the fundamental) in dB. This last column is what is plotted.

Interesting stuff happens when the fretted length is an integer multiple of
the picking position. In the example that's shown when you open the
spreadsheet, the scale length L0 is 24.5", the picking position x0 is 3.5",
the fretted position is 0 (meaning L = L0), so L/x0 = 7. As you can see
from the plot, the 7th, 14th, and 21st harmonics disappear!

Now choose x0 = 6.125" and leave everything else the same. Now L/x0 = 4,
and the 4th, 8th, 12th, etc. harmonics disappear.

Now go back to L0 = 3.5" and gradually change the fretted position (first
fret, second fret, etc.) and watch how the harmonics evolve.

The second worksheet, called Rectangle, solves a similar problem except with
different initial conditions. Rather than specify the initial string shape,
it assumes that the portion of the string where the pick strikes acquires an
initial velocity, and that the rest of the string is initially motionless.
The inputs are similar to those on the Triangle worksheet, except that
there's an extra parameter called w which is the width of the pick at the
point where the pick hits the strings. As you can see from the graph, the
distribution of harmonics is similar to the Triangle case except that they
don't decrease as much as the harmonic number increases.

Fool around with this if you like, and let me know what you think! I've
highlighted ones, so you can't accidentally change something important.

What I eventually want to do is extend this so that it includes the effects
of the pickups: they don't "see" the whole string, of course, but only the
portion immediately above and adjacent to the pickup. Also, I want to
include multiple pickups, either in or out of phase.

If I get really ambitious, I'll add in the effect of the guitar circuitry
(including pickup impedance) and maybe even the amp's tone stack.

Thanks RichL for this. Very nice tool to have here.
I am not sure though how string gauge might relate to differences.
For example, 2 different size strings might produce 440 fundamental
but will they produce the same over and under tone frequencies (?)
Or in other words - the chart does not take into account for varying
string types (wound or plain) or different gauges playing the same note.
I know that it is very different sound playing 009 string versus .011 string
even though both are tuned to high E on the guitar. So there must be
something going on there since each is tuned to the same fundamental
but there has to be major differences in resonant over and under tones
occuring. It's hard to quantify - but most any guitar player would know
the difference in sound in standard tuning playing a light 009 string set
versus a .012 heavier string set even though both sets are tuned the same.

Good points. I'm not sure how that enters into the physics/math at this point but you can be assured I'll look into it.

.

• References:

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