Copyright © 2018 by Wayne Stegall
Updated August 21, 2018. See Document History at end for
details.
Guitar Shape
Mathematical
modeling
of
guitar
shape
leads
to
modeling
program
Introduction
Some time ago I wondered if the shape of a classical guitar was
mathematical and further if a mathematically modeled shape would be
better in some way. I imagined this to be a polar plot.
Manual Attempts
When I began to explore these ideas, I first tried for a peanut (or
hourglass) shape with the following equation and succeeded. This
resulted in upper and lower bouts of same width with the expected waist
in between.
(1)

r = a_{0} + a_{2}cos2θ

I already had in mind that if I added a
_{1}cosθ to
the equation the lower bout could be larger than the upper.
(2)

r = a_{0} + a_{1}cosθ
+
a_{2}cos2θ 
Now the width of the outline was unavoidably narrow relative to the
length without some further change. The required change was to
stretch the y axis of the polar plot.
(3)

x = r·cosθ 
(4)

y = y_{mult}·r·sinθ 
Now that could get the right bout widths, the outline seemed too
roundish. Adding a sixth harmonic then allowed the familiar shape
with considerable tinkering with the coefficients.
(5)

r = a_{0} + a_{1}cosθ
+
a_{2}cos2θ + a_{6}cos6θ 
Computer solution
When I came to want a program to calculate the coefficients and plot
the curve, I was stuck on the idea of and empirical method of refining
the coefficients but failed to imagine a final algorithm. Then I
thought of the min/max algorithm used to calculate digital filters and
thought it right.
Min/Max algorithm process
 User specifies length, lower bout, waist, and upper bout
dimensions and a_{4} and a_{6} coefficients
 Choose reasonable θ values for minimum and maximum points
 Calculate coefficients a_{0}, a_{1}, and a_{2}
from current min/max θ values by solving simultaneous linear equations.
 Find actual min/max points for next calculation.
 Repeat 3 and 4 until convergence.
Equations on which simultaneous linear equations are based
r = a
_{0} + a
_{1}cosθ + a
_{2}cos2θ + a
_{3}cos3θ
+
a
_{4}cos4θ
+ a
_{5}cos5θ
+
a
_{6}cos6θ
x = r·cosθ
y = r·sinθ/a
_{y}
r = a
_{y}·y
/sinθ
a
_{0} + a
_{1}cosθ + a
_{2}cos2θ + a
_{3}cos3θ
+
a
_{4}cos4θ
+ a
_{5}cos5θ
+
a
_{6}cos6θ  a
_{y}·y/sinθ
=
0
Length = 2(a
_{0} + a
_{2} + a
_{4}
+ a
_{6})
Matrix input to Gaussian elimination
a_{0} 

a_{1} 

a_{2} 

a_{y} 











2


0


2


0


Length – 2(a_{4}
+ a_{6}) 
1 

cosθ_{0} 

cos2θ_{0} 

–Lower/sinθ_{0} 

–(a_{3}cos3θ_{0}+
a_{4}cos4θ_{0}
+ a_{5}cos5θ_{0}
+
a_{6}cos6θ_{0}_{}) 
1 

cosθ_{1} 

cos2θ_{1} 

–Waist/sinθ_{1} 

–(a_{3}cos3θ_{1}
+ a_{4}cos4θ_{1}
+ a_{5}cos5θ_{1}
+
a_{6}cos6θ_{1}) 
1 

cosθ_{2} 

cos2θ_{2} 

–Upper/sinθ_{2} 

–(a_{3}cos3θ_{2}
+ a_{4}cos4θ_{2}+ a_{5}cos5θ_{2}
+
a_{6}cos6θ_{2}_{}) 
_{
}
Program and its operation
Program name is
guio.exe
Download
program version
1.1.0. (New version, adds new functionality, see document history)
Download
program version
1.0.0. (Old version)
EULA (End user license agreement found in about box of program).
I retain copyright. You may
freely use this program for personal, noncommercial, use (that is you
cannot offer it for sale). You may share it with others as long
as it is provided to them complete and unmodified as you got it from my
website.
Because this program is free, although it works, I do not guarantee its
operation or application.
Run program and a default program shape appears.
Menu>Edit>Input Shape
brings up dialog to define your own shape loaded with default
parameters.
Figure
1:
Input
Shape
dialog.


Enter all required values experimenting with small values of a
_{4}
and a
_{6}.
in>cm and
cm>in buttons convert units.
Press
Calc to calculate and
draw shape.
Press
OK to do the same and
exit dialog.
Figure
2:
Guitar
outline
in
horizontal
orientation


Menu>View>Vertical
toggles between vertical and horizontal modes with vertical mode
indicated by check mark.
Figure
3:
Guitar
outline
in
vertical
orientation


Help in adjusting coefficients a_{3} through a_{6}
Note: These plots relate to horizontal guitar shape drawing. Use
with vertical drawings would rotate the help plots 90° clockwise.
A
_{3} through a
_{6} should only have small experimental
values.
Figure 4 below
shows positive and negative values of a
_{4}
and a
_{6} added to circles to help you to understand their
effects. Starting from the inside working out they are:
 + a_{4}.
 – a_{4}.
 + a_{6}.
 – a_{6}.
Figure
4:
Graph
of
the
effects
of
coefficients
a_{4} and
a_{6}.


Figure 5 below shows positive and negative values of a
_{3}
and a
_{5} added to circles to help you to understand their
effects. Starting from the inside working out they are:
 + a_{3}.
 – a_{3}.
 + a_{5}.
 – a_{5}.
Figure
5:
Graph
of
the
effects
of
coefficients
a_{3} and
a_{5}.


To add a soundhole to drawing
Menu>Edit>Input Soundhole
brings up dialog to define or calculate a soundhole to add to guitar
shape..
 Fixed calculation: Enter Diameter and Position for a fixed
calculation (Position is measured from neck end of body).
 Classical calculation: Enter Diameter, Scale, and
Fretboard/Fret width at the 19th fret (Position is calculated from
these) or
 Classical calculation: Enter Diameter, Scale, and
Fretboard/Fret width at the nut and 12th fret. (19th fret value
are calculated from these.)
The classical hole position is calculated so that the hole intersects
the 19th fret so as to put the middle third of that fret inside the
hole. Please doublecheck the position of the hole by your normal
methods if the drawing finds actual use.
Figures
6
and
7:
Soundhole
dialog
before and after classical calculation.



Figure
8:
Guitar
shape
with
soundhole added.


Additional program notes
 Units for soundhole and guitar shape must be the same. If
using different units a conversion must be made at some point.
This seems most likely to arise from body dimensions in inches but a
scale length in centimeters.
 Use centimeters and not millimeters for metric units or grid
would be too tight and the resulting dots per millimeter too small to
be useful.
 Conversions provided in input dialogs only affect soundhole or
shape variables.
To convert the entire drawing do so from the edit menu.
 Menu > Edit > in>cm
 Menu > Edit > cm>in
Lutherie use
For those who want life sized drawings of guitar shapes the number
displayed in the upper left hand corner of the plots indicates the
number of pixels per unit measure. Then follow this procedure.
 AltPrtSc with program selected to get its image.
 Paste into a graphics program.
 Scale the image to the indicated dots per unit.
In GIMP this is done as follows:
Menu>Image>Scale Image
Change X resolution and Y
resolution to <your value> pixels/in (or pixels/cm)
 Use graphics program to create subimages that when printed can
be taped or glued together to create a complete blueprint.
(guitar bodies are larger than standard letter paper.)
Dreadnoughts too
Although I originally sought to model the Spanish guitar the following
shows an attempt at a dreadnought. The addition of coefficients a
_{3}
and a
_{5} to the new program (version 1.1.0) enable a shape
flat at neck and tail as shown below.
Figure
6:
Input
and
results
of
attempt
to model a dreadnought.



Document History
July 24, 2018 Created.
August 21, 2018 Added additional coefficients a_{3} and a_{5}
to program to add more versatility (i.e. to emulate more possible
steelstring shapes). Added a soundhole plot.
August 21, 2018 Added comment clarifying orientation of a_{3}
 a_{6} help plots.