Copyright © 2018, 2019 by Wayne Stegall
Updated January 10, 2019. 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 an 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
1.4.0 (New version, adds new functionality, see document
history)
Download
program version
1.3.1. (Old version)
Download
program version
1.3.0. (Old version)
Download
program version
1.2.0. (Old version)
Download
program version
1.1.0. (Old version)
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}.


Beginning in program version 1.4.0 these help plots can be displayed
inside the guitar shape.
 Enter the Input shape dialog ( Menu>Edit>Input
Shape ).
 Click check box Show help for coefficients a_.
 Enter the coefficient number for which help sought in the edit
box at the end of the same sentence.
 Click Calc to update the shape and show help figure.
The help plot displayed is centered with the guitar shape to have
direct relevance. The plot is inverted if the current coefficient
is negative from its positive form just as represented above. Any
visible soundhole or fingerboard plot will be suppressed while help
is on.
To turn help off:
 Uncheck the previously activated check for the coefficient help
from the Input shape dialog,
 or turn off from menu ( Uncheck Menu>Help>Coefficient
help ).
Figures 68: Input
shape dialog with
help activated for coefficient 6, and horizontal and vertical plots
with help figure shown. 



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
9
and
10:
Soundhole
dialog
before
and
after
classical
calculation.



Figure
11:
Guitar
shape
with
soundhole
added.


Fingerboard display
Menu>View>Fingerboard
will
display
fingerboard
(and
uncompensated
saddle
position)
if
scale
and
fingerboard
width
at nut and 12th fret were entered correctly in
Soundhole dialog. The fret at the jeck join is not plotted to
prevent interference with display of shape. The program
presumes neck join at 12th fret for a classical soundhole calculation
and 14th fret for otherwise. A fingerboard can only be shown if a
soundhole is already displayed. This function is to help position
a steelstring soundhole relative to the highest fret or to visually
verify the correct placement of the classical soundhole relative to the
19th fret.
Soundhole dialog functions for fingerboard
 Fingerboard can now be activated from the Soundhole dialog when
the soundhole and fingerboard are calculated.
 Neck join fret override
edit control can now override the default 14th fret join to the body
for the fixed calculation (Left blank is as if 14 entered).
Caution: Fingerboard
display may not have enough resolution to place frets accurately and is
only for use in overall design. I think 100 dots/cm (254 dots/in)
or more – as indicated in upper left corner – would be required for
accurate fret placement.
Figure
12:
Vertical
guitar
shape
with
soundhole
and
fingerboard
added.


Figure
10:
Horizontal
guitar
shape
with
soundhole
and
fingerboard
added.


Miscellaneous functions
Menu>View>Side Length
will
display
a
popup
dialog
showing
length
of
one
side.
Figure
11:
Program
with
Side
Length
dialog popped up.


Menu>View>Show Data
will
print
primary
parameters
on
upper
left
of
plot.
In
smaller
window
sizes will overlap drawing.
Figure
12:
Guitar
shape
data
displayed.


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.
 Maximize program to get greatest plot resolution.
 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.
Figures
13
and
14:
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.
November 28, 2018 Program version 1.2.0.
Added fingerboard, data, and side length displays. Internal
changes increase resolution of shape plot and reliability of program.
December 5, 2018 .Program version 1.3.0. Extended
horizontal guitar outline for improved appearance. Added neck
joiin fret override for fixed calculation. Now can activate
fingerboard from Soundhole dialog. Made corrections to figure
titles.
December 5, 2018 Program version 1.3.1. Added
more
input
validation
and
error checking to Soundhole dialog.
January 10, 2019 Program version 1.4.0 Added graphic help
for coefficients..