Copyright © 2015 by Wayne Stegall
Updated November 14, 2015. See Document History at end for
details.
Turntable Geometry
Theory,
calculator,
and
protractor
useful
to
setup
your
turntable's
tonearm
Introduction
Vinyl records you play are cut on a machine where the cutter is always
lined up with direction of the playing groove. Ideally a record
should be played back the same way. However, the great majority
of turntables play with a pivoted tonearm, one where the angle to the
groove changes with the playing radius. Any error of this type
adds distortion that a linear tracking tonearm does not
introduce. Because of this, turntables with pivoting tonearms
must be set up for minimum angleinduced distortion in order to play
correctly.
Some Theory
Figure
1:
Diagram
illustrating
turntable
geometry


Refer to
figure 1
above. Analysis of turntable geometry begins with the concern
that the stylus be inline to the record groove to be played.
However with a pivoting tonearm it is necessary to tolerate some angle
error between the stylus and the groove. Then the task would be
to align the cartridge to minimize this error for the range of radii in
which playable grooves lie. Then high school trigonometry reminds
us of the law of cosines as the first formula relating distances in the
triangle between the stylus point, platter spindle, and the tonearm
pivot to the angle indicated in
figure
1 as
θ.
(1)

d^{2} = r^{2} + l^{2}
2rl·cosθ 
However
θ is only related to
the angle we desire. Substituting the sine of the complementary
angle
φ for the cosine of
θ
gives a related formula more suited to the task.
(2)

d^{2} = r^{2} + l^{2}
2rl·sinφ 
where
φ = 90 
θ
Now the formula can be rearranged to solve for the angle of interest.
(3)

φ
= arcsin

r^{2} + l^{2} d^{2}
2rl 
Then values of l and d are calculated which give a minimum variation in
φ. Usually one value is
fixed by the design of the tonearm and the other is calculated from it
by means not necessary to detail now. Then if the cartridge is
turned toward the spindle by the median value of
φ through its range called the
offset angle (
φ_{offset})
only
a
lesser
error
angle
called
tracking
error
with
equal
magnitude
minimums
and
maximums
remains.
(4)

φ_{err}
= arcsin

r^{2} + l^{2} d^{2}
2rl 
φ_{offset} 
Figure
2:
Typical
plot
of
φ
optimized
for
minimum
angle
range.
(equation
3)


Figure 3: Typical plot of
tracking error angle after stylus has been rotated by offset angle.
(equation 4)


Although it seemed sufficient to minimize tracking error for a time, it
was later determined that distortion was not determined by tracking
error angle alone. It was further determined that distortion for
any given angle error was also inversely proportional to the radius.
(5)

distortion ∝

φ_{err}
radius 
Therefore minimizing
φ_{err}/r
became
the
new
criterion
for
determining
the
correct
tonearm
setup.
(6)

distortion ∝

φ_{err}
radius 
=

arcsin

r^{2} + l^{2}
d^{2}
2rl 
φ_{offset} 
r 
Once the correct geometry variables have been calculated, you see from
figure 5 below that optimal
calculations result in all maximum values
being the same. Then radii at two zero crossings become the main
focus for correct alignment as they represent the only points where the
stylus is perfectly inline with the groove of the record.
Figure
4:
Plot
of
φ/r
optimized
for
minimum
distortion
range
resembles
tracking
error
plot
except
that
zero
crossings
are
closer
to
spindle.
(equation 6) 

Figure 5: Plot of φ/r
for optimized setup shows equal distortion maximums.


Note
Although not necessary for the context of this article, another figure
of merit to optimized turntable geometry is overhang. Overhang is
the difference between the effective arm length
l and the pivottospindle distance
d. It is often given in place
of pivottospindle distance along with effect arm length in turntable
specifications.
(7)

overhang = l  d 
(8) 
d = l  overhang

The use of overhang as a specification suggests aligning a cartridge by
bringing the tonearm to the spindle until the stylus is in line with
both the pivot and spindle and setting overhang as distance from
spindle to stylus along with an offset angle presumed parallel to the
lines of the headshell. Some early turntables specify a setup
similar to this and produce results incorrect compared to more modern
calculations and methods.
Software
I provide here a turntable geometry calculator oriented toward
minimizing
φ/r.
Figure
6:
Turntable
geometry
calculator 

To get the new version 1.1.0 of the program with the φ
_{err}/r
graph,
download
here.
The old version 1.0.0 of the program without the φ
_{err}/r
graph is still available
here.
Operation
 All numbers specified must be in the same units. The
default values given for inner and outer radii are in millimeters for
12 inch LPs. If you want to use inches or want to specify radii
for 45 records, you have to supply other values.
 The calculation of zero radii from inner and outer (minimum and
maximum) radius is required for all calculations, others are optional.
 Choose which aspect of the tonearm adjustment is fixed:
pivot to spindle distance or arm length.
 It is then best to then specify the fixed parameter then press calc.
 Sometimes tonearm specifications specify the two calculated
values instead of the fixed one. My turntable, for example, has a
fixed pivot to spindle distance but specifies the arm length and
overhang instead. In these cases enter the two values you have
press calc. The program
will calculate the fixed value from these and recalculate the given
values.
 As of version 1.1.0, the calculator plots a φ_{err}/r graph with from the calculated
parameters to to give feedback that calculations are correct.
The graph is scaled to fit the graph area. Therefore only the
shape of the graph with equal maximums provides information.
 I have provided to change the precision of calculated number
values which may need to be increased if you change to inches or
centimeters. Simply enter a C language number format
specification in an empty input box and press calc. For example %.4f will specify 4 digits
after the decimal point. Format
specifications using e or g instead of f are allowed but they will
likely produce undesirable results. Enter %.2f to restore the default format.
 Press reset to clear all
calculations. Any change in number
format will remain however.
 Pressing help will open
this web page in your default web browser.
Terms of use
I retain copyright. You
man freely
use the 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 its application.

The program has been verified on Windows 7, on Linux under WINE (
Windows
Emulator), and by providing data
for the plots of
figures 4 and 5
above.
I no longer recommend downloading VC6 run time DLLs from my
website. Instead they can be downloaded from WINE with the
provided utility winetricks.
Protractor
This protractor is set for a playing radius of 58146.3mm. As
such it has zero points at radii of 63.6mm and 119.6mm. The files
were created at 10 dots/mm for precise millimeter drawing and should
print well. For now I recommend downloading the PDF version of
the protractor as Adobe Acrobat is a universial program. On the
downside the PDF version seems to suffer a little from much higher
compression. The JPEG version may print best from within an
appropriate
graphics program.
Notice: From Windows 7 the PDF
file prints to correct dimensions from Adobe Acrobat. The JPEG
version also prints correct from my graphics program but not from
Window Preview. Linux will print the PDF version of the
protractor with Adobe Acrobat 9.5.5. This version of the acrobat
reader is no longer offered by Adobe but is still available on the
internet. Since Mac OS X and Linux share much code, you might try
Acrobat for Mac. I haven't yet. Outside of Adobe Acrobat
you may have to experiment
to get the correct results.
Figure
7:
Turntable
Geometry
protractor
with
Offset
Angle
Guides 

Alternative
PDF
version
Use instructions
 Download file and print file.
 If using the JPEG file, right click on the file, select Save
Image As, and choose a directory, and click save.
 If using the PDF version, simply click the link and follow the
directions to save, or right click on the file, select Save As, and
choose a directory, and click save.
 Print file with print options set to exact size rather than fit
or scale.
 Verify millimeter scale with ruler to ensure correct print
sizing. 1 inch = 25.4mm.
 Glue the printout to a record thickness cardboard substrate if
desired.
 Cut the spindle hole.
 If only using paper, a hole punch punches a smaller hole than
desired but you can still nibble the hole out to the correct size.
 If the protractor is mounted to a thicker substrate requiring
drilling, the official specification for spindle holes is 0.286
+0.001 0.002 inches. Some have said that a 9/32" drill will
create the desired hole. Apparently they expect vibration to ream
a slightly larger than 9/32" hole.
 The cartridge is then aligned so that the stylus is perpendicular
to the radius line from the spindle at the two zero points.
Ordinarily this would be a trial and error process requiring adjustment
of overhang values until the cartridge aligns at both
points.
 However offset angle indicators are provided for each zero
point. These would allow alignment with little trial and error if
the line from the zero point to the tonearm pivot is made to pass the
appropriate angle indicator at the correct offset angle calculated for
the cartridge. It might be useful to draw lines on the protractor
with a straightedge to help the angle alignment. If the alignment
at one zero is done at the correct angle with respect to the pivot the
other zero will be correct as well and need only be verified for
accuracy.
 It is more important that the inner zero radius be aligned
correctly than the outer due to the increase in distortion as the
radius is shortened for any error angle.
Final remarks
Updates may be made later, such as the detailed calculations used in
the calculator and other things.
Sources
Source 1 was for exact mathematics, except that I rederived
the zero point calculations to determine full accuracy for constants
specified only to four digits. Source 2 was used to verify the
correct application of the mathematics.
^{1}J. K. Stevenson, "Pickup Arm
Design," Wireless World, May
and June 1966.
^{2}Noel Keywood, "Arm Geometry," updated
from HiFi Answers, October 1979, HiFi
World,
www.hifiworld.co.uk
Document History
October 26, 2015 Created.
October 26, 2015 Only PDF protractor prints correctly, and only
on Windows, troubleshooting.
October 29, 2015 Added explanation of overhang to the end of the
theory section.
November 14, 2015 Added φ_{err}/r graph to calculator
to give feedback that calculations are correct.