Home  Audio Home Page 

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


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 angle-induced 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.

d2 = r2 + l2 – 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.

d2 = r2 + l2 – 2rl·sinφ
where φ = 90 - θ

Now the formula can be rearranged to solve for the angle of interest.

φ = arcsin
r2 + l2 – d2

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 (φoffset) only a lesser error angle called tracking error with equal magnitude minimums and maximums remains.

φerr = arcsin
r2 + l2 – d2
 – φoffset

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.

distortion   φerr

Therefore minimizing φerr/r became the new criterion for determining the correct tonearm setup.

distortion   φerr
r2 + l2 – d2
 – φoffset


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.


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 pivot-to-spindle distance d.  It is often given in place of pivot-to-spindle distance along with effect arm length in turntable specifications.

 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.


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 |φerr/r| graph, download here.
The old version 1.0.0 of the program without the |φerr/r| graph is still available here.


Terms of use

I retain copyright.  You man freely use the program for personal, non-commercial, 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.


This protractor is set for a playing radius of 58-146.3mm.  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

Final remarks

Updates may be made later, such as the detailed calculations used in the calculator and other things.


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.

1J. K. Stevenson, "Pickup Arm Design," Wireless World, May and June 1966.
2Noel Keywood, "Arm Geometry," updated from Hi-Fi Answers, October 1979, Hi-Fi World,

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 |φerr/r| graph to calculator to give feedback that calculations are correct.