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A Treatise on Cartridge Alignment  Part IPreamble.
When I started listening to records again recently, the mysteries of cartridge alignment still alluded me. Thanks to the many discussions which have taken place on this bulletin board, and to the technical articles sent to me by Klaus Rampleman and Helge Gunderson, I now believe I understand the topic very well, and have been planning an article that treats the subject comprehensively for a couple of months. Many of the regular posters won't find anything particularly new or enlightening in the rest of this text, but it may serve as a useful piece for those who are attempting to set the their first turntable, or it may be of some help to those who never really understand what alignment was all about. Also I have tried to gather most of the useful formulae and data together, so it can be used as a source of reference.
Where possible I included equations as an appendix so as not to weigh the text down with lengthy mathematical formulae. However, I must stress that parts of what follows are still a little heavy, and certainly if I could have included diagrams, some of the more wordy explanations would have been trimmed.
For those who don't like using scientific calculators, I have made an excel worksheet that calculates arm parameters and null radii etc based on the equations below. It is available on request by email.

Definitions.
Effective length (Le): The distance from the centre of the tonearm pivot, to the stylus tip.
Mounting Distance (Lm): The distance from the centre of the tonearm pivot to the platter spindle.
Overhang (D): The distance between the stylus tip and the platter spindle when the tonearm is positioned so that a straight line can be drawn through the stylus tip the spindle and the centre of the tonearm pivot.
Note that D = Le  Lm.
Offset angle (theta): The angle between a projection of the cantilever on the record surface, and a line which passes through the centre of the tonearm pivot and the stylus tip.
Tracking Error (e): The angle (in degrees) between a projection of the stylus cantilever on the record surface, and a tangent to the record groove at the point of contact of the stylus.
Linear offset: The perpendicular distance between a line through a projection of the cantilever on the record surface, and the centre of the tonearm pivot.
Inner Null Radius (N1): The groove radius nearest the centre at which the tracking error is zero
Outer Null radius (N2): The groove radius furthest from the centre at which the tracking error is zero
Outer Groove Radius (R2): The groove radius, at which the modulation on the record starts.
Inner Groove Radius (R1): The groove radius at which the modulation on the record finishes.
Note in the text below, A*B means A multiplied by B; and R^2 means R squared.

Introduction
The whole issue of alignment (from a geometrical point of view) is based on selecting three variables. These are the tonearm effective length, the mounting distance, and the offset angle; these will be referred to as the arm parameters in the rest of the text.
In general the user cannot adjust all three arm parameters (though there are some exceptions such as the Graham arm). The most common type of setup includes a tonearm which has a fixed pivot position and slots in the headshell; this arrangement allows movement of the position and angle of the cartridge. In this case it is not possible to adjust the mounting distance (without making a new mounting board) so only two of the arm parameters are adjustable: effective length, and offset angle.
Another type of setup involves tonearms with adjustable pivot positions (all SME arms). These typically have holes in the headshell (no slots), and in this case the user can adjust the mounting distance, and can also make limited range of adjustments to the offset angle, but cannot change the effective length. It should be noted that with this type of arm, the effective length depends on the cartridge used, as the distance from the stylus tip to the headshell mounting holes on the cartridge vary from one manufacturer to another.
The overhang is just the difference between the effective length and the mounting distance. Therefore in case 1 above, adjusting the effective length by a given amount, or in case 2 above adjusting the mounting distance by a the same amount both give rise to an identical change in the overhang. As a result the parameters often quoted for setting arm alignment are the overhang and the offset angle.

Calculating tracking distortion.
E LÃ¶fgren and H.G. Baerwald both provided a detailed treatise on tracking distortion in their papers on the subject of tonearm alignment in 1938 and 1941 respectively [1] [2]  the later work is in English. The result is rather complex, and an exact solution is only given for the case of a pure sinusoidal signal with a frequency f.
To summarize the salient points of these calculations, for a pure sinusoidal signal with a frequency f, the predominant distortion due to a tracking error (e) is 2nd harmonic (ie at a frequency 2f). This is referred to as tracking distortion. The tracking distortion is proportional to the tracking error (e), and inversely proportional to the groove radius R. The distortion is also proportional to the peak groove velocity (v) (ie the velocity of the stylus moving perpendicular to the record groove, due to the recorded signal), and inversely proportional to the angular velocity of the record (omega).
ie tracking distortion = epsilon = (v * eta)/(R*Omega)(1)
where v = Peak Recorded Groove Velocity (mm/sec)
eta = Tracking Error (***converted to radians***)
R = Groove Radius (mm)
Omega = Angular Velocity of Platter (radians / sec)If the 2nd harmonic distortion is wanted as a percentage of the recorded signal, and for the case of an LP record which rotates at a rate of 33.33 RPM; and if the tracking error is given in degrees, then the above equation becomes.
Tracking distortion (percent)= 0.5 * (v*e)/R(2)
Hence it can be seen, from equation (2), that for a tracking error of 1.75 degrees (which is a typical for the edge of the record), at a groove radius of 145mm, and a recorded groove velocity of 100 mm/sec, will give rise to a 2nd harmonic distortion signal with an amplitude of 0.6% of the recorded signal. This distortion will decrease to 0.3% for a recorded signal of 50mm/sec etc. The tracking distortion can be calculated for any groove radius if the tracking error is known. The tracking error can be expressed in terms of the arm parameters; however the equation is rather long, so I have included it in appendix (1).
The above calculations do not take into account the effect of RIAA equalization on 2nd harmonic distortion. Since the RIAA equalization in a phono stage attenuates with increasing frequency, the distortion at the output of the phono stage will be less than the values given above. Using a linear approximation for the RIAA curve, the attenuation is approximately 3.87 dB per octave [3], and this effect will reduce the distortion fed to the speakers to 64% of the values given above.

Selecting the null radii.
For a given set of tonearm parameters, provided these are chosen with a little care (see appendix 2), there will be two groove radii at which the tracking error is zero. These radii are called the null radii. Since the tracking error is zero at these radii, the tracking distortion is also zero (though there may be distortion from other effects). Provided two null radii exist (appendix 2), a graph of tracking distortion versus groove radius has curve which is shaped something like a distorted W. That is the distortion begins with a high value at large groove radii, drops to zero at the outer null radius, rises to a peak between the two null radii, drops to zero again at the inner null radius, then rises again as the groove radius decreases.
The object of aligning a cartridge is to choose the null radii so that the distortion between the inner and outer radii is minimized. But before this can be done, the inner and outer groove radii must be selected, and the method of minimization must also be selected. Unfortunately there is no universal standard for inner and outer groove radii: the IEC standard specifies an inner radius of 60.325mm, and an outer radius of 146.05mm; the DIN standard specifies an inner radii of 57.5mm and an outer radius of 146.05; and audiophile records tend to have larger average inner groove radii than mass market records. So the weakest point in the cartridge alignment procedure, is the arbitrary choice of the groove radii between which the tracking distortion is to be minimized.
A further complication arises from the method of minimizing the distortion. LÃ¶fgren [1] and Baerwald [2], both proposed a system based on Tchebichef's method. The Tchebichef method has its most common application in RF filter design and is a system for minimizing the difference between the actual filter response and the desired response. The method involves defining some error function, and selecting parameters so that the error function oscillates between its maximum and minimum value as many times as possible over the frequency range selected. For the case of tracking distortion, Tchebichef's method selects arm parameters so that the distortion reaches the same maximum value at three groove radii. The distortion maxima occur at the inner groove radius, the outer groove radius and at a point between the null radii. The radius of peak distortion between the null radii is given below
R Max Distortion = 2*(N1*N2)/(N1+N2)(3)
The above is a rather mathematical way of putting it. To use more simple terminology, the Tchebichef method, applied to tonearm alignment, minimizes the maximum distortion between the inner groove radius and outer groove radius, and adjusts the shape of the distortion curve so that the three maxima have the same value. In the rest of this text this method is referred to as the "Peak Distortion Equivalence" method.
Another system is to calculate the RMS distortion between the inner groove radius, and the outer groove radius, and to minimize the result. This method was proposed as an alternative by LÃ¶fgren [1]. The benefit of this method is that it leads to a lower distortion between the null radii, but at the expense of about two times greater distortion at the inner radius. In my opinion the object of HiFi reproduction, is to achieve optimum playback 100% of the time, so this method is unsuitable for cartridge alignment. Often people judge the performance of their systems by the number of times it fails to perform well, minimizing RMS distortion, will guarantee higher peak distortion, even if the duration of the higher distortion is shorter.
A further method was proposed by Stevenson [4]. Stevenson pointed out that other distortions (such as tracing distortion caused by the difference between the stylus shape and the shape of the cutter tip) increase as the groove radius decrease. He argued that aligning a cartridge so that the three peaks in the tracking distortion are equal (at the inner groove radius, the outer groove radius and between the null radii), will actually result in a larger distortion at the centre of the record than at the edge of the record, after taking into consideration distortions due to other phenomena.
This is a valid point, however Stevenson did not derive new null radii based on this fact. Instead he proposed, as a compromise, that the cartridge should be aligned so that the inner null radius occurs at the inner groove radius, and that the distortion at the outer groove radius should be equal to the distortion between the null radii. Using Stevenson's method, the distortion curve has only two peaks between the inner and outer groove radii,  as opposed to three peaks for the "Peak Distortion Equivalence" method or the "Minimum RMS distortion" method.

Formulation of the equations
The equations in the text below give the arm parameters etc in terms of the null radii. This represents a departure from the standard approaches [1] [2] [4] which give equations for the overhang and offset in terms of the inner and outer groove radii. The are two reasons for this new formulation: firstly the equations below are more simple, and even those with a complete aversion to using scientific calculators can calculate the arm parameters for their own setup.
Secondly I think it is more logical to use the null radii as the variables in the equations. Expressing the tonearm parameters in terms of the inner and outer groove radii, is a bad idea, as those equations depend on the method used for minimizing distortion. It is preferable to give the arm parameters in terms of the null radii, as these relationships are based on geometry alone. The null radii can be calculated from the inner and outer groove radii after the method of minimizing distortion has been selected  see below.
A third point to note is that, when Baerwald and LÃ¶fgren wrote their papers on the subject of tracking distortion, turntables came with tonearms attached, and without the opportunity of adjusting any of the tonearm parameters. Therefore, at that time it was logical to define the overhang and offset angle in terms of the groove radii, as these parameters were set in the factory. At that time, the null radii had no practical significance. In fact Baerwald gave equations for the null radii only as an incidental point in his lengthy paper.
The idea of interchangeable cartridges, and user set alignment, came more recently, and with that the need for some kind of alignment tool. The most practical method for setting aligned is to use the null radii. For the user, setting overhang directly is difficult as this requires measurement of effective length and mounting distance, and setting the offset angle directly is almost impossible. Today almost all alignment tools are based on setting alignment at one or both null radii.
For the above reasons, it seems more logical give the equations in terms of the null radii.

Equations for the null radii.
Peak Distortion Equivalence
Using the "Peak Distortion Equivalence" method of minimizing tracking distortion, the equations which give the null radii in terms of the inner and outer radii, are given below.
2/N1 = [1+sqrt(0.5)]/R1 + [1sqrt(0.5)]/R2(4)
2/N2 = [1sqrt(0.5)]/R1 + [1+sqrt(0.5)]/R2(5)For the IEC standard (ie inner groove radius of 60.325, and outer groove radius of 146.05mm), these equations give null radii of 66.00mm and 120.89 mm.
For the DIN standard, the null radii are 63.10mm and 119.17mm.
It is a fortunate coincidence, that equations (4) and (5) are not a function of the arm parameters themselves. This means that the same null radii apply regardless of the tonearm parameters, and hence a tonearm can be aligned without knowing the arm parameters.
Equations (4) and (5) can be rearranged to give the inner and outer playing radii in terms of the null radii.
2/R1 = [1+sqrt(2)]/N1 + [1sqrt(2)]/N2(6)
2/R2 = [1sqrt(2)]/N1 + [1+sqrt(2)]/N2(7)Minimum RMS Distortion
Using LÃ¶fgren's alternative method of minimizing the RMS distortion, the null radii are a little different. In fact LÃ¶fgren did not provide a complete solution the "Minimum RMS Distortion" method. He gave the optimum overhang for a particular linear offset in terms of arm length and inner and outer groove radii. However he didn't calculate the optimum linear offset, and didn't provide equations for the null radii. Graeme Dennes [5] (to the best of my knowledge) was the first person to calculate the null radii that result from this method. He calculated the null radii that result for a linear offset of 93.445mm  ie the same linear offset as that for an arm aligned using the "Peak distortion Equivalence Method" for the IEC standard inner and outer groove radii. The results are 70.29mm and 116.60mm.
I have supplied equations for the null radii using LÃ¶fgren's "Minimum RMS distortion" method as an appendix (3). These equations are given in terms of the groove radii and the linear offset.
A numerical solution for the null radii, independent of mounting distance, can also be obtained for LÃ¶fgrens "Minimum RMS Distortion" method using proprietary software such as Microsoft Excel. This method has the advantage of being independent of linear offset, but does not give equations for the null radii, just the values for a particular arm effective length, or arm mounting distance.
A latterday expert on the subject of tonearm alignment by the name of John Elison has drawn up an EXCEL spreadsheet to do just that. This spreadsheet can be downloaded from the internet at [6]. Using an EXCEL spreadsheet based on his method, applied to the IEC standard groove radii, the null radii which give minimum RMS distortion for typical tonearm dimensions are 70.15mm and 116.23mm.
Remarkably, for the IEC inner and outer groove radii (60.325mm and 146.05), Graeme Dennes result (which uses a mounting distance of 93.445mm) and John Elison's numerical calculation (which is independent of linear offset) are almost the same. Whether LÃ¶fgren realized it or not at the time, his method of providing null radii for "Minimum RMS Distortion" using the same linear offset as "Peak Distortion Equivalence" appears to be valid. In other words the same linear offset will give correct alignment (almost) for either LÃ¶fgrens "Minimum RMS Distortion" method, or LÃ¶fgren and Baerwald's "Peak Distortion Equivalence Method".
The significance of the above point is that the user can switch from one alignment to the other relatively easily [7]. And this fact has been used by cartridge alignment tool designer Wally Malewicz, whose arc type protractor can be used to set alignment for "Peak Distortion Equivalence" or "Minimum RMS Distortion".
For the DIN standard the "Minimum RMS Distortion gives null radii of 67.41mm and 114.86mm using LÃ¶fgren's calculations. Using EXCEL numerical calculation, the null radii are 67.29mm and 114.53mm.
Stevenson's Method
Using Stevenson's Method, the inner null radii occurs at the same radius as the inner groove radius  equation (8), and the outer null radius is given by equation (9) below.
N1 = R1(8)
[1+sqrt(0.5)]/N2 = [1sqrt(0.5)]/R1 + sqrt(2)/R2(9)For the IEC standard groove radii, Stevenson's alignment method gives null radii of 60.325 and 117.42mm. For the DIN standard groove radii, Stevenson's alignment method gives null radii of 57.5 and 115.53mm.

Calculating the arm parameters
Once the null radii have been selected, and one tonearm parameter is known, it is possible to calculate the other two parameters. The following equations give the tonearm effective length, and the offset angle in terms of the null radii and the mounting distance.
Le^2 = Lm^2 + N1*N2(10)
Sin(theta) = (N1+N2)/(2*Le)(11)
Equations for arm parameters are often presented in a different form, however I have selected the above form as it is the most simple  see section entitled "Formulation of the Equations".
If the known arm parameter is not the mounting distance (for example most SME arms have a fixed effective length, and an adjustable mounting distance) then it is straight forward to rearrange equations (10) and (11) to give the required result.
Similarly equation (10) can be used to give an expression for overhang, D, using the fact that D = LeLm
D = Lesqrt(Le^2N1*N2)(12)

Significance of linear offset
Another useful geometric property of tonearm geometry is called the linear offset. The linear offset can be visualised by drawing a line along a projection of the cantilever on the record surface. If this line is extended back past the arm pivot, then the linear offset will be the perpendicular distance from that line to the arm pivot.
The linear offset may seem to be a rather irrelevant arm parameter. It is not an independent variable, being equal to the effective length times the sine of the offset angle. But the explanation which follows will show that in many ways the linear offset is the key arm parameter.
The linear offset is given by equation (13) below
Linear Offset = Le*Sin(theta)(13)
Using equation (11) above, it can be seen that
Linear Offset = (N1+N2)/2(14)
That is, the linear offset is equal to the average of the two null radii.
The most common type of tonearm design has slots in the headshell, and when the cartridge is aligned according to the designers intentions, the cartridge is squared up in the headshell. Moreover if a line is drawn through a projection of the cantilever on the record surface, this line is will be parallel to the headshell slots. In this case, sliding the cartridge backwards and forwards in the headshell slots, so that it moves along the above line, does not change the linear offset of the arm.
The importance of this point is that that neither mounting distance nor the effective length are such a critical values for setting up a tonearm correctly. For example, with tonearms which have headshell slots, the effective length is adjusted to match the actual mounting distance according to equation (10) above, and the linear offset is set according to the manufacturers design when the cartridge is squared up in the headshell. Changing the mounting distance does not effect the angle of the cartridge in the headshell that is needed for correct alignment. The only condition for mounting distance in this case, is that it should be selected so that there is sufficient room in the headshell to set the correct effective for a range of different cartridges. SME arms have neither a fixed effective length or a fixed mounting distance: the effective length depends on the distance between the stylus tip and the headshell mounting holes in the cartridge being used. For SME tonearms the mounting distance is adjusted to match the actual effective length in accordance with equation (10). The linear offset is constant, provide the cartridge is mounted so that it is squared up in the headshell.
In cases where the manufacturers recommended linear offset does not conform to one of the standard systems of alignment above (for example 93.445 for the IEC standard), the user may choose to ignore the recommendations and in this case the cartridge will be skewed in the headshell no matter what mounting distance or effective length is chosen.
Of course, all tonearms do have a recommended effective length, and this length determines the arm effective mass, and the calibration of the tracking force dial. If the manufacturers recommended effective length is not used, this will throw the calibrated tracking force dial off slightly. However, a typical arm length is 240mm, so provided the user keeps to within a few millimeters of the recommendations this effect will be in the order of 1%, which is hardly measurable and certainly not significant.

Manufacturers intended null radii  to ignore or not to ignore.
If the arm parameters are given, it is possible to calculate the null radii, by using the above equations in reverse, the results are
N1 = Le * Sin(theta)  Sqrt{[Le * Sin(theta)]^2[Le^2  Lm^2]}(15)
N2 = Le * Sin(theta) + Sqrt{[Le * Sin(theta)]^2[Le^2  Lm^2]}(16)By substituting the null radii given by equations (15) and (16) into equations (6) and (7), the corresponding groove radii that give "Peak Distortion Equivalence" in accordance with the manufacturers recommendations can be calculated. It should be noted here, that the results of the above equations and calculations are often rather strange. Some manufacturers (like SME and Graham) adhere rigorously to one of the conventional systems, but frequently the manufacturers intended null radii seem to have been selected arbitrarily. The corresponding inner and outer groove radii that give "Peak Distortion Equivalence" have no meaning.
For example many tonearm designs place the inner null radius near 60mm, and the outer null radius around 110mm, but this gives peak distortion equivalence at 54.8mm and 132.9mm, and these numbers have no apparent relationship with the dimensions of a LP record. Null radii of 60mm and 110mm are close to the results of applying Stevenson's method to the IEC groove radii, though not exactly correct.
In fact Stevenson's method is more common than people realize. When the accompanying literature with a tonearm specifies tracking error as being from minus one angle to plus another angle, it often refers to the case when the tonearm is aligned so that the tracking error at the inner groove radius is zero (Stevenson's method). The following is a list of tonearms with recommended null radii at or near the IEC inner groove radius (60.325mm): Audio Technica AT1009; Audio Technica AT1010; Dynavector DV 505; Hadcock Super Unilift MKIII; Infinity Black Widow GF; Keith Monks M9BA Mk3; Series 20 PA1000. In addition the Rega tonearms (perhaps), and many integrated turntables from Japan appear to conform to this system. Perhaps the popularity of Stevenson's method relates to the predominance of British engineering in turntables and tonearms.
Regardless of the above, the user can ignore the manufacturers recommendations, and align in accordance with whatever system they choose. However if the manufacturers intended null radii are not selected, then the linear offset may differ from that intended by the manufacturer. Remembering that linear offset is the average of the two null radii. This will mean that the cartridge may be slightly skewed in the headshell, and this will give rise to two slight effects.
The first effect is that the antiskating force may vary slightly from the settings on the tomnearm scale  see below.
The second effect relates to alignment of the vertical bearings. Modern tonearms are generally designed so that the vertical bearing axis is perpendicular to a line through a projection of the cantilever on the record surface. The significance of this point is that variations in record thickness, or up down movement of the stylus due to warp, will not change the azimuth alignment of the cartridge. If the user sets the cartridge so that the linear offset is different to that intended by the manufacturer, then variations in the height of the stylus on the record surface will have a slight effect on azimuth. However this effect should not be overestimated. It is very small indeed. Moreover it should be remembered that older arm designs (SME 3009 SII) and unipivot arms, do not have aligned bearings to begin with. The SME vertical axis is approximately 30 degrees displaced from perpendicular. Even with such grossly misaligned bearings, the effect on azimuth due to a change in stylus height of 1mm is less than 0.25 degrees. While this effect may be worthy of consideration when choosing between tonearms using aligned or misaligned bearings, the effect on azimuth of an alignment error of a few degrees (which would be typical of selecting an alternative linear offset to the recommended value) is absolutely negligible.

Continued Below
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Topic  A Treatise on Cartridge Alignment  Part I  bkearns 04:42:42 04/20/01 (0)