Time to choose: Baerwald, Lofgren, Stevenson ?

I'm generally in the same camp as Lewm on this one: I align the cartridges based on the alignment the tonearm was designed for. Since most of my tonearms use removable head shells and were designed for a 52mm distance from the head shell connection to the stylus, this has the added benefit that the cartridge needs minimal if any realigning when put into another tonearm.

Thom,

If the cartridge is aligned as designed by manufacturer, then the cartridge is straight in the headshell. Now move it another tonearm and align it according to that tonearm’s design: it will again be straight in the headshell. To the extent that the tonearms are designed to have different offset angles, those are manifest in the arm tube. So the only thing that could change then is how far in the headshell the cartridge is. But if both tonearms are designed to have the stylus 52mm from the tonearm connection, that doesn’t change either.

If you have difficulty visualizing this, then visualize moving an SPU from one tonearm to another. This is just replicating that with a normal cartridge & headshell.

Oh no, that's not at all necessarily the case. As others have pointed out, there's often a slight misalignment of the cantilever within the cartridge itself. Proper cartridge alignment aligns the cantilever to the record, and not the cartridge to the headshell. 

As, lewm already indicated, that doesn't change the conclusion at all. If you align by the cantilever, then the cantilever (rather than the body) will be aligned straight in the headshell and will remain so in any arm if you use the alignment that the tonearm was designed for.

Raul, I think you're missing my point and talking about something different. I’m not talking about Baerwald, Loefgren, or Stevenson, but whatever the alignment was that the tonearm in question was designed for and the resulting position of the stylus in the headshell.

Dear @sampsa55 : """ I align the cartridges based on the alignment the tonearm was designed for. Since most of my tonearms use removable head shells and were designed for a 52mm distance from the head shell connection to the stylus... """

that’s true only if those tonearms has the same effective length.

Ah, no. The effective length of the tonearm has nothing to do with this. You can have tonearms of very different length that are still designed to have the same 52mm distance from stylus to tonearm connection or tonearms with the exact same length that are designed for a different distance (e.g., 50mm) from stylus to tonearm connection.

Again, think about the SPU: it can be used in short and long tonearms. 

To illustrate, here are snippets from the manuals for the original tonearm of Yamaha GT2000, Ikeda IT407 tonearm, and Denon 308/309 tonearms on how to align a cartridge. They all say that the cartridge has to be straight in the headshell with the stylus 50mm from the tonearm connection. Once a cartridge is aligned in one of these tonearms according to the alignment that the tonearm was designed for, the headshell can be moved to any of these other tonearms and the cartridge will automatically be aligned as the designer of that tonearm intended.
https://imgur.com/a/Xtdg3

Not all tonearms use this 50mm distance. Some other examples:
- Sony PUA1600L: 52.5mm
- Sony PUA9: 49mm
- Technics (all of them): 52mm

Lewm & Fleib:

The RS-A1 is a bit different as the cartridge is able to rotate and (possibly) maintain tangency to the groove. In Viv Lab and the Yamaha YSA-2, the cartridge is held straight and thus the cartridge will at the beginning and end of the record be at a ~10 degree angle to the groove.

Some thoughts:

1) The distortion resulting from the stylus being at an horizontal angle to the groove is 2nd harmonic, which I believe is the least harmful distortion.

2) It is not clear to me how different stylus shapes affect the distortion resulting from being at angle to the groove.

3) The calculations only cover horizontal alignment and it’s not obvious to me what the resulting distortions in the vertical part of the stereo signal are and whether these are dependent on the frequency (highly likely) and stylus shape (also highly likely). In particular, it is not clear whether these are 2nd harmonic or something else.

4) This horizontal alignment is the only kind of distortion for which I have seen a calculated estimate. Does anyone know of estimates for the distortion resulting from skating forces or vertical misalignment and whether these are 2nd harmonic or something else? I recall reading a discussion of the design of the Viv Lab where the designers seem to have been concerned about skating forces and considered the related problems worse than those related to horizontal alignment.

5) I’ve only seen one graph claiming to show the actual measured distortion from playing a test record. I came across this on the web and do not know if it is authentic. It was claimed to be from a Yamaha test of their YSA-1 (~Stevenson) and YSA-2 (straight) tonearms. Does anyone know of actual measurements of the distortions?

You say alignment error causes 2nd harmonic distortion. Is this documented?

It's in Baerwald's original derivation from 1941, where he shows the resulting distortion to be primarily 2nd harmonic.

I wasn't referring to a test, but to Baerwald's math models where he derives the distortion resulting from the stylus being at at angle to the groove and the results show the 2nd harmonic dominates. I don't have Loefgren's original paper, so I don't know if the result is even older. (Stevenson wrote his in 1966.) I'm sure it has been tested too.

Found the Löfgren article from 1938 (and a more recent translation from German to English).

One key assumption Löfgren made in calculating the distortion is that the lateral stylus tip velocity is 10cm/s. It seems others followed this assumption and for instance the Ellison Excel sheet is based on it. This will scale all the calculations and hence is key in understanding the magnitude of the problem. This assumption was made in the time of 78s without RIAA correction, so I’m wondering about the extent that it’s still valid.

Here’s the quote in Löfgren (p. 355):

Die Schnellenamplitude überschreitet wohl selten 10 cm/s (11), entsprechend einer Lichtbandbreite (12) von etwa 25 mm.

The citations (11) and (12) are:

(11) H. J. von BRAUNMüHL und W. WEBER, Einführung in die angewandte Akustik. Leipzig 1936, S. 106.

(12) G. BUCHMANN und E. MEYER, Eine neue optiselle Meßmethode für Grammophonplatten. Elektr. Kachr.-Techn. 7 (1930), H. 4, S. 147—152.

Just came across Gilson’s 1981 article in Wireless World that is quite interesting. I’ll just leave these quotes here without any comment (p. 61):

It shows simply that the lowest possible tracking angle errors can be achieved only at the cost of increasing the values of F and t; and conversely forces F and t can only be reduced by accepting increased angular errors. In the absence of published information on the audible effects of the opposing factors, the optimum balance is anybody’s guess, but it is hard to see justification for the assumption that the lowest possible angular error must necessarily be the best condition.

In thinking about these problems, it is necessary to keep a sense of proportion; tracking error is only one source of distortion and possibly a minor one. Probably the worst source is tracing error, which can easily run into double figures percentage at the inner grooves, particularly with slight stylus wear. Then there is vertical tracking angle error, which is difficult to avoid. Another source is that due to any longitudinal compliance in the stylus/armature system; it is usual to mount the cantilever in an elastomeric grommet or block, and this is not adapted to providing much rigidity in the longitudinal direction.

Lewm:

I've been looking for that RS-A1 white paper. Do you have a link to it?

Lewm:
I recall seeing that white paper online, but have not been able to find it again. If you have a PDF, I'd appreciate it.

The RS-A1 unipivot it supposed to be positioned so that the cantilever and tonearm are parallel or ideally on the same straight line. Since not all cartridges have the same angle for the cantilever (I think they range from ~15 to ~22), this will be an approximation. The limited design info I've found doesn't mention torsion, but speaks of "downforce":
http://www.eifl.co.jp/index/export/rs-a1.html

When we usually talk about skating force, we assume that the cantilever is tangent to the groove and pulled along this tangent (only true at those null points naturally). Since the resulting force is at an angle to the line going from the stylus to the pivot, a sideways force results. If the cantilever is not tangent to the groove, the friction force would still be tangent to the groove and thus straight arms would also have a sideways force proportional to the angle between the stylus and the groove tangent.

I've seen reviews of the Viv Lab in The Ear, Stereo Times, and Audio Beat. All reviewed the 7" version and none of them complained about distortion. Audio Beat said they also had the 9" version, but just mentioned that the 7" and 9" sound different "and not in the ways you might predict". Also Audio Beat mentions that the order was the opposite you mentioned: The designer started with the very long version to reduce tracking distortion and then realized that the shorter ones sound better.

Fleib:
I'd appreciate links to those comparisons of Viv Lab versions too.

In defense of the arc type protractor, the converse of your criticism of it is that if the P2S distance is not precisely correct, the arc type will tell you that, and you can fix the problem.

If the tonearm is 1mm too far and you mount the cartridge 1mm forward so that the overhang is correct, can you really tell the difference between an arc with radius of 270mm vs. 271mm? The difference is going to be tiny. Notice that this is different from having the tonearm mounted right and having the overhang off by 1mm.

The good news is that the effect on the tracking distortion is also very tiny, shifting the null points by less than 1mm if the tonearm is 1mm too far or too close but overhang is right:
http://www.vinylengine.com/tonearm_alignment_calculator_pro.php?arm1=Arm+1&l1=ps&a1lv=271&am...