The Principle of the Dennesen Sountractor (and the Acoustical Systems Smartractor, and Dr Feickert Protractor)

Firstly, to avoid confusion, the Dennesen and the others like it are not suitable for setting up the Odyssey arms and SME arms which have unslotted headshells and are adjusted at the base. For this a two point protractor is needed. Also Arc protractors are not suitable as they are designed for arms with fixed mounting distance (pivot to stylus).  However the Feickert has  two null points and can be used without the gauge arm. So the following is just for information.

The principle of the original patented 1981 Dennesen alignment protractor  and its later derivatives is primarily based on a concept that goes back to Percy Wilson in 1924 and before him to Bela Harsanyi in 1907.
Both were attempting to find a method to minimise tracking error in pivoted arms, and came up with a way of doing so by using overhang and offset. The overhang allowed the arc of the stylus to be such that it minimised the variation in error, and the offset allowed the error itself to be minimised for that overhang.

Once the limits of the arc are chosen, there is a constant which follows from this. This constant (which is also related to the radii of the two null points) was named by Wilson as the Linear Offset

For any pivoted arm, the inner and outer recorded radii of a record determine the extent to which the tracking angle varies. As overhang is increased, tracking angle also increases from inner to outer radii. until there is a "best fit" arc which crosses a disc of given recorded radii such that, with overhang and with increasing cartridge offset to compensate for the tracking angle, the arm can cross the record with reducing tracking errors. By angling the cartridge, the errors are minimised, and at two points on the arc the error is zero. These are the nulls. These points on the arc can be refined so that it is distortion rather than tracking angle error that is minimised. This gives rise to the various alignments - Lofgren A (Baerwald), Lofgren B, Stevenson, and others, which minimise distortion in different areas of the record depending on the desired outcome.  Each of these alignments has its version for particular recording standards - IEC, DIN, etc, or can be modified to any recorded radii.

The Dennesen U.S.patent number 4,295,277 cites Baerwald (who follows Lofgren) and uses nulls of 2.6" and 4.76" which are nulls resulting from the equations for IEC inner and outer recorded radii, of 2.375" and 5.75" For the Dennesen, like the Acoustical Systems Uni-protractor, now called the Smartractor, and the Dr Feickert Analogue Protractor(both of which use the Dennesen principle but don't particularly credit him),  the arm mounting distance (pivot to spindle distance, P2S) can be obtained from two dimensions:

First:
The distance to the arm pivot along the axis of the protractor's offset arm (alloy in Dennesen, acrylic in Smartractor) from the point where the null radius crosses it, is determined by sliding it to coincide with the arm pivot point.
Call this X.

Second: 
The distance from the centre of the spindle extended along the null radius to the centre line of the alloy or acrylic protractor arm. 

This distance varies depending on the alignment, and must adjust for each. However, you don't have to measure it if the device is set for a particular alignment (such as  LofgrenA/Baerwald IEC in the Dennesen), as it is fixed and given by:

Outer Null minus Inner Null, then divide by 2, 
Call this Y.

(For LofgrenA.Baerwald IEC this is 27.45. If the device is set for another alignment, then the same calculation applies with the appropriate nulls.) 

This gives a right angled triangle, where the mounting distance (P2S) is given by: 

the square root of: X squared plus Y squared. 

So, if this distance has been set on the protractor by sliding the arm until the pointer is over the arm pivot, then the nulls must be in the correct position for the appropriate effective length relative to the pivot point, wherever it is. See the photo below from the web site of Origin Live:

All that remains is to slide and/or twist the cartridge in its slotted headshell to align with the grid.

This applies to the Dennesen but only for the LofgrenA/Baerwald IEC alignment for which it is set up, unless modified. The Smartractor and Feickert have the facility to set up for other alignments.

This aspect (and the Acoustical Systems UNI-DIN alignment, which is, in effect, just the result of the Lofgren equations for different recorded diameters) is covered in more detail with an illustration of the basic geometry rectangle in another post here, concerning arm design geometry.

The accuracy of the method depends on how well you can position the protractor arm over your estimate of where the tonearm pivot is, and then, as with any protractor, how accurately you set up the stylus/cantilever/cartridge to the grid lines. Consideration of the degree to which small differences affect alignment can be found here, in a post about variations in spaindle size and small errors in alignment

The whole point of the Dennesen principle and its big innovation was to allow correct alignment with an existing (and, more importantly) an unknown mounting distance. If the distance has been set wrongly for a particular arm which requires a particular mounting distance, this would be corrected (if using a slotted headshell) by adjusting the effective length and cartridge offset (assuming enough adjustment) to match the null on the protractor.  

Which begs the question, though, of why anyone using a protractor of this type would ever need to  know the actual mounting distance. A device for measuring this is unnecessary if using the Dennesen, Smartractor/Uni-protractor or Feickert.

The principle of the Dennesen and the Smartractor/ Feickert could be refined to allow even simpler, more universal, and more accurate setup, particularly for SME arms which are adjusted at the base.

The SME Protractor

As I don't own one, I don't have a vested interest in promoting SME or otherwise, but I have been sent a pic of their protractor for the SME V, so I can comment on it.

So let me say that SME owners can be assured that the concept is not at all crazy, nor inaccurate.

Although, like all protractors, it depends on a number of things.
 
The first is that the SME arm is correctly designed and the headshell is as specified at the appropriate angle for a Baerwald IEC alignment at a particular nominal effective length (as given in the SME spec for their various arms)

Hopefully this is so. Then, as the quality of SME manufacture will be top class,  we assume the arm is made to the specification.

The second thing is that the cartridge and therefore the cantilever is set exactly on the headshell axis. For this a depth gauge should be used to align the the cartridge body, if it is cuboid, although the cantilever is a problem, as always (but that should really be an issue for cartridge manufacturers). So it is possible to align the cartridge to within 0.05mm along its length, ie 0.15 degrees or better. This is at least quantifiable, not guesswork.

The third thing is that one has to align the arm tube edge with the outline of the arm tube on the card. This isn't so clear - they should have omitted the outline of the arm entirely, and simply drawn some closely spaced lines at the same angle as the arm outside edges, as the principle is to align the arm outside edge parallel with a line, any line, at that angle when the stylus is on the null point. If a cartridge has a longer (or shorter) cantilever, the angle of the arm, and hence the edges (even though they taper) remains the same but moves sideways.

Using the arm tube as a guide, it is possible to obtain an accuracy far greater than with a cantilever view - perhaps as much as a factor of 10 (assuming generator on the same axis as cartridge and cantilever on same axis as generator.... we have to assume something ...!)

The fourth thing is that the protractor is made  accurately.

So, therefore, the potential for correctly setting up an SME using their protractor is very high. The areas where it is lacking are the same as other protractors, namely the accuracy of manufacture of arm and protractor, but it scores in angle alignment accuracy, as in this area it is potentially more accurate. Also any errors in the protractor will be minimised by the fact that the arm is optimising alignment at the base not at the headshell.

Of course it never hurts to double check using an accurate two point protractor with suitable nulls. But this applies to all protractors.

Arc Protractors

Regarding arcs, there is an argument that says that a second null is pointless for arcs, as differences in mounting distance lead to tiny unobservable variations in the outer null.

However this makes the erroneous assumption that the inner null will be set exactly. It is almost certain this will not be the case, due to observation errors in offset, therefore a second null allows errors in offset to possibly become apparent and allows a mean setting.

It also avoids a false sense of security being generated in the user, as it acknowledges the possibility of an erroneous setting.

While I do not quibble with the ease of use of arcs, I have yet to see any proof of their increased accuracy. The usual argument is that if the new arc shows a difference in alignment  from the old alignment using a two point,  the old alignment was inaccurate (rather than vice versa). Or that the new alignment with an arc sounds better, which simply begs the question of whether the old alignment was not so well set up.

All arc protractors should have two nulls as a simple means of cross checking, as, by definition, a correct alignment should be square to both nulls. If not, then something is wrong with the arc, the mounting distance or the cartridge offset.