The Guru Protractor Analysed
Ok, let’s take a look at this rather unconventional design.
The first step is to align the cartridge to Lofgren A null points, also known as the classic Baerwald alignment.
I'll assume here that the user is rotating the protractor during the first stage.
The angle formed by the two lines is equal to 15° - with a tonearm mounted at 222 mm that angle should be 14.22°, with a tonearm mounted at 215mm that angle should be 14.7° (both results come from the rega baerwald and technics baerwald arc protractors drawing). To obtain the indicated 15° your tonearm would need to be very short (the length could be calculated but it’s not important), so one must presume that the protractor is meant to be rotated during alignment.
Let’s assume you’re trying to set up a cartridge on a tonearm mounted at 220mm from the spindle center. You align it perfectly (with only one pair of eyes) during the first step. Your overhang will be in that case equal to 17.442 mm (the effective length will be 237.442 mm) and the offset angle will be 23.175° from Lofgren A / Baerwald equations.
During the second step, the protractor requires the cartridge to be moved back by 1.27 mm. I’m assuming here that the angle formed by the cartridge body on the headshell is conserved: you’re moving your cartridge back in the headshell, without giving it a twist.
First consequence: your offset angle is no longer equal to 23.175°. The offset angle is formed by the line passing by the stylus point and the pivot of the tonearm and the line passing through the cantilever. The line passing through the cantilever is just the same, not the line passing through the stylus point and the pivot of the tonearm.
A little drawing will help here:
Let’s add another line of this drawing and resume what we already know:
The AO distance is known and equal to the effective length: 237.44 mm
The original offset angle is known and equal to 23.175°
AC is then 237.44*cos(23.175°) = 218.28mm
OC is then 237.44*sin(23.175°) = 93.44mm (the linear offset)
BC = AC-1.27mm = 217.01 mm
OB (the new effective length forced by the move back of the cartridge) is (217.01²-93.44²)^0.5 = 236.275mm
And the new offset angle is arcos(217.01/236.275)=23.296°
What are the null points you’re finally obtaining?
Well, the mounting distance is constant (220 mm), the offset angle is known, the effective length too. Let’s take a look at the Graeme Dennes paper to see if there is a formula that could give us the null points corresponding to the data…
Equation (4) page 96 of the PDF file will help :
NP = eff. length * sin (off. angle) +/- ((eff. length*sin(off. angle))² – overhang*(2*eff.L-overhang))**0.5
A few millimeters this way (237.44-236.275= 1.167mm), a few ° this other way (23.296-23.175=0.121°) …
What about having some curves to compare the “guru” method with the classical one?
Well, may be the way to measure distortion according to the guru is not of our world but, from a classical point of view, the Baerwald (Lofgren A) method looks more efficient. Now, when dealing with a guru, it’s never a matter of “science”, it’s only a matter of faith.
Let’s generalize now
This last table is a simple generalization of the calculus presented here for various mounting distances. It shows that there are no universal guru null points since the null points you obtain depend on the mounting distance. You’ll note that the longer the mounting distance is, the worse it is.
You’re having the right to be convinced by the guru method. As I indicated before, this is a matter of faith.