# Arc Protractors

**Seb**

#### Let’s think again about arc protractors

You may be a happy user of arc protractors like the wallytractor, or one of mine you can download here. They are very simple to use: make sure that your stylus is following the arc (engraved on a mirror as on a wallytractor, or simply drawn on a piece of paper like stephlouv’s or mine) and then, go to the grid to give the cartridge the correct angle for the cartridge body to be parallel to the lines of the grid (paper protractor – with Wally’s one, you’re aligning the cantilever – the advantage of a mirror…)

Now, this simplicity has a price: an arc is designed for a given mounting distance and you’re not supposed to use an arc protractor designed for a given mounting distance with another mounting distance.

I drew my first arc protractors in 2000 or 2001 and on every arc I’ve designed, I always insist you respect the given mounting distance. That is why I am only proposing arc protractors for certain turntables with a known mounting distance (Thorens, Rega and Technics). My drawings are directly inspired by the wallytractor and at the time I started to draw this kind of tool, I read very carefully the recommendations Wally Malewicz gave and considered them engraved in stone: the real mounting distance must be precisely measured – an error of +/- 0.5mm is a maximum.

I said to myself: Wally certainly thought more than I did and never decided myself to see what is behind this very strict recommendation.

And then along came Stephlouv with lots of questions about the limits of arc protractors especially regarding the use of arc protractors when the mounting distance is not the one stated on the paper.

So, since the questions were good, I said to myself, “Come on, use your brain and think more about the solutions you’ve been proposing for years!”

In fact, I think now that I should have thought about this kind of protractor a long time ago… shame on me.

#### The basics

The idea is to calculate the alignment parameters (mounting distance, effective length, angular offset, null points) when using a tonearm with a given mounting distance different to the one stated by the tool.

#### First step: let’s draw an arc protractor

In fact, we’ll do more than draw an arc protractor. On a preliminary version of this paper, I tried to calculate the consequences using measurements made on a draw (and Illustrator showed its limits... I’m afraid the circle made by Illustrator was not exactly the circle I wanted; I needed 0,001 mm precision). Well, since things weren’t that clear I decided to make them clear with a little calculus.

Not very complicated... I’m only using tools I learnt when I was 18.

So, we must be able to express all the elements we’ll need in an X, Y plane.

The tonearm I had in mind to draw the tool has a 235mm effective length. The guy who mounted the tonearm did a very good job since he mounted the tonearm at the exact distance recommended by Baerwald calculus: that is at 217,36237... mm from the spindle center (the guy was helped by 12 guys from the NASA).

Let’s put the spindle center at a known dot: (0,200)

we’ll use the 66mm Baerwald inner null point: its position is (0,134)

To represent the mounting center, we need to draw two circles: the first one centered on the spindle center has a diameter equal to 2*the mounting distance, the other one, centered at the null point has a diameter equal to 2*235mm. At the intersection of the two circles (the one on the right) is the mounting center.

Here we have a system of two circle equations to solve: the solution is (215.623,227.447). Centered on this point, let’s draw another circle with a diameter equal to 2*235mm: here we have the stylus path.

This first drawing will show you where we are:

What are those crosses?

In fact, when you have one of my arc protractors in hand, read the explanations and you’ll see that, to help you to be sure that you’re following the arc, I put two crosses: one very close to the spindle center and one close to the beginning of the LP. Let’s give values to the dots at the center of the crosses:

For the A cross, X is -15, Y is calculated using the equation of the circle centered on the mounting center, that is Y=182,300. Same method for the B cross: X=45 (fixed), Y=65.852 (calculated).

#### Let’s use this arc protractor badly...

What will happen if you’re using this protractor with a mounting distance different to the one specified on the paper?

In fact, since the user has read the instructions, he’ll try to follow the arc by moving the cartridge in the headshell to be precisely on A and B cross without moving the paper on the platter. Be sure he’ll succeed!

Err.. not exactly because those two points will be the only two points his real arc will have in common with the one I drew.

Suppose the actual mounting distance is such that, when using the arc protractor, he’ll obtain a say 225 mm effective length. (I could have ordered my calculus centered on the mounting distance and not on the effective length – the conclusions will remain).

Our goal is to draw a 2*225mm circle passing through A and B crosses.

The center of this circle will be on a line perpendicular and passing through the middle of the line passing through A and B cross.

Let’s first have data about that line: I know the coordinates of A and B crosses, so I can calculate the equation of that line. That is: Y=153.817 – 1.9408 X

The coordinates of the dot at the middle of the line can be deduced too: (15 , 124.076). The shape of the perpendicular line is 1/1.9408 = 0.515251 and the distance between A and B is 130.996 (=65.498*2)

Let’s make another drawing to see where we are:

We now need to calculate the position of the mounting center of the tonearm which doesn’t have the good mounting distance (let’s call it N).

We’ll first calculate the distance between M, the middle of the AB line and N: (225²-65.498²)^1/2 = 215.255 mm

Since we now know the slope of the line passing through M and N, we can use it to calculate the position of the N dot.

We need to find the value of dX and then to solve the equation:

dX²+(0.515251dX)²=215.255²

dX=191.349 and dY=98.592

The position of the N dot is now known

X = 15+191,349 = 206,349 and Y = 124.076 + 98.592 = 222.668.

The N dot is the center of the 2*225 mm diameter circle. This circle shows the real path followed by the stylus. The stylus is passing through A and B crosses.

Let draw it (in fact that arc was already there in the other drawings...) and clean up this mess.

Well, the two arcs are there but you cannot see the two arcs... let’s take a 6400% binocular.

We’re somewhere in the middle of the LP : 0,45 mm is the appreciative distance (because this is the distance I measured from my drawing – the exact distance could be easily calculated but it’s not that important) between the right arc (the one you follow if you’re having tonearm mounted at 217,36) and the left one is the arc you follow if your tonearm is mounted at... err... what is the mounting distance of this tonearm we’re forcing to “follow” an arc not designed for him?

Well since we know the coordinates of the spindle center and the coordinates of the real mounting dot (N), we can calculate this mounting distance (Pythagoras again... I swear that when I was 13, I didn’t know it could be that useful). That mounting distance is: 207,59 mm and by using this tool, we’re forcing an effective length of 225mm.

Now let’s calculate the angular offset. Once this offset is known, since we already know two parameters (mounting distance and effective length), we’ll be able to calculate the other parameters.

Let’s go back to the point we once called the null point.

Well, the wrong arc is not passing right in the middle of the cross (in fact here, there is the grid you normally use to give the good angular offset).

We’re making here a hypothesis: the user won’t see that he is not right in the middle of the grid: he’s (a little) after the middle of the grid.

How far?

Since we know the center and the diameter of the circle that define the arc, we can calculate the coordinates of the Z point. Y=34 and then X= -0,444 mm

Remember that the X coordinate of the null point is 0: the horizontal difference between Z and the null point is 0.444mm.

Our fellow user is that happy to be following exactly the arc that he won’t see this 0,444 mm difference. I think this is not a strong hypothesis.

Whatever, our user is now trying to align his cartridge body with the lines on the grid (and we’ll suppose that the cantilever is perfectly parallel to the cartridge body – and yes, antiskating is set to zero).

The offset angle is the angle formed by the line passing through the stylus and the pivot of the tonearm and the projection of the cantilever onto the record’s surface. The projection of the cantilever is in fact in our drawing the horizontal line passing though the middle of the null point (which in fact is not the null point for the tonearm we’re trying to set up). We can now see the offset angle on our drawing and calculate it:

Offset angle = ArcCos ((206,349+0,444)/225) = 23.208°

#### Damn it! Where are my null points?

We can now calculate the null points you’ll finally have. I simply used the formulas given in the John Elison Excel file.

Enter the effective length, the overhang (effective length-mounting distance), the angular offset and you’ll have the null points associated with you data. In our case, the null points are: 70.49 mm and 106.85 mm.

Well, we’re really far from the pair of null points we wanted to reach! (66mm and 120.9mm).

Two little curves to compare the % distortion you’ll obtain with the use of the adapted protractor with a tonearm mounted at 207.59mm and what you’ll obtain if you’re using the arc protractor adapted to your mounting distance :

The null points you’ve forced here are really bad!

Normally, with such a mounting distance, your effective length should be 226mm and your angular offset should be 24.4°. By using an improper protractor, you’re obtaining an effective length of 225mm and an angular offset of 23.2°. This is because between the mounting distance indicated by the protractor and the real one, there is 9.7mm difference. You now know that a 10mm difference will drive you nowhere!

#### How low can you go? (Limbo Rock – Chubby Checker)

The calculus I’m presenting here can be systematized – Excel would do the job for us. This table is proposing to see what could be the maximum error between your mounting distance and the one stated on the arc protractor.

This table is having no practical use since the calculus we’re providing are based on the position of the two A and B crosses: if the position of the crosses (or the position of the points you’re using to see if you’re following the arc) are not exactly the one I indicated here, the values will be different. I just want to show the kind of errors you’ll make if you use badly an arc protractor.

One would say: “well, since the position of the crosses are important, let’s choose a pair that could do things go well”. That would be a waste of time. The crosses should have the position I indicated because the far there are from each other, the more accurate you’ll be in setting the overhang. With an arc protractor adapted to your exact mouting distance, with the kind of crosses I indicated, I can guaranty you a +/- 0,25mm (certainly less) precision in the setting of the overhang. Don’t ask me to draw a tool for people who are not following my recommandation that won’t be that good for the people who are following my recommandation!

Back to the table: On the left part of this table, you see what you’ll obtain with our arc protractor designed for a 217,36mm mounting distance if you’re using it with another mounting distance. I computed the distortion at the second Baerwald null point in the last column (dist at 2nd NP ie 120,89mm).

My point here was to see if it is useful to add (as Stephlouv and I did on our arc protractors) a grid at the second null point as a control. I thought that by adding this second grid, if your actual mounting distance is not the one stated on the protractor, you would be able to see it.

What you normally observe is, at the second null point, the cartridge body should be parallel with the lines. In fact I’m afraid that most of us won’t see the 0.xx° error. This control is in fact having no use.... (sorry guys...).

In the right part of the table are the Baerwald alignment parameters you’ll obtain if you’re using the correct arc protractor. As you can see, very small errors in the overhang (less than one millimetre) associated with very small errors **of same sign** in the angular offset can give you very bad pairs of null points (and then distortion, roughly at the end and at the beginning of the LP if your actual mounting distance is inferior to the one stated on the protractor – and in the middle of the LP if your actual mounting distance is superior). If the errors would have been of different signs, one may think that some kind of “compensation” could occur (but this question will be for another paper...)

So, is there something that may interest you in this paper if you really don’t care about geometry?

YES

If there is something important here, it’s the fact that when you’re aligning a cartridge, make sure you’re doing things properly. Small differences between what should be and your results could lead you away from the audiophile nirvana. Of course, there is always a difference between geometry and real life but, by doing things properly; you could certainly minimize those errors.

So,

Get the good tool,

Take your time

And when it says, “you must follow the arc”, or “the cartridge body must be parallel with the lines”, please, do not consider those “must” as produced by an insane (audiophile) mind.

To me an arc protractor, if adapted to your mounting distance, is a very good tool – I will work on this soon in another paper but I’m convinced that errors on the overhang should be very small when using this kind of drawing. Setting the angular offset could be the difficult (and dangerous) part of the job...

For sure, your exact overhang and angular offset, even if you were helped by some NASA guys when you mounted it at the recommanded value (remember, 217,36mm in this exercice) won’t be 17,64mm and 23,43°. We’re only having a pair of eyes (and that damned piece of paper – print on scale by the way ?) to set up our cartridges. There always be some errors but by using the GOOD tool, I’m convinced that those errors will be minimized.

Three little remarks to conclude:

- 1- About Lofgren arc protractors: we all know that Lofgren null points leads to a higher % tracking distortion at the end of the LP. If now, you’re using a Lofgren arc protractor designed for a mounting distance inferior to your real mounting distance, distortion at the end of the LP would be really high!!!

- 2- About Wally’s recommendation (remember, a +/- 0.5mm error is a maximum). If the error in the mounting distance is within that range, no problem. One could say that a +/- 1mm error should be of no consequence but an error superior to 2mm will be synonymous of troubles...

You’ve been warned (twice!)

- 3- A special note now for my Rega arc protractor users

I’m proposing two Rega Baerwald arc protractors: one for a 219.5 mm mounting distance (optimal mounting distance for a 237 mm effective length tonearm) and one dedicated to Rega TT (the Rega mounting distance is 222 mm).

Make sure you’re using the correct one!

Imagine, your tonearm is mounted at 219.5 mm and you’re using the 222mm one Check this table:

If your tonearm is mounted at 219.5mm, even if you’re a NASA guy, you’ve created some errors that move the inner null point by 1mm and the outer null point by 3.3mm. And since you’re not working for the NASA, add to those errors, the ones you’ll make by using a pair of eyes to align your cartridge.

Many thanks to jas for his reading of my bad English - there are certainly some other faults - just PM me and we'll correct those faults - and don't yell at me! or show me that your French is better than my English ;-)

## Comments

## Phono cartridge offset angle and mounting distance

Thanks for that really interesting article about arc protractors.

As stated in the article, the overhung tonearm with the offset head has two points in its arc from the outer grooves towards the middle when the tracking error will be zero (the two null points).

The process for calculating the correct overhang and offset calculates the overhang first and then calculates the "optimum" offset angle.

The Baerwald method makes the positive deviation from ideality equal to the negative deviation. In other words, it is based on the "range" of the positive and negative deviation. This is one kind of optimum. A different approach is to find the mean deviation to calculate the offset angle - one gets asymmetric deviation (for example +0.81 degrees and -1.61 degrees rather than +/- 1.21 degrees for the Baerwald method), however the cartridge spends more time with a lower tracking error.

I'd like to compare some calculation results with your results:

Turntable spindle to tonearm pivot distance = 205 mm

Calculated tonearm pivot to stylus tip distance = 226 mm

Offset angle = 25.7 degrees

Nulls at 74.4 mm and 121.6 mm from turntable spindle

Greatest positive error = +0.81 degrees

Greatest negative error = -1.61 degrees

The Baerwald values are:

Offset angle = 26.1 degrees

Nulls at 70.5 and 128.4 mm fro spindle

Error = +/- 1.21 degrees

Assumption inner and outer radii of grooves are 63 mm and 145 mm.

On a less mathematical subject, what does misalignment error sound like? I'm not sure if it can be called "distortion" exactly. I imagine it will result in a slight channel volume imbalance.

Your thoughts on these topics would be greatly appreciated.

Best Regards,

Tubewaller

## thanks for your interesting

thanks for your interesting comment.

a few notes :

IMO, analysing a pair of null points only by its consequences of peak tracking error is not a good thing since a given tracking error is not having the same consequence if you are at the beginning of the LP or is you're at the end. To measure the consequences of a given tracking error, you can calculate the % tracking distorsion : |50.165 * tracking error|/groove radius.

The Baerwald null points (in fact Loefgren A null points) are calculated to minimise peak distorsions, ie % tracking distorsion at the beginning of the LP, somewhere in the middle and at the end of the LP. With your given effective length offset angle and mounting distance, your outer and inner most grove, peak tracking error are respectively 1.6° (at 145mm), -0.8° (at 95.1mm) and 1.5° at 63mm but the % tracking distorsion is 0.56% at 145mm, 0.42% at 95.1 mm and 1.2% at 63mm.

with the baerwald null points, peak distorsion is minimized and equal to 0.59% whatever the point you're considering (145mm, 63mm or somewhere between the two null points).

Your idea of calculating a pair of null points based on the mean distorsion is in fact the idea behind the loefgren B solution. the idea is here to minimize overall distorsion across the record length.

What is the good approach ? don't know...

tracking distorsion, if it's not reaching a "certain" level can not be heard, beetween that "certain" level and "another" level, may be you could hear it, may be not - in fact, I think you could live with that level of distorsion... After the "another" level, you'll hear it. The problem is that I don't know the exact levels we're talking about... as a "security", minimizing the peak distorsion is something to consider !

Now, regarding the other question (about your DGG LP with no music after 70 mm from the spindle center - see the VPI jig paper) there is certainly something to do... the problem is to obtain a lot of data to see if there is something to do.

what is distorsion ? something you'll certainly hear with your null points at the end of the LP with a innermost groove of 60 mm (a 2% distorsion is something you should hear).

best regards

Seb