Fretboard Geometry

[Nick JD]

Warning: Mathematics content! Anal levels of analysis content! This is not intended to change any techniques or minds, rather to prompt discussion and deeper understanding.

Two things that I'd like to bring up. Firstly, I think neck relief is a band-aid solution for an incorrectly set up (made) neck. Secondly, that while most level and crown techniques improve a set up by miles, they can almost always be done perfectly with one simple tweak.

A Stew Mac article talking about neck geometry here gives a few simple equations to calculate the optimum neck geometry for a perfect neck set up. I'd like to discuss these equations and their outcome with regard to neck set up methods and leveling and crowning techniques.

For a long time I was puzzled by a few of the methods and in general the geometry of the whole neck. Since delving into it I have had a couple of revalations that I'd like to share if you can bear with me.

Here's the first equation (I've linked it off their site because it's got the diagram there too).



This first equation above is not really important, but it's needed to solve the next equation. So we plug the numbers in and solve X.

T = half the scale length = 12.75
St = the width at the 12th fret = 2.05
Sn = width at the nut = 1.67

When we plug those into that equation we get X = 56

Now we have X, we can use it to find what the radius should be at the 21st fret in order to have the strings all the same distance above their frets. What is the optimum conical radius for the lowest possible action without buzzing in a standard Fender 9.5" nut radius neck with zero relief?



We need to solve equation number 2. We need to find Rd for a Fender neck.

Rn = the radius at the nut = 9.5
X = 56
D = the distance between the nut and the 21st fret = 17.92

When we plug these numbers into the equation we get 12.54 ... or a 12.54" radius at the 21st fret being the optimum conical radius for a 9.5" nut radius Fender neck.

That was a lot of numbers, but what does it mean in the real world? Here goes:

For the E string and the D string to both have the same action as each other off the frets, the neck radius needs to change from 9.5" at the nut to 12.54" at the 21st fret. In other words, if the radius is a constant 9.5" all the way up the neck, the strings simply can not have the same distance above the frets above each fret - one can (either the D or the E), but not both. You can either adjust the trussrod to have the E string's frets underneath it exactly where they need to be, or, the D strings ... unless you have a conical radius.

So armed with your fret leveling beam, do you adjust the neck to be straight in the middle (the D string path)? Most people do. But the math above clearly shows that it is impossible to level the string paths for all the strings unless you have a 9.5 - 12" conical radius.

So how to get a prefect level and crown on a constant radius - put a conical radius in it. Adjust the neck so it is straight -- at the E and e string paths -- and is slight off straight in the middle, at the D and G string paths. Then the leveling process will remove excess material from the 12th to 21st frets effectively giving them a 12.5" radius and allowing your strings to be exactly evenly the same distance off the frets all the way up the board.

Continued in part Two: how does this relate to a 7.25" vintage radius with low action fretting out with bends in the 12th to 21st frets? And how does this relate to the radius to which we set our saddles?

[LeroyBlues]

Good stuff. Thanks Nick

[Nick JD]

If we run the numbers again for a 7.25" radius neck, we get an optimum 21st fret radius of 9.57".

I've made these comparison pictures of a 7.25" and a 9.5" radius overlaid to illustrate the difference between the two. Quite a lot. You can see that above the 12th fret, where optimally you want an increasing radius to 9.5" ... you are bending up a big 7.25" hill, and it's causing the note to deaden on the fret in front.

It's not really possible to put a conical radius in a vintage radius neck with a level and crown - the difference between 7.25 and 9.57 is too great.



Let's compare this same 2" difference in radius with a 14" and a 16".



Virtually no difference. The bigger the radius gets, the less the difference becomes - and the less important a conical radius becomes for a low action.

So, if the 21st fret should be a couple of inches flatter (roughly, for Fenders) than the nut, what should the saddle radius be?

Using the equations above for a 9.5" Fender neck we get 13.8".

[Nick JD]

Originally Posted by LeroyBlues

Good stuff. Thanks Nick

No worries - I'm only just coming to terms with all of this. My brain hurts .

[Keyser Soze]

Ken Parker talks about this in an interview he did with ToneQuest Report. The ideal relationship is defined by the radius at the nut and the radius at the bridge, or really, any other single defined point along the string length. But once you have chosen those two points they determine what everything else should be.

http://www.tonequest.com/articles/article23.htm

TQR: How did you arrive at the fretboard radius for your guitars?

On any stringed instrument where the string spacing is wider at the bridge than it is at the neck, the surface of a correct fingerboard is a cone and not a cylinder. It’s a section of a paper coffee cup, not a soda can. In that event, the arc on the fingerboard is different everywhere. It changes.

TQR: You’re saying it should change, ideally…

It has to change…in order for it to be correct. The big manufacturers have never used conical surfaces – they never have. One of he reasons to refret a guitar is to flatten the end of the fretboard so it doesn’t fret out when you bend strings. They fret out because they’re too curved up there and if they are built correctly they won’t fret out. So in common guitarspeak, there are two phrases that are used to describe the arc of a fingerboard. Geometry is pretty cut and dried…how about the phrase compound radius? This is a term of nonsense – it has absolutely no meaning. A radius is the straight line measurement between the center of a circle and the outside of a circle, as you know. It is no more or less than a straight line, and you cannot compound a straight line. It’s geometrical nonsense, like saying a flat curve or a straight circle. What it’s intended to convey is the notion of a conical surface. Every violin that was ever made has this combination of shape. On a guitar that’s correctly set up, the arc is from a smaller circle at the nut than it is at the bridge, and that’s what people are trying to say when they use the term compound radius.

TQR: So what are you using on the Fly? I was at a shop earlier today and asked what the fretboard radius on a Strat I had with me was, and the guy just pulled out a Stew-Mac plastic template gauge – I was expecting something more interesting and complex as an answer…

If you had a good guitar repairman and he refretted your guitar, when he was finished you would have a conical section. Not because he would get out templates necessarily (although if you subscribe to the Stew-Mac style you have to have a template to blow your nose), but just to accommodate the string. If you corrected the fretboard surface so that all the strings were happy, you’d end up with a conical surface. The reason is that there is no string on a six-string guitar that is parallel to the centerline.



Where things become less than ideal is when we acknowledge that each string is not identical to the others and therefore each will have a different amplitude of vibration. Heavier strings simply demand a higher action, or some relief. But there is nothing wrong with this, we are simply treating each string as it should be treated - individually. We do this when cutting nut slots too.

In my own experiece when doing a setup I will start with a totally straight neck and try to achieve a good action by slightly flattening the radius at the bridge (thereby raising the action of the strings with the largest amplitude) and only if that is not working out (when the strings get too high far up on the neck intonation can be compromised) will I start adding any relief.

Not that relief is necessarily bad, but as you allude to, it does significantly complicate the geometry. Once you go down that route the whole thing becomes very seat-of-the-pants.

[Sharp5]

Makes sense. I think.

So how does one do this? With different radius blocks or after it is fretted with a beam?

I'll have to check out the article to see if it says how Parker does this. I do respect the fact that he is not tied to tradition and is advancing the field.

[piece of ash]

Boy... he sure gets worked-up about the term compound radius... too much coffee maybe... or doesn't like interviewers.

[piece of ash]

This is how I do it...

1) The level is being used as a sanding beam... simply because it has a nice machined straight surface. It has sticky sandpaper on the bottom.

2) Notice the rollers and support glide at the far end of the sanding beam. This keeps the beam traveling through a single point at all times. The distance from the pivot glide to the nut is calc'd using the formulas Nick JD has provided.

3) The neck is checked periodically throughout the sanding process using a a radius gauge at predetermined location.

4) I use dressmakers chalk to put a series of hash-marks across the fretboard... so you can determine easily when the sanding has leveled (or "coned" we should say) the entire surface.

5) I shaved my wookie hand for the photo.

[Jack Knife]

That's a nice jig you've rigged up, piece!

[piece of ash]

Thanks... I'm going to build a "real" one once I give it some more thought. That was sort of a thrown together kluge... just to see if'n i'd work. That's a Jazz bass neck getting a fretless Ebony fretbo... oops FINGERBOARD installed.

And BTW... those levels are available at Home Depot... maybe Lowe's too. They actually have a machined edge that my precison rule says is straight. There is a Momma bear version as well... so another alternative for a fret leveler. A little on the spendy side though.

[Keyser Soze]

Originally Posted by piece of ash

Boy... he sure gets worked-up about the term compound radius... too much coffee maybe... or doesn't like interviewers.

Yeah, but his crack about needing a template to blow your nose being the 'Stew-Mac' way is priceless.

FWIW, I've only re-radiused existing fretboards - using a method very similar to Piece of Ash's. You work along the position of each string, moving up and down the fretboard. Which actually creates a series of bevels that can then be smoothed into something approaching a cone. It's not exact, but it really doesn't need to be exact. Again, primarily because we are actually treating each string independently, but also because ultimately it is the frets we are concerned with (and through the fretting process each fret naturally forms a 'better' curve.

Heck, if you are one who tends to press harder with the beam on higher frets during the levelling process that alone can create sufficient 'compound' if you are starting from a relatively flat radius neck (e.g. 12" or such.)

Edit: and those are the exact same beams I use. Buy two, and a small jar of 150 grit valve grinding compound. Slather one with the compound, lay them side by side on a flat surface, mate the two together, then using long even strokes rub back and forth to polish them both into perfectly smooth flats.

[e-merlin]

So, how do arthritic fingers and wrists play into the numbers? How about personal preference?

This is why eggheads will never solve the world's problems.

[piece of ash]

Keyser... I have been wondering if one should proportion the radius to match the spread of the string... by doing so... you would make the strings parallel to the "axis" of the cone.

e-merlin... the 2011 AM DLX Strats and Teles have compound fretboards... I'd go try one out. Not to mention that the quality of Fender AM necks is pretty danged high as well. You could get one off Ebay or Warmoth makes them too.

[Jack Knife]

Originally Posted by Nick JD

But the math above clearly shows that it is impossible to level the string paths for all the strings unless you have a 9.5 - 12" conical radius.

Maybe that's why I liked my Mighty Mite compound radius neck so much? They have a 9.5-12 compound.

[murrmac123]

Originally Posted by Keyser Soze

Edit: and those are the exact same beams I use. Buy two, and a small jar of 150 grit valve grinding compound. Slather one with the compound, lay them side by side on a flat surface, mate the two together, then using long even strokes rub back and forth to polish them both into perfectly smooth flats.

This is a great thread, and I will be coming back to it, but, Keyser, forgive me for being pedantic, but I have to point out that the process you describe is not actually guaranteed to give you two absolutely flat surfaces. You could conceivably end up with one concave and one convex surface. To guarantee straightness and flatness, you would need three beams, and lap them all one against the other in the order A:B, A:C, B:A, B:C, C:A, C:B, A':B, A':C, B':A, B':C, C':A, C':B, where A' denotes A reversed. If the process is repeated a couple of times (perhaps with finer abrasives) , you will inevitably end up with three perfectly flat, straight surfaces.

OK, back to fretboard geometry ...

[e-merlin]

Originally Posted by piece of ash

Keyser... I have been wondering if one should proportion the radius to match the spread of the string... by doing so... you would make the strings parallel to the "axis" of the cone.

e-merlin... the 2011 AM DLX Strats and Teles have compound fretboards... I'd go try one out. Not to mention that the quality of Fender AM necks is pretty danged high as well. You could get one off Ebay or Warmoth makes them too.

I've owned several guitars with Warmoth or other compound radius necks. I speak from experience.

[Nick JD]

My point in this thread was more along the lines of: you can put a perfect conical radius in a 12" constant radius guitar's frets.

Without realising, on fretboards with a radius greater than 9.5", a fret level is also a conical radiusing ... because the only way to take a tiny bit off each and every fret (marking the sharpie line) while following the string paths is to make the neck conical.

This can be done in two ways: either by decreasing the radius towards the nut, or increasing it towards the heel. Otherwise the neck isn't being leveled properly.

This is why an L&C plays so nicely, as much as getting rid of those few high frets.

The OP wasn't about compound radius boards. But it does show that the perfect compound radius is a 9.5 to 12.54, or a 7.25 to 9.57...

...or a 14 to 14 with a quick wipe of the leveling beam .

[piece of ash]

Originally Posted by e-merlin

So, how do arthritic fingers and wrists play into the numbers? How about personal preference?

This is why eggheads will never solve the world's problems.

Originally Posted by e-merlin

I've owned several guitars with Warmoth or other compound radius necks. I speak from experience.

You speak from experience... What are you speaking of? I'm confused...

[piece of ash]

I see what you're driving at Nick.

Because if we were truly maintaining a perfect constant cylinderical radius, the leveling beam would have to be over/past the ends of the frets, off the neck completely on the nut end, when leveling the ends of the fret on the heel end.

Good point!

[e-merlin]

Originally Posted by piece of ash

You speak from experience... What are you speaking of? I'm confused...

If you weren't fishing for an argument you would have quoted yourself when you said:

Originally Posted by piece of ash

e-merlin... the 2011 AM DLX Strats and Teles have compound fretboards... I'd go try one out. Not to mention that the quality of Fender AM necks is pretty danged high as well. You could get one off Ebay or Warmoth makes them too.

Oh, look how easy that was. Fret no more, I got you covered. So as not to further impinge on the integrety or intent of this thread, I'm out.