Accurate Fret-board design

Jeff Strouse · Post by **Jeff Strouse** » 12 Jul 2004 3:41 pm

With the wide variety of scale lengths in the guitar world, I'm curious as to how fret- board accuracy is determined. When builders start from scratch, how do they determine placement of the frets, in order to be on pitch with the actual tone of the string at that particular location on the neck?

For a 22 inch scale, the precise distance between fret 1 and 2 would be "x", but on a 22.5 inch, it would be different value. And the distances would change again for a 23, 24.5, 25, and 26 scale...there would be a different fret alignment all the way up the neck.

The simple answer to the question (especially for those of us who hate math), would probably be to just go buy a pre-made fret board. But how did that guy figure out where to draw the little lines? It's interesting to look at a T-Logo neck...it looks like the frets are part of the bakelite mold...one piece, possibly?

And, could factors such as string spacing and a tapered neck have an effect on fret spacing; or, is fret placement solely determined by the location of the nut and bridge (say if 7 or 8 strings are crammed on a 6 string neck, would the fret location stay same because it's still the same scale length)?

[This message was edited by Jeff Strouse on 12 July 2004 at 04:49 PM.]

Rick Aiello · Post by **Rick Aiello** » 12 Jul 2004 4:19 pm

This is a real cool site ... with tons of tools ...

DGB Studios

Scroll down on the right and you will see what you're lookin fpr.

Ignore the Japanese "installation" stuff ... just choose English

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www.horseshoemagnets.com [This message was edited by Rick Aiello on 12 July 2004 at 05:21 PM.]

Terry Farmer · Post by **Terry Farmer** » 12 Jul 2004 4:23 pm

There are a number of free fret spacing calculators on the internet. They will give you your fret spacing for any scale length and number of frets. The one I used to build my steel was Fret Calc 3.10. Don't worry about string spacing and neck taper. Hope this helps.

Jon Light (deceased) · Post by **Jon Light (deceased)** » 12 Jul 2004 4:34 pm

Fretcalc is excellent. You can download it from here:
http://www.dougsparling.com/software/fretcalc/

You can buy rulers that are marked to the 1/100th of an inch for when it's time to execute those calculations on a real live fretboard.

C Dixon · Post by **C Dixon** » 12 Jul 2004 4:44 pm

If you want them more than accurate enough for any steel player, simply mutliply (or divide) any scale by the following "constant":

1.05946

As an example on a 23 inch scale the 1st fret would be at 21.71 inches. The second fret would be at 20.49 inches. How did I arrive at this? I simply divided 23 by the constant, then divided that number by the constant, etc, etc, etc, for as many frets as you want.

For any scale length follow the above procedure. If you do not end up at the 12th fret with EXACTLY half the scale lenth, it is due to rounding off. But you should be very very close. Closer than any steel player would ever need since we do not fret the steel guitar.

Good luck and may Jesus bless you in your quests,

carl

Jeff Strouse · Post by **Jeff Strouse** » 12 Jul 2004 5:50 pm

Here's a pic I came across that got me wonderin' bout it...

Interesting, huh?
[This message was edited by Jeff Strouse on 12 July 2004 at 07:15 PM.]

Ron Bednar · Post by **Ron Bednar** » 12 Jul 2004 7:46 pm

The best and easiest method of fret calculation that I have seen is based not on straight measurement but geometry. It's way easier to lay out the frets this way. It looks confusing at first but becomes very fast after you get into it.

The scale length divided by 17.817 gives the distance from the nut to the first fret. Instead of going on with a mess of fractions for each fret do this:
1. On a heavy sheet of paper or a flat piece of aluminum, draw a perpendicular line for the nut and another for the saddle.
2. Draw a baseline between the nut and saddle lines.
3. Set a compass from the perpendicular nut line to the 1st fret distance, that you calculated above, on the baseline. Scribe an arc from the baseline to the perpendicular.
4. Draw a tangent line from where the arc meets the perpendicular to the point at which the baseline meets the saddle line.
5. Draw a perpendicular from the baseline to the tangent at the 1st fret position.
6. Reset the compass to the distance from the 1st fret to the tangent and scribe another arc. Where the arc meets the baseline is the 2nd fret.
7. Continue drawing perpendicular lines at each fret position and scribing arcs to find the next fret position.
8. The 12th fret will be exactly 1/2 the scale length.

[This message was edited by Ron Bednar on 12 July 2004 at 08:57 PM.]

Rick Collins · Post by **Rick Collins** » 13 Jul 2004 10:12 am

If you like doing all the math, that's OK; but, if you take any fretboard to a graphic arts shop that has a camera with a horizontal bellows, they can either reduce or blow-up the fretboard onto a piece of litho film and make you a copy of any size scale you desire.

...seems the most expediant way to me.

Rick

Bruce Clarke · Post by **Bruce Clarke** » 13 Jul 2004 10:24 am

That's ok if you are absolutely sure that the fretboard you are using is correct, but Ron has it right. Ideally you need a drawing board, tee square, set square and a draughtsman's compass, plus sharp pencils, say 2H.

Al Sato · Post by **Al Sato** » 13 Jul 2004 10:39 am

Ron,

That's pretty neat - I will have to think about that. One point I'd like to make, though. When you draw that diagonal line, you call it a tangent line. If the line were actually tangent to the circular arc you drew for fret 1, it would intersect the perpendicular nut line slightly above where the circular arc intersects that line. It's not a large amount, but for accurate fret placement it's important to distinguish because it will throw the whole thing off after the first fret. Did you mean tangent to the first circular arc or just the line between the place where the circular arc intersects the perpendicular line and the place where the base line and saddle line intersect?

Thanks.

Oh, on edit, the answer is clear to me. It is not the tangent line, it is the line from where the first circular arc meets the perpendicular nut line and the point where the base line meets the saddle line. Just a minor but important (to me) point.[This message was edited by Al Sato on 13 July 2004 at 11:43 AM.]

Ron Bednar · Post by **Ron Bednar** » 13 Jul 2004 12:55 pm

Al, I'm not sure if you have a question...

I like this method, it seems to work really well. It was given to me by a old luthier that built classical guitars. He learned his craft in Spain, where he grew up, before coming to the States. This method came from a book he had, but I think he also said it was way old, like from the 1500's or there abouts. Hope you all find it useful.

[This message was edited by Ron Bednar on 13 July 2004 at 01:57 PM.]

Roy Ayres · Post by **Roy Ayres** » 13 Jul 2004 2:20 pm

When I first went to work for Fender Musical Instruments in the mid 60's as "Director of String Instrument Development" I was assigned by my boss (Paul Spranger, Engineering Director) to "shadow" Leo Fender (who had stayed on as a consultant after selling the company to CBS) for purpose of learning as much as possible about his design philosophy, etc. Leo called the fret-scale design rule "The rule of eighteenths" -- meaning that the distance from a given fret to the next higher fret is one eighteenth of the distance from the previous fret to the bridge. As pointed out above, the true factor is slightly less than 18. The problem with this type of measurement is that cumulative error builds up as you proceed up the fingerboard. A better and more accurate method is to always measure from the bridge to the posititon of the next fret, thus precluding cumulative error. If I knew how to do so, I would put the formula in this post, but since I can't do the mathematical symbols here, I'll state the formula in words. The equation is usually stated in terms of the frequency (f)of a given note, but can easily be solved for the length (l) from the bridge to a given fret. The equation in its basic form reads: [The frequency (f) of a given note] is equal to [the reciprocal of 2 times the length (l) of the string (measured from the bridge to the fret)] multiplied by [the square root of the tension (T) in the string divided by the linear density (r) of the string]. This gives the theoretical distance (l) to each fret from the bridge, but ignores a small factor called the "stiffness" of the string. To be complete, the stiffness (s) of the string must be added to the Tension (T) under the radical. The reason that (s) is usually ignored is that the designer never knows what gage strings the guitar owner may use -- and the stiffness is only a small contributor to the measurement. This stiffness factor is why many standard guitar bridges are "offset" where the strings are of sligntly different lengths. It is also the reason that adjustable saddles are required in order to keep a guitar from having intonation problems. While Leo's "rule of eighteenths" was actually an approximation, and his secretary actually laid out the fretboards on all of the early models, one can calculate out a given fret scale using the proper equation and compare it to Leo's guitars (Broadcaster, Telecaster, steels, etc.) to find that the differences are in the thousandths of an inch -- usually imperceptible to the player. If you don't believe this, tell a Tele player that his fret scale is "off" and see what happens.

As was typical of Leo, he always did the right things -- although not always for the right reasons.

Hey, who can criticize success? [This message was edited by Roy Ayres on 13 July 2004 at 03:31 PM.]

Rick Aiello · Post by **Rick Aiello** » 13 Jul 2004 3:33 pm

This is the best I could do with MS Paint ...

Ron, I think Al was discussing the use of the term tangent in #4 in your description.

The two points on each circle that would generate a common tangent line ... would not be at 12 o'clock position ...

Therefore when it (common tangent) crossed the perpendicular ... it would have a greater positive value than a tangent line drawn from the 12 o'clock point on the first circle.

At least that's what I gathered from his post ...

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www.horseshoemagnets.com [This message was edited by Rick Aiello on 13 July 2004 at 04:36 PM.]

C Dixon · Post by **C Dixon** » 13 Jul 2004 3:42 pm

"Accumulative error", as you correctly state, CAN and will add as you add more and more frets. However IF your "constant" is carried to enough places, this error is for all practical purposes non existant. I will explain.

If you want to double OR halve ANY number; but you want to do it in 12 equal "log" steps, the formula is the 12th root of the digit 2.

If you want to triple or cut any number by 1/3 in 17 steps the formula is the 17th root of the digit 3 and so on.

In our western style of music our octaves are doubled (or halved) in 12 equal log steps. Therefore we use the "constant" 12th root of the digit 2 to calculate not only all of the notes expressed in HZ, but the distance the frets are from the bridge.

Now, in order for it to be as accurate as possible, engineers have determined that the 12th root of the digit 2 must be carried to 12 places to the right of the decimal point.

Most scientific calculators are designed around this by the way; EVEN though the LCD readout may show less.

I said all the above to tell you that the actual constant "divider" to determine extremely accurate placement of frets is:

1.059463094xxx.

Where xxx is I don't know on my 9 place calculator. However the arithmetic chip inside my calculator calculates to full 12 places beyond the decimal point.

So If I store the above number in mem on my calculator, then continue to divide ANY scale length by that number, fret placement on any fretted instrument can be placed extremely close IF one does the following:

1. Round the answers to just 2 decimal points.

2. Use a 1/100th divided ruler.

Here is an example:

Scale=25.5 inches.

Where is the 1st fret?

1. Divide 25.5 by the constant above.

2. It comes to 24.06879997

3. Round this number off to 24.07 ("Round" means, 5 or above add 1; 0-4 leave it alone)

4. Using the ruler divided into 1/100's, locate the first fret at exactly 7 slashmarks above 24 inches from the bridge.

To determine the 2 fret, repeat the above steps. Continue this for every fret you want.

This will place every single fret on a 25.5 inch scale soooooo close, your eyes would not be able to see the error IF it was pointed out to you. IE, less than than the width of a human hair.

Note: The simple test is; if you get to the 12th fret and it is not precisely 1/2 the distance from the bridge to the nut, you have done something wrong. Same thing if you get to the 24th fret and it is not precisely 1/2 the distance from the Bridge and the 12th fret. And so on.

carl

Roy Ayres · Post by **Roy Ayres** » 13 Jul 2004 4:17 pm

Carl,

You said:

This will place every single fret on a 25.5 inch scale soooooo close, your eyes would not be able to see the error IF it was pointed out to you.

I agree with you -- and the above statement can also be applied to the fret scales laid out by Leo and his non-technical secretary in the 40's. Although his methods were crude and his knowledge of the physics of vibrating strings was zilch, he nailed it well enough that his axes became the "standard" in the world of solid bodied guitars.

Thanks,

Roy

Bill Brummett · Post by **Bill Brummett** » 13 Jul 2004 5:17 pm

Jeff...

that's a really weird picture of the 4 neck Stringmaster. It looks like the guitar (perhaps a 26" model) has been modified to a shorter scale and has the "professional" 22.25 inch fretboards installed. As long as the bridges were adjusted proerly, i guess this is OK. But why would anyone go to all that trouble to DECREASE the sound quality of the 26" Stringmaster???

Ron Bednar · Post by **Ron Bednar** » 13 Jul 2004 5:46 pm

Al & Rick, The discription of the line as a "tangent line" comes from the old luthier's book. That's how it is noted there. Don't know who wrote it. A look in the dictionary comes up with this for "tangent": A line, curve, or surface meeting another line, curve, or surface at a common point and sharing a common tangent line or tangent plane at that point.

Seems to me that's exactly what is going on with that line. But then again, I'm no geometry mental giant...

[This message was edited by Ron Bednar on 13 July 2004 at 06:48 PM.]

Jody Carver · Post by **Jody Carver** » 13 Jul 2004 6:37 pm

Like Roy said in essence.....If Leo were here today he could learn a lot from this thread and perhaps made something of himself.

Oh Boy

Mark Herrick · Post by **Mark Herrick** » 13 Jul 2004 7:13 pm

A tangent is a tangent is a tangent.

The drawing with the two circles is misleading. Yes, the two circles share the same tangent line, but the centers of both circles are not on the same perpendicular line to the left side vertical line.

If the two circles had their centers on the same line (perpendicular to the left vertical line), and the radius of each circle was equal to the distance from the center of each circle to the tangent line, then you would have the same drawing that Ron started with. Ron's drawing just doesn't show the complete circles, which would necessarily overlap each adjacent circle.

Al is right, however. The line that is defined by the center of each circle and the point on the circle where it intersects the tangent line is perpendicular to the tangent line, not the horizontal base line. So the tangent line would intersect the vertical line slightly above the first circle.

Where the initial vertical line intersects the circle or the tangent line is not important. We are only concerned with the points on the horizontal base line that are defined by the length of each successive radius to the tangent line.

Now I need a donut...D'oh!

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[This message was edited by Mark Herrick on 13 July 2004 at 08:37 PM.]

Rick Aiello · Post by **Rick Aiello** » 13 Jul 2004 7:38 pm

Ron ... a line tangent to a curve touches a curve at 1 particular point only ... it does not cut thru the curve.

<BLOCKQUOTE>quote:<HR>4. Draw a tangent line from where the arc meets the perpendicular to the point at which the baseline meets the saddle line.
<HR></BLOCKQUOTE>

from where the arc meets the perpendicular.

If you drew a tangent line as described here ... it would be parallel to the base-line .

Your line certainly is a tangent line ... and a marvelous way to map a fretboard ...

But that particular tangent line ... isn't the one described in #4 ...

They have differnt slopes.

Rick Aiello · Post by **Rick Aiello** » 13 Jul 2004 7:43 pm

So the tangent line would intersect the vertical line slightly above the first circle.

Mark, that was what I hoped to show in my MS Paint .gif ...

I certainly was not trying to mislead anyone by using an inaccurate model ... simply thout a visual might help.

Jeff Strouse · Post by **Jeff Strouse** » 13 Jul 2004 7:49 pm

I know one thing, I'm in need of a "Geometry for Dummies" book.

Definitley a long scale (26), turned into a short. I'd be curious to hear the sound of that paricular instument..."before" and "after"...

Mark Herrick · Post by **Mark Herrick** » 13 Jul 2004 7:50 pm

You beat me to it Rick. I was just getting ready to edit my post again to say that both you and Al are correct.

You guys know more about this stuff than I do anyway!

Ron Bednar · Post by **Ron Bednar** » 13 Jul 2004 8:00 pm

Rick, oh, o.k. now I understand. So, essentially the idea is sound, it's just the discription or terms that are off. Wish I could see that book again, I got to xerox the page with the diagram and that's all I got of it.

Seems I remember the old luthier telling me that Leo Fender invented this method...

[This message was edited by Ron Bednar on 13 July 2004 at 09:09 PM.]