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Man... my math skills have seriously atrophied. I used to be able to do stuff like this without even thinking hard... but lack of use has put the kibosh on my math brain.

I am looking for a formula that I can build into a simple algorithm that will let me calculate the width of the fingerboard at any fret and for any given scale length, nut width and saddle spacing.

So... for instance, if I am going to do a fb that is 1 7/8" at the nut, 2 3/8 ***SADDLE SPACING*** and a 25.5" scale length what would the width of the fb be at the 14th fret. (I am sort of making the assumption that the 12th fret width is the same spacing as the saddle for the sake of simplicity).

I can get the answer either by drawing it out, or doing it in CAD, but I am looking to come up with a snazzy formula so I can just plug in the numbers and get the answer.

I know a couple of you guys are math gurus... any help?

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Brock Poling
Columbus, Ohio
http://www.polingguitars.com

There's no solid formula that can come from those dimensions that I could think of, unless one is willing to assume the desired distance from the edge of the board to the string. Overall width at one end and string spacing at the other, means one would have to be changed to put it in to formula.

If you were just dealing with either string spacing at the nut and bridge, or nut width and related to fingerboard with (imagining the lines continued to the bridge) it would be simpler.

The formula I would use would be indifferent to the scale length. In the example of string spacing :

saddle string spacing = ss
nut string spacing = ns
fret number = fn
string spacing at chosen fret = fs

ss-[(ss-ns)/(12th root of 2)^fn] = fs

I don't know if my notation is right there, but let's say you wanted to check string spacing for the 14th fret. Find the difference between the saddle spacing and nut spacing. Divide that by the 12th root of 2 to the 14th power. Take that answer and subtract it from the saddle spacing, and it will give you the string spacing at the 14th fret. No need to consider scale length at all.

The catch I think is that most neck do not maintain a constant distance of the string in from the edge as it goes up. It may sometimes be proportional to the increase in string spacing as you move up, but I don't even think that's often the case. The inset of the strings typically increases faster than the spacing. This could make it tough to come up with a consistent formula that mixes the two.



Edit: I just re-read your post and saw you make the assumption of the saddle spacing width being equal to 12th fret board width. I guess that combined with an assumed inset of strings at the nut, and you probably could come up with a "simple" formula. I'm a bit tired to try to tackle that one myself - that would actually require whippin' out the pencil and paper. There's probably a simple answer, but it's just not coming to me right now.

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Eschew obfuscation, espouse elucidation.

Thanks David.

Actually I would find this tool useful in a number of situations, but in this particular one, I am looking to calculate the size of the headblock for a cutaway guitar with a 1 7/8" nut and 2 3/8" saddle. (assuming the fretboard edge lies flush with the cutaway).

This isn't the first time I have had to get the answer to questions like this and it would be really handy to either have an equation or a spread sheet I could plug the numbers into. Right now I am just drawing it out in CAD and taking a measurement at the 14th fret.

What I need though is the width of the fretboard at certain locations.

Thanks again for your help.

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Brock Poling
Columbus, Ohio
http://www.polingguitars.com

Edit: (Brock, you came in again while I was rambling away. Can you elaborate? Why is it that you can't extrapolate the fretboard width from the string spacing at a given fret?) Back to original post.

If you can live with string spacings rather than actual widths of wooden bits, this should work. I like David's better, but I don't yet understand it. This is another way to look at it.

The strings make a section of a triangle. If my math isn't too rusty, the dimensions of similar triangles are directly proportional. So, if your triangle height changes 10%, the sides all change 10%. The base of the triangle created by your strings changes a certain percent from saddle to nut. That is (s-n)/s. So the total height of the triangle (h) created by your strings is scale length (L) divided by that percent like so: h=L*(s/(s-n)).

There must be some formula to tell you the location of the frets and that would be useful here. I assume it has something to do with that 12th root of 2 stuff David was talking about. You could use that or just measure it.

However you get your fret location, figure it as distance from the saddle (f) and then the height (h) of the imaginary triangle minus the distance (f) divided by h gives the percent you've shortened the triangle. Multiply by the string spacing at the saddle (s) and you get the string spacing at your fret.

As an equation, it should simplify to this. (but as I keep getting different answers, it seems my math brain has atrophied as well and I could be completely off)

string spacing at a fret = (SL-FS+FN)/L

where S=string spacing at the saddle, L=scale length, F=distance from saddle to fret, and N=string spacing at the nut.

For fretboard width, it should be a simple matter to decide how much wider you want your fretboard than your strings at any particular location and add that amount to your calculated string width. You could do a similar equation to figure your fretboard overhang if it tapers evenly from the nut to the body.

This ignores any compensation and I don't understand enough about guitar stuff yet to tell you what to do about it. As I understand it, the saddle has a theoretical location that is then moved slightly to lengthen the strings. If so, the formula would be based on the uncompensated saddle. I'd guess it was the scale length minus the distance from the nut to the fret.

Now that I've looked at it a little more, assuming that the ratio of scale length to resonant string length is 2^(n/12), this is the same as Davids He just said it better. But since I went to trouble of typing it, I'm posting anyway. I hope it's useful to someone.

Miek

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Mike Lindstrom

Brock,

If your scale length is SL, nut width is NW, and the length along the centre of the fretboard from nut edge to centre of the fret is FL, and the fretboard width at the 12th fret is WT (in your assumption the same as saddle string spacing) then the fret board width at any length FL is given by:

Fretboard Width = NW + 2*FL*(WT-NW)/SL

Where * is the multiply by symbol and / the divide by symbol.

A more general formula where you want to set the saddle string spacing to be the fingerboard width at the n'th fret would be:

Fretboard Width = NW + 2*FL*(SSS-NW)/FLN

where SSS is your saddle string spacing and FLN is the distance along the fretboard centre-line to the middle of the n'th fret.

As the others have said, where you choose to inset the outer strings at the nut will determine how the string lines follow the fretboard's outer edges.

If you pm me with your e-mail address, I can send you an Excel spreadsheet that lets you put in scale length, nut width and saddle string spacing and gives you each width at each fret for the saddle string spacing equal to the 12th fret width.

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Dave White
De Faoite Stringed Instruments
". . . the one thing a machine just can't do is give you character and personalities and sometimes that comes with flaws, but it always comes with humanity" Monty Don talking about hand weaving, "Mastercrafts", Weaving, BBC March 2010

I do this without formulas at all by just making the fingerboard first. Then you measure from it to get the dimensions for the neck block/cutaway and everything else that relates to the fingerboard.

Of course, you have to decide in advance how much fingerboard width you want from the E strings to the edge, but that is a subjective dimension anyway. Just decide on that, build the fingerboard and then measure from it to make the rest of the guitar fit right.

Mark

I thought a bit more this morning. I like to keep formulas simple as possible, and I think the most practical way to do this is to come up with one more constant to eliminate string spacing from the formula. You have your nut width, and string spacing at saddle. To simplify that you need to remove the saddle string spacing and replace it with the projected fingerboard width at the saddle were the edges to be continued to that point.

You could add a formula that would take the desired distance from the string to edge at the 12th minus the string to edge distance at the nut, multiply that by two and add that to the saddle string spacing. I have a hunch though that you would find that increase beyond saddle string spacing to be nearly constant on all guitars you find comfortable. So what I would do is simply measure the difference between the saddle spacing and projected fingerboard width at the bridge on some comfortable guitars, and make that number a constant.

Once you have that (let's call the constant c), the formula becomes very simple. No need to include scale length or nut width to string spacing, or any of that stuff.

c = projected board width - s (this "c" is constant you'd have to determine)
n = nut width
s = saddle spacing
f = fret at which width is to be calculated
w = fingerboard width at chosen fret

And to simplify the text in the formula here, I'll say
a = 12th root of 2

w = (s+c) - [(s+c-n) / a^f]

a and c would be constants, so with this formula you could just plug in your nut width, saddle spacing, and fret at which width is to be determined, and you will get the width at that point. I've no idea how to plug something like this in to excel, but I can punch it in easily enough on my cheap $20 calculator.

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Eschew obfuscation, espouse elucidation.

Unless you are planning to cut each fret to length, I really see no need to have the FB width at every fret. No matter what scale length you pick, the 14th fret is always the same ratio between the nut and saddle = 0.5546, or the 12th is always 0.5, 20th is always 0.7028. You have to decide what the distance from the string to FB edge is , no matter what formula or spreadsheet you use. So, lets assume its 1/8. Add 1/4 inch (1/8 each side to your saddle width) and get 2 5/8. With a 1 7/8 nut, the difference is then 3/4. Take .5546 x 3/4 = .4195, add that to the the nut width, and the answer is 2.2910. Thats the FB width at the 14th fret.

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Tony Karol
www.karol-guitars.com
"let my passion .. fulfill yours"

Yeah, I don't know that I see the need for a ready-to-use formula either, but c'mon - you know math riddles are just plain fun whether there's a need for them or not.

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Eschew obfuscation, espouse elucidation.

I was behind the bleachers smoking.... during math class in High School, so I'm apt to take the simple empirical route to these kinds of situations.
My neck width at the 14th is the same as my string spacing at the bridge therefore the distance from the centerline of the neck block to the cutaway side of the neck block is 1/2 that measurement (that level of math I can handle), less the thickness of the side.
So in your case that's 2-3/8"/2 - .085(or your side thickness) =1.1025

If it's true that ignorance is , I'm one happy camper...

-C

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Freeborn Guitars
and home of BeauGuard©

Tony,

I'm probably dim and have this wrong, but if Brock want's a string with at the saddle of 2 3/8" (2.375") and the fret board to be this wide at the twelfth fret, then how can it be narrower than this at the 14th fret?

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Dave White
De Faoite Stringed Instruments
". . . the one thing a machine just can't do is give you character and personalities and sometimes that comes with flaws, but it always comes with humanity" Monty Don talking about hand weaving, "Mastercrafts", Weaving, BBC March 2010

yes Dave .. I noticed that too .. so its fairly obvious that one of two things needs to is going on .. either the 1/8 string offset is slightly too small (by about 85 thou), or the string to edge distance tapers with the FB.

Or you can look at it this way .. if you have 1/8 inch string to FB edge, then the 12 fret FB width is NOT the same as the saddle string spacing.

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Tony Karol
www.karol-guitars.com
"let my passion .. fulfill yours"

Or - if you use the formula I quoted above - the answer is 2.43". Once you have set the width of the fretboard at the nut and the twelfth fret together with the scale length, that defines the fretboard (assuming it is symetrical about the centre line and has straight edges). Outer string distance from the edges of the fretboard is irrelevant.

If on the other hand you want the outer strings to be a constant distance from the fretboard along it's length then you need to set the gaps you require and calculate the apropriate width of the fingerboard at the nut (or set the nut width and calculate the appropriate saddle string width). For example if Brock wanted 1/8" gap at each edge then the nut width of the fretboard needs to be 2.1274" for all of the other variables he set. Or if he wants the nut width to be 1 7/8" then the saddle string spacing will be 2.122".

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Dave White
De Faoite Stringed Instruments
". . . the one thing a machine just can't do is give you character and personalities and sometimes that comes with flaws, but it always comes with humanity" Monty Don talking about hand weaving, "Mastercrafts", Weaving, BBC March 2010

There was a thread earlier where I laid down 90% of what you need for this sort of thing for both constant string-to-edge distances and a tapered string-to-edge distance with some guidance from Howard and Mario on the taper style they were using. I'm pretty sure you can plug in the ratio Tony's talking about with that other stuff to get what you're after. I'm up to my neck right now in cabinets, guitar necks, and wooden shoes (no kidding!) so I'm too burned out right now to do the whole thing from scratch but the above should get you there.

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Bob Garrish
Former Canonized Purveyor of Fine CNC Luthier Services

And not all guitars connect at the 14th fret... I was thinking it would be nice to know this for other fret positions too (remember I also build electrics)...

but you're right for 99% of these calculations I will be looking @ the width @ the 14th fret.

Thanks again guys for all your thoughts on this.

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Brock Poling
Columbus, Ohio
http://www.polingguitars.com

I never got past 10th grade math, so I'm probably of little help to you. But, I recently looked at several FB tapers on different guitars to reach a decision about exactly what taper I wanted to establish as a standard on my instruments. In figuring this out, it helped me to calculate what each taper was per inch of fingerboard length, and that was ultra simple math, something I could handle. It seems to me that if you just calculate what your taper is per inch, you could then measure exactly what the length of your FB is from the nut to any given fret, and, knowing what your nut width is, do a simple calculation to arrive at what the FB width is at that fret.

Now, whether the actual FB width at a particular fret on a particular guitar turns out to be exactly as predicted depends on how accurately you make your FBs, including the binding... maybe a good idea to just make the FB first and measure it, as has been suggested.

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Todd Rose
Ithaca, NY

https://www.dreamingrosesecobnb.com/todds-art-music

https://www.facebook.com/ToddRoseGuitars/

The way I'm going about it is establishing my FB taper first (as taper-per-unit-length, i.e. a certain angle relative to the centerline), and making that a constant on all my instruments (with possible exceptions, of course). The string spacing at the saddle, then, is a function of the scale length and nut width (as well as the distance between the outer strings and the edges of the FB, which is also pretty much constant on my guitars). If you're not going about it that way, then my simple math method won't do diddly squat for you.

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Todd Rose
Ithaca, NY

https://www.dreamingrosesecobnb.com/todds-art-music

https://www.facebook.com/ToddRoseGuitars/

Brock .. until you define exactly how you want the FB to be, ie constant string to FB edge of tapered, the answer cant be defined. Note the different answers and approaches listed !!!

So, if how you do this now, is that you want the 12 fret (or any other like 14) Fb width to be the same as the saddle spacing, then its pretty easy (and the edge spacing becomes irrelevant), and as mentioned since the fret spacings are all ratios, scale length is not required. A simple Excel sheet can produce the results.

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Tony Karol
www.karol-guitars.com
"let my passion .. fulfill yours"

Yeah, that is the plan. That is how I do it when I draw it out by hand.

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Brock Poling
Columbus, Ohio
http://www.polingguitars.com

Ok Brock .. so you any good at Excel ... ????

you need one cell to input the nut width, and one with the saddle width (spacing), which will end up as your 12th fret FB width. Now take any fret scale you have (preferably in mm, it will be easier to do), and divide each distance from the nut to a given fret by the scale length - you want the percentage of the length of FB from the nut to a given fret. Put the fret numbers down one column. Put the percentages in the next column. Now the formulas will go into the next column, and its easy ... take the saddle spread and subtract the nut width, multiply this by 2, then multipply that by the percentage for each fret. Add this number to the nut width and you now have the FB width for any fret, and in any scale length (irrelevant to the calculations).

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Tony Karol
www.karol-guitars.com
"let my passion .. fulfill yours"

Thanks Tony - I wasn't thinking of the 12th fret = saddle spacing when adding that edge to string factor in to the equation.

To use the same characters as before -

"n = nut width
s = saddle spacing
f = fret at which width is to be calculated
w = fingerboard width at chosen fret

And to simplify the text in the formula here, I'll say
a = 12th root of 2
"

So the formula is...... (drum roll, please)

w = (2s-n) - [2(s-n) / a^f]

Plug in your saddle spacing (s), nut width (n), and fret number (f), and "w" will be your width at that fret. And best of all, the formula's independent of scale length or even units of measure (so long as you use the same for all the variables you enter of course).

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Eschew obfuscation, espouse elucidation.

Thanks David... this is a big help.

But I have a question -- and remember it has been a while on my math skills -- but how can it be scale independant? The 14th fret (or whatever fret you want) will be at a different point along the path depending on the scale length? A 27" baritone will be different than a 24" parlor guitar... no? Won't the angles of the taper, and distance between the frets be sufficiently different to give you different results?

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Brock Poling
Columbus, Ohio
http://www.polingguitars.com

Brock,

It is independent of scale length as the length from nut to middle of the n'th fret is a function of scale length and so this cancels scale length out of the equation. The width of the fretboard at each fret will stay the same for the 27" and 24" scale lengths in your example but the fretboard angle of taper will be different.

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Dave White
De Faoite Stringed Instruments
". . . the one thing a machine just can't do is give you character and personalities and sometimes that comes with flaws, but it always comes with humanity" Monty Don talking about hand weaving, "Mastercrafts", Weaving, BBC March 2010

Right. The easiest way to think of it is as a section of a triangle. The formula uses the 12th root of 2 so as to calculate the width at the proportionate point in the triangle. For example, if you have a triangle that is 2" at it's base it will always be 1" wide at 1/2 distance to the top, regardless of what the angle or height is.

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Eschew obfuscation, espouse elucidation.