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I know there are plenty of fret scale calculators around, but this was mostly a mental exercise during the commute today.

C&N has a table of constants to calculate the offset of each fret from the nut:

Fret Constant
1 0.0561
2 0.1091
3 0.1591
4 0.2063
5 0.2508
6 0.2929
7 0.3326
8 0.3700
9 0.4054
10 0.4388
11 0.4703
12 0.5000
13 0.5281
14 0.5546
15 0.5796
16 0.6032
17 0.6254
18 0.6464
19 0.6663
20 0.6850
21 0.7027
22 0.7194
23 0.7351
24 0.7500

Presumably, everyone already knows about the "Rule of 18, " (modern value 17.817) in which fret positions are calculated by progressively dividing the remainder of the scale length by that value.

It's fairly easy to create a spreadsheet with a couple columns to make a running total of each fret's offset from the nut using this method, but I was still curious.

Recognizing the fret position as a geometric progression, intellectual curiosity prompted me to wonder "What is the exact formula for the location fret N?"

According to R.M. Mottola (http://liutaiomottola.com/formulae/fret.htm), that constant can ultimately be derived by the formula based on the 12th root:

distance of fret n = scale length – (scale length / (2 ^ (n / 12)))

Before I found this formula, I had worked out the formula based on a geometric series whose value is calculated by:

distance of fret n = a(1-r^n)/(1-r)

where a is the distance to the first fret, calculated by dividing the scale length by 17.817, or now the formula above, r is the multiplication constant 1-1/17.817, and n is the fret number.

The results are significantly identical.

I suspect the two formulas are different forms of the same equation, but my feeble brain isn't able to make the connection just yet.

If anyone wants to use or play with the spreadsheet, I've made it available here:

https://www.dropbox.com/s/gto280t2owre1 ... .xlsx?dl=0

The spreadsheet I built a few years ago that is at the top of the jigs and fixtures page has both fret position and cumulative (It is posted under Lances name). I believe I even added the formulas at the bottom of the sheets. For those who just want the numbers you simply add the scale length and the rest calculates for you. I built the sheet because I too was curious one day and I also wanted a simple way to do a series of scale lengths to cnc some fret boards.

Shane

_________________
Canada

Glen you might want to engage my business partner and friend Dave Collins.

Dave has been studying fret spacing for nearly a decade now and the study includes spread sheets, graphs, and a database of hundreds of instruments that he has measured with long calipers and graphed the spacing errors.

What resulted is a digital finger print of sorts where we can verify to some degree the authenticity of an instrument, when and even where it was made based on similar errors produced by say a gang saw.

It's pretty interesting and he's got this subject down with lots of data, tools, and personal experience. Hopefully he will be along shortly.

By the way when interesting instruments come into our shop for service we may take a bit of time to measure them and record the results. This database includes lots of examples of vintage instruments as well.

It's a fascinating subject learning what manufacturers did and currently do and in some cases why. One also learns that fret errors are abundant so much so that many of them are audible and noticeable....