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Rule of 18 vs Rule of 17.817

Rule of 18 vs Rule of 17.817

by James Buckland

 

 

At the 2014 GAL Convention, I conducted a lecture/presentation entitled “Mythbusting the Rule of 18”. The intent was to explore, and possibly refute, some of the misconceptions concerning the Rule of 18.

As conventional wisdom goes, the old rule of 18 was, at best, an approximation on how to calculate fret positions. In truth, it is a better formula than it’s generally given credit. Unlike many historical predecessors, such as Juan Bermudo’s approaches using Pythagorean ratios, the Rule of 18 is based on the concept of equal temperament, probably before the term existed in the vernacular. But, most importantly, Rule of 18 includes its own compensation factor in regard to the position of the bridge.

Most commonly used today, is the square root of two, or 17.817, as the factor with which to divide string length. The result is believed by many to be more accurate based on the fact that it places the 12th fret at the exact midpoint of the vibrating string length. Since we all know that the 12th fret is the octave above the open string, it makes common sense that it should be in the exact middle of the vibrating string length.

However, it is also well known that a guitar built this way will play out of tune, with intonation problems increasing the further one plays up the neck. The solution is to compensate by increasing the string length slightly, generally by moving the bridge position. But, by how much? Examine enough classical guitars fretted with a “650MM scale”, and you’ll find the actual vibrating string length is generally longer, by 2 to 4MM. There doesn’t seem to be much more than vague empirical evidence in just how much to use. In other words, a little bit of “Kentucky Windage” is considered good enough. (For you non-shooters, Kentucky Windage is the practice of adjusting your aim to compensate for wind, without the use of any mechanical features on the weapon.) To me, this is a seemingly strange attitude considering the derision generally cast upon the good old Rule of 18. Maybe “17.817” just sounds more precise than saying “18”?

So, here’s a practical example of the similarity between the two approaches when bridge compensation is taken into account. I began by calculating a fret scale for a 565MM string length (as might be used for the terz guitar in GAL Instrument Plan 80) using the “Rule of 18”. The results are shown in Table 1. Then, I took the resulting value for the distance between the nut and 12th fret (280.446MM) and multiplied it by two. Next, the resulting value (560.892MM) was used to calculate a fret scale using the contemporary 17.817 factor. The calculations are shown in Table 2.

Table 1
Table 2

Notice the outcome! Although the vibrating string lengths are different (560.892MM vs 565MM) the results for the fret placements are generally from about a tenth of a millimeter at the first fret, approaching one millimeter towards the higher frets. To put that into context, consider real world variabilities typically introduced during fretboard fabrication, from layout, to slot cutting, to fret dressing and crowning. Or, compare this to the guesstimate made by many luthiers when choosing bridge placement compensation.

But, what about the issue of the difference in string lengths? Well, as stated above, most luthiers know that to satisfactorily use the 17.817 approach, one must add their own bridge compensation (hence the “Kentucky Windage” analogy).

So, if you add 2-4MM bridge compensation to the 560.892MM string length, you can see that resulting vibrating string length gets pretty close to 565MM!

But, there is another way of looking at the data that shows even more surprising results. The tables of values reflect the string length from the nut to the respective fret(s). What about the other length of the string, the vibrating string length from the fret to the bridge, the part of the string we actually hear?

The greatest discrepancy between the two tables is found with the 24th fret. In the case of the Rule of 18, the distance from the 24th fret to the bridge is 143.312MM. In the case of the 17.817 factor, the distance is 140.221MM. Add 2MM compensation, and the value increases to 142.221MM. Add 3MM compensation, and the value increases to 143.221MM. Add 4MM compensation, and the value increases to 144.221MM. In other words, the discrepancy between fret placement values calculated with either the Rule of 18 or the Rule of 17.817 is less than the variables introduced through the subjective choice of bridge compensation which must be made when using the latter rule. When comparing the other fret values, the difference between the two table of calculations is even less.

Does the use of the Rule of 18 vs 17.817 result in a significantly discernible difference, given other contributing factors? Maybe the old guys knew more than they’ve been given credit?

Thanks to Nitin Arora for writing the fret calculating program
“Eighteen Rules” found on his website.

At the 2014 GAL Convention, author James Buckland also demonstrated the use of proportional dividers for marking fret positions. Set the big end of the dividers to the scale length. The little end now shows you the distance from the nut to the first fret. Re-set the long end to the distance form the first fret to the bridge. You are off and running. Ron Fernandez tries it. All photos by Tom Harper.