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Finite Element Simulation of Guitar Top Vibration

Finite Element Simulation of Guitar Top Vibration

by Phil Banks

Originally published in American Lutherie #18, 1989



The use of engineering finite element analysis software to determine modes and natural resonant frequencies of a guitar top can be a useful (albeit lengthy) process which, if used judiciously, can yield useful information to the guitar maker.

As a graduate mechanical engineer and a guitar maker, I’ve always been interested in marrying the discipline of the luthier’s craft with that of science. I got that chance last year at the University of Sydney. While working as a programmer developing a Finite Element package, I was asked to produce a demonstration of the program’s capabilities. I decided to analyze a guitar top.

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Tuning the Guitar

Tuning the Guitar

by Ian Noyce

Originally published Guild of American Luthiers Data Sheet #56, 1977 and Big Red Book of American Lutherie Volume One, 2000



Because the guitar has fixed frets set to an even temperament, tuning it properly is not the cut-and-dried process that many people believe. And due to various factors that we’ll get to shortly, if the guitar’s bridge is placed exactly where the nominal scale length says it should be, the instrument may not play in tune at all.

The two most common methods of tuning are: (1) the 4th- and 5th-fret method and (2) the harmonic method. Both of these methods are often misunderstood through confusion regarding perfect (or Pythagorean) intervals and even-tempered intervals.

The 4th- and 5th-fret method. Theoretically, this is the simplest method as it simply involves tuning unison intervals. The A string can be tuned to an A tuning fork, then the bass E is fretted at the 5th fret and tuned in unison with the open A. The D string is tuned in unison with the 5th fret of the A, the G string is tuned to the 5th fret of the D, the B string is tuned to the fourth fret of the G, and the high E is tuned to the 5th fret of the B. In practice this can be difficult because any errors are cumulative. It’s also true that many guitars tuned this way will not play in tune in all keys.

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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.

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Free Plate Tuning, Part One: Theory

Free Plate Tuning, Part One: Theory

by Alan Carruth

Originally published in American Lutherie #28, 1991 and Big Red Book of American Lutherie Volume 3, 2004

See also,
Free Plate Tuning, Part Two: Violins by Alan Carruth
Free Plate Tuning, Part Three: Guitars by Alan Carruth



I started learning free plate tuning on violins and violas more than ten years ago from Carleen Hutchins. For those who have not had the pleasure of her acquaintance, Carleen is one of the founders of the Catgut Acoustical Society and its permanent secretary. She is an able scientist, a great teacher, a fine luthier, and a self-confessed mediocre violist. While working with physicist Frederick Saunders almost thirty years ago she helped rediscover and update the old Chladni method of visualizing the vibrations of plates. Her subsequent research, using Chladni patterns as a window into the differences between good and poor violins earned her a silver medal from the Acoustical Society of America.

Violin makers have traditionally used some variant of “tap tone” tuning to guide them in working out the final graduations of the top and back plates. Although the technique seems simple and organic on the face of it, it is in fact very complex. It takes a long time, as well as a good ear and a lot of talent, to learn to tune plates by tap tone. Even those who are good at it don’t always succeed. Felix Savart, back in the 19th century, tried to adapt Chladni’s method to research on violin acoustics, but the technology wasn’t there. Now we have the means, and as we gain more understanding of how the instruments work, we also gain more control over the sound.

And it doesn’t only work on violins. Fred Dickens, Graham Caldersmith, and Gila Eban have all done major work in applying the principles of violin acoustics to guitar construction. Of course, there are differences and it takes time and effort to sort them out, but physics is physics, or, as a friend of mine said, “it all comes down to F=mA in the end.” I have found these techniques to be useful, and sharing useful techniques is what the Guild is all about.

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In Search of the Perfect Cone

In Search of the Perfect Cone

by Tim Earls

Originally published in American Lutherie #30, 1991 and Big Red Book of American Lutherie Volume Three, 2004



I think I've got it. I have here an untested method of finding the exact, correct multiple radius for any given fingerboard using simple barnyard geometry and no computer. Danny Rauen and Tim Olsen wrote interesting articles on multiradiused, or conical, fretboards in American Lutherie #8. (See Big Red Book of American Lutherie Volume One, p. 298.) Great stuff! Let’s talk about cones for a moment.

A cone is a tapered cylinder extended up to a point. Or a tapered cylinder is a cone with its point lopped off, take your pick. You knew that. Bear with me. In a two-dimensional view, this looks like Fig. 1. The circular base of the cone is seen as a horizontal line, since you’re looking at its edge. The height of the cone, what I call “true length” is measured on the centerline from base to point. The side line of the cone I call “true distance.” The radius at any spot on this cone can be found by drawing a horizontal line from the centerline to the true distance line and measuring it. You probably knew that too.

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