Intonation Research ...
Here is a very brief description of the main issues and conclusions of my work. The transcript of my 1995 GAL lecture on this subject is available as an article in American Lutherie (number 47, Fall 1996). It is also available in the Guild of American Luthiers compendium volume, The Big Red Book of American Lutherie, Vol. 4.
As a fretted instrument, the guitar is constrained to operate with equal temperament, like a piano. The governing equation for fret placement, then, is "X subscript n equals X subscript zero times 2 to the minus n over 12 power". If you can write this in math shorthand, do it now. Here "X subscript zero" is the distance from nut to saddle and "X subscript n" is the distance from the saddle to the nth fret. From this equation comes the oft-published factor of 17.817, which is the approximate number by which string length is divided to find the distance to the next fret. For ideal strings you could set fret placement with this equation and your guitar will play in tune. Unfortunately real-world strings have certain properties that work against such an easy solution. Two properties in particular complicate the situation.
One is string elasticity, or "stretch". Every time a string is fretted it is stretched. As you stretch the string it will rise in pitch. This sharpness is not equal for each fretted note, but, more particularly, the pitch of the open string is not raised at all, since it is not stretched. It is then necessary to compensate for the fact that the pitch of the open string is flat, relative to all the other fretted pitches. To do this the nut position can be moved forward to shorten the open string length (thereby raising its pitch) without moving any of the fret positions.
The other property of interest to us is string inharmonicity or "stiffness". An ideal string will have an "harmonic" overtone series, which, by definition, consists of the fundamental and a series of overtones which are exact multiples of the fundamental. Real strings have an overtone series where each succeeding overtone is progressively sharper. Moreover, as a string is shortened by fretting, its inharmonicity increases. Our ears/brain do complex calculations that render the perceived pitch sharper than the fundamental. This means that as we fret higher on the fingerboard, the notes will sound progressively sharper than they should. The traditional way to compensate for inharmonicity is to move the saddle back, thereby rendering the relative string length increase greater for higher fret positions.
This, in a nutshell, is why compensation is necessary at both nut and saddle, though traditionally it has only been practiced at the saddle. The complex interaction between stretch and stiffness can be measured experimentally. Here is a thought experiment. Imagine a device where a string can be brought to pitch over a moveable fretboard. The nut and saddle are fixed in position. Their distance apart is precisely measured. Nut and saddle heights are set as they would be on a guitar. The fretboard can move back and forth. Position it so that a particular fretted note plays exactly in tune with the open string (use an equal tempered pitch-measuring device). The distance to the saddle is measured precisely for each fretted note - the position of the fretboard will vary slightly from note to note due to stretch and stiffness of the string. If you make these measurements for many or all of the notes on the fingerboard, you will have a set of fret positions for this string with the scale length established by nut and saddle positions.
This set is, in theory, precisely in tune, but it does not necessarily precisely conform to equal tempered fret placement. Consider, for a moment, our original equation for fret placement. You can generalize this equation by replacing X sub zero with "a" and add "b" at the end: "X sub n equals a times 2 to the minus n over 12 power plus b". In this form, which I have called the canonical form of the equation for fret placement, if b equals 0 you are left with the first equation given above. When b has a positive value, it represents saddle setback from the nominal position. "a" equals scale length. With the help of this equation we can optimize nut and saddle positions. The trick is to use statistical methods to find the equal-tempered set of fret placements that most precisely matches our experimentally determined set. Your computer can do this work for you, matching your experimental set to the canonical equation given above. The result will be values for a and b, where b is the change in saddle position, or saddle setback, with respect to the "nominal" saddle position, and a is the nominal "zero fret" position (scale length) for the statistically fitted equal-tempered set of fret positions. This is not the same as the nut position measured on our experimental apparatus. In fact, there is a shift in nut position forward (I call it "nut setforth") on the apparatus compared to the value of a. This setforth is equal to the total measured distance from nut to saddle (our original X sub zero) minus (a plus b). This is probably the hardest concept to grasp in all of this. Try to draw a little model of all this, and I think you can do it.
Now you need to go through this procedure for each string. You will get different results because the strings differ in their properties. Consequently, the values for the fitted canonical equation will differ. In order for all this information to be useful, we have to scale the 6 equations so that a is identical for each. This allows us to use one fretboard for all 6 strings. Meanwhile, the computed values of saddle setback and nut setforth can be scaled too, though the differences are negligible. If you actually do all this you will have established optimal nut setforth and saddle setback for each string, given the experimental conditions. These can be built into the nut and saddle of your guitar but will require cutting the fingerboard short at the nut end by about 1 mm to accommodate the nut setforth of the G string, which shows the most extreme behavior. I also use a 3 mm wide saddle to accommodate the range of breakpoints (from about 1 1/4 mm setback for the high E to about 3 mm setback for the G). Naturally, different strings give different results, as do different saddle heights, left hand pressure, etc., etc., so in practice, it is not easy to get useful results. It is also difficult to measure precisely enough. Even so, the precision of fit between the experimental values and the canonical equation can be remarkably good, though not perfect. All in all, intonation is improved with this system. Even a little improvement would seem to be worth the trouble (which isn't much really). Guitars with accurate fret placement can be retrofitted by cutting a bit off the end of the board, making a new faceted nut, and perhaps redoing the saddle in a wider slot (to accommodate string to string differences).
In the table that follows, I present values for nut setforth and saddle setback. These values are based on measurements of a set of Augustine Regals with blue label basses, scaled for a 650 mm scale length. These work reasonably well for most nylon strings but are probably a little on the high side for Savarez Alliance and various "carbon" strings.
|string||∆N - nut setforth (mm)||∆S - saddle setback (mm)|
|E - 1||-0.3||1.3|
|E - 6||-0.5||2.9|