Stay in Tune vol # 103
By LuthierPatrick Podpadec
As I have looked back at many of the issues that I have written for the Voice I haven't mention very much about String compensation or Intonation .This oddly enough is the whole reason our instruments "Stay In Tune" These to factors are what make our instruments "sound in tune as best as our ears can hear it. Although the theory can be a little intimidating if your not a math major , but I will attempt to explain it to you in layman’s terms the best that I can.
Many, many years ago, Pythagoras, the Greek mathematician had figured out a formula that would divide a string length so that the 12th fret would land precisely at the exact center of a given string length. This would produce a perfect octave above the note being played by each string. This mathematical formula is called the 12th root of 2. In essence how it works is when you divide the string length by a constant denominator (17.817152) you will reach the the distance from the nut to the first fret. When you subtract that distance from the string length and again divide the remaining length by 17.817152 you will arrive at the distance to the 2cd fret, and so on and so on. This theory and formula has been used by luthiers for centuries with a certain amount of relative accuracy. The problem that occurs is that his "equal tempered tuning" does not account for the many other variables that are in the equation of a vibrating string.. The formula assumes an "ideal" or "perfect" string - one which only exists in theory, not in the real world. It assumes, firstly, that the strings have no stiffness. Secondly, it assumes that all strings behave identically, regardless of their thickness, whether they are plain or wound, and the material they are made of. Thirdly, it assumes zero string height - and completely ignores what happens when the strings are pressed down on the frets.
The frequency of a vibrating string is determined by three factors: the string length, its mass, and the tension applied. All three of these factors are affected to varying degrees when a string is pressed down on a fret. Along the neck, the length and mass decrease by 50% per octave. Changing the length affects the stiffness. The tension is affected by fretting the string, as the string height is not zero. Pressing the string to the fret stretches the string slightly, increasing the tension and thus sharpening the notes produced.
The strings themselves vary considerably in diameter and construction (plain or wound), and thus react differently to being fretted. One single adjustment per string at the bridge ("intonation") cannot possibly fully compensate for all these variables at once, as they all vary in different degrees on different strings.
The way that luthiers have attempted to solve this problem is by adding a little length on to the thicker strings ( because of their mass) so that when you play notes farther up the scale that they are"compensated" . this is why saddles on most stringed instruments are set at an angle. This unfortunately only solves some of the problem. To most musicians this has been accepted as the "norm" and players have learned or trained their ears to work around the slight intonation variables that occur during normal play.
Many new attempts have been made in the past to correct this problem. Some manufactures are trying to "compensate" the nut along with the saddle to fix the problem. Again, this only fixes certain frequencies of certain strings at certain areas of the fingerboard. I'm also a little leery about the accuracy of this system once you have added a capo. It seams to me that any compensation at the nut would be eliminated once you added a straight capo and play in a different key. Do keep in mind that all of these very small changes are for the most part not able to be heard by the average ear. There are some musicians though that this minor intonation problem drives them crazy.
The latest attempt that I have come across is to bend each individual fret to compensate for each frequency. I guess that might work until you decided that you might want to change the diameters of your strings, not to mention the nightmare that it causes for the luthier to try to install the fret system. There is a company in Europe that has developed a system called Dynamic Intonation™, and Curved Frets™.and they will customize your frets for about 800.00 to 900.00. That is usually out of the ballpark for any of my clients.
If you are interested in this sometimes very complicated discussion of "True" intonation I have listed a few websites that may be able to shed some light on the subject.
I do understand and applaud the many luthiers and musicians that have spent years of their lives trying to reinvent the "intonation wheel", but I personally don't think that many of the attempts that have been made to create a better intonation system for the modern day guitar have improved the problems to the degree that I would want to change what Pythagoras figured out in 500 b.c..
Although the guitar's intonation has been a problem for some musicians with "perfect pitch" it never seemed to cause such a fuss for other musicians such as Segovia and the likes of him. As for now I'm still gonna stick to the traditional "Equal Tempered Tuning " It works for me. So until next time , please try to "Stay in Tune".
Patrick from Liam guitars/ Wood-n-Strings