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I've described how the nut, frets and bridge saddle must be in correct relative position for the guitar to play in tune. But in our 2-dimensional intonation problem, the height of each string above each fret is the other factor affecting intonation. Let's talk what happens when you bend the string by fretting it, on a guitar whose fretboard is less than perfectly straight.
Let's assume that we have a guitar whose nut, frets and saddle are placed at the theoretical ideal. That is to say, the frets are placed relative to one another according to the 12th root of 2 formula, and the nut and saddle are compensated to account for the general tendency for a string to go sharp when bent out of a straight line (all described in Intonation VI). One might think that the guitar should play perfectly in tune, right? Unfortunately, there are still at least three factors that may cause some notes to play out of tune:
We all know that bending a string out of a straight line raises its pitch, and we've seen that we do just that by simply fretting a note. That's why the nut and saddle must be compensated. But nut and saddle compensation apply globally to every fret along a given string.
For example, filing the saddle back a little under the high E string moves that string's saddle contact point away from every fret, and therefore compensates each fret by the same distance. Since the frets get closer together as you move up the fretboard, this distance of compensation is more significant the closer the frets get to one another. The 1st fret is compensated very little, but each successive fret gets a little bit more compensation up to the last fret, which gets the most compensation. This is just dandy, because the string gets further away from each successive fret as you go up the neck, so each successive note actually needs a bit more compensation than the last.
So if we assume a gradual and consistent increase in distance from each successive fret to the string, the saddle compensation applies correctly to each fret, and every note plays in tune. But what if this increase in distance is not gradual and consistent? If the neck has a dip or a hollow in it, or if one fret is worn more than its neighbors, the distance between the string and a given fret may be more or less than expected for the given compensation. And clearly, bending the string more or less means more or less sharpening of the pitch: a note that isn't in tune.
This is the kind of intonation problem found on older instruments whose necks have warped over time and whose frets need work. The solution is a complete refret, where the frets are removed, the fretboard is planed true, and new frets are installed, leveled and crowned. Players are often pleasantly surprised to find that a complete refretting not only makes their old friend play without buzzes, but play better in tune.
Change Those Old Ropes!
Modern guitar strings are a marvel of technology. We all take for granted that we can go to the corner music store and buy a high E string that's exactly .012" in diameter along its entire 30" length, and that's a pretty remarkable feat, especially for less than the price of a cup of coffee! But your guitar's intonation depends upon the perfect consistency of the diameter, tensile strength and density of each string, as it is exactly those properties which make it vibrate predictably.
The above is the "best case scenario", with a brand new set of strings. Fretting the string permanently kinks it slightly at each fret. Finger oils accumulate in the windings, thereby increasing its mass, but only where you actually touched the string. Abrasion and corrosion wears away tiny bits of the string's surface in similar places on the string. After playing them a bit, our technological marvels have entered the real world, and are no longer performing quite up to specs. If a guitar won't play in tune, the first thing to try is a new set of strings, and maybe a different brand or gauge, since they all intonate differently.
But even a brand new string has limitations. Any string gauge plays best in tune within a relatively limited range of scale length and tension. This is why medium gauge strings usually play better in tune that extra lights: a medium set on a guitar with standard scale length in standard tuning is in its optimum range to play in tune (that's also why it's so hard to bend blues notes on a medium set). The extra light set is really a bit floppy when tuned to standard tuning, so it's easy to play those blues bends, but also easy to unintentionally bend a note out of tune. This set is at a shorter scale length, or a lighter gauge, or a lower pitch, than optimum for intonation purposes; those gauges could play better in tune with a longer scale length or a higher pitched tuning.
At the other extreme, a heavy string gauge at high tension may run into another limitation: the stiffness of the string itself. One of my readers is exploring chordal playing in the high reaches of the bass guitar and is having a lot of intonation difficulties. While I have advised him how to adjust the instrument, I suspect that the stiffness of standard bass strings at such a short effective scale length (played well above the 12th fret) may cause them to play sharper than standard intonation techniques can compensate for. In fact, I have found it nearly impossible to intonate a medium-gauge low E string on a guitar so that it plays in tune above the 12th fret, while still playing in tune lower on the neck. String stiffness is a reality that unfortunately may limit what we can do at the extremes of our instrument's ranges.
In the next article...
I'll save the body resonance problem for Intonation VIII. There I'll tell you my weirdest intonation horror story to date, and my completely hypothetical explanation of it. Until then, I'll look for your questions and suggestions in my mailbox...
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