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Question: What effect do compensated nuts have, and why would you want to compensate both the nut and saddle? In short, why are compensated nuts a thing?

Now for some physics-based background on intonation:
In principle, if strings were negligibly small, there would be no need to compensate them, or it adjust their lengths so that their notes are aligned with the frets. The fundamental frequency they would produce is a simple function of their length, tension, and mass (see http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/string.html.) However, the finite thickness of strings introduces additional effects:

A) Strings are held fixed at both ends of a guitar at the fulcrum of the saddle and nut. Consider the saddle: here, the portion of the string going to the headstock can vibrate, but the portion going the other way is fixed. Now, string wire does not like to bend at acute angles -- it will instead make a radius. The thicker the string, the larger the radius (that's the reason you buy string that's coiled in a loop, and bass strings are larger loops than guitar strings). This radius effectively makes the string stiffer near the saddle and nut, making its effective length for calculating the fundamental frequency slightly shorter. This effect scales with the thickness of the string, i.e. the thicker the string, the shorter its effective vibrational length.

B) There's a secondary effect related to (A), which is that a string typically makes a small angle as it passes over the fulcrum of the saddle toward the headstock. There is some radius to this angle, and the thicker the string, the larger the radius. This contributes to the effect of thicker strings having shorter effective vibrational length.

For (A)-(B), there is an interesting caveat for wound strings vs solid ones, the former having a smaller bend radius compared to a solid one with the same thickness, but in practice the effect on intonation is still present.

C) Finally, there is the effect of intonation of thick vs. thin strings when fretted. The action on thick strings is typically higher because they have larger deflection, so they must be pressed more to touch the fret. For a fixed string length, the higher the action, the higher in pitch the string will sound when fretted. Thus, thick strings will sound sharp relative to thin ones. However, this is a relatively small effect, it probably contributes an order of magnitude less than effects (A)-(B), if at all.

Given this, I don't see why there would be any reason to have a compensated nut over a compensated saddle, or why you would want both compensated. There may be aesthetic or practical reasons, but not physics-based ones. Your thoughts?

I'm not convinced compensated nuts are worth the trouble, but if the client wants one... okay. Don't feel the need to put one on any of my stuff.

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"Act your age, not your shoe size" - Prince

There have been several articles in American Lutherie about the theory behind compensated nuts (and possibly moving the first couple of frets). I understand the physics but don't see any compelling reason to do it. I keep thinking that as soon as I fret a note the nut is out of the picture (of course the saddle isn't).

Partly true as the length of the string is still the length given by the compensated nut!

Ignoring all of the physics and the like... Nut compensation does two things, It helps deal with the difference between a fretted note and an open note. The fretted note first goes to the fret and then up to the saddle, the open note is straight. The second way it helps is to spread the intonation error more evenly across the frets. When doing 12th fret intonation the 12th fret is correct but as you go down the neck toward the nut the intonation error increases. Intonating on both ends allows one to spread the intonation errors evenly across all of the frets.

Most guitars either do not have nut compensation or have a little bit if the nut nut end of the fretboard is cut at the zero fret. People get used to hearing the intonation errors when playing the first few frets. I can see it with a tuner though.

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http://www.Harvestmoonguitars.com

Nut compensation has been discussed previously on the forum. Perhaps the best treatise on the overall subject is within the Gore/Gilet books — worth the read even if you have no intention of messing with nut compensation.

The objective of any approach to compensation is to minimise the errors in pitch along the string at each fret. If haven’t already done this experiment, it’s worth using a good tuner to establish the error (up or down in cents) for any given guitar at EVERY fret. If you’ve intonated the guitar using the familiar method at the 12th fret, you should of course have no intonation error for that fret, but you will be amazed how variable are the errors at the other frets. You’ll also be amazed at how different those errors are with different string gauges/materials (especially with classical guitars).

Gore describes an approach to intonation (with or without nut compensation) that attempts to minimise overall variance in intonation errors across all frets. I have that programmed into an Excel spreadsheet, but I only use it for classical guitars, where the 12th fret harmonic approach is notoriously bad.


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A guitarists technique is worth considering as well. I realized quickly that how I attack the strings with my fretting hand can drastically affect how the guitar registers at any given fret.