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Traditionally a builder selects a preferred scale length from nut to saddle, then based on that scale length, calculates fret placement according to “The Rule of 18” or more accurately “The 12th root of 2”. The instrument is strung up and a string is pressed down to a fret. When a string is pressed down to a fret it stretches. The stretching increases the tension causing the note to play sharp. The builder compensates for this by moving the saddle away from the frets, increasing the vibrating length and thereby causing the note to flatten enough to play in tune. At least in theory.

In reality each string requires a different amount of compensation according to its stiffness (not tension). Of course action also plays a role in all of this, but it is stiffness that has the greatest effect on compensation. Strings of higher stiffness require more compensation. The higher the stiffness the greater the tension increases when fretted. When I refer to stiffness I am referring specifically to the stiffness of the plain wire or core wire of a string. Mainly there are two things that determine stiffness in a string. First is a material property called the modulus of elasticity. And second is the physical size of the plain wire or core wire. And by size I am not just referring to diameter but also the length of the string itself from tuner to tailpiece (tension length). With all things being equal a longer string will be less stiff than a shorter string.

Instruments are traditionally compensated based on the pitch at the 12th fret only. A note is plucked at the 12th fret and compared to the open-string harmonic plucked near the same fret. The saddle offset is adjusted until both the fretted note and the harmonic play at the same pitch. So a fretted instrument plays in tune at the 12th fret. Unfortunately everywhere else it plays either a little sharp or a little flat.

The reason for this, and hence the Catch 22, is that during the compensation process the builder inadvertently increases the scale length when they shift the saddle offset to increase the vibrating length. This means that the open-string scale length is now longer than the scale length the frets were based on. A longer scale length string on a shorter scale length fret placement. The frets north of the 12th fret (towards the nut) are now a little too close to the saddle so they play sharp and the frets south of the 12th fret are a little too far away from the saddle so they play flat. And the farther away from the 12th fret you play the more pronounced this effect becomes. It is the process of traditional compensation that causes this. A better approach would be to move the nut offset and saddle offset together, maintaining the exact scale length distance between them. Though an improvement, this is not perfect either.

The compensation system I developed (MCS) starts with a basic desired scale length and string set. It looks at each string individually based on the string stiffness, fretting geometry, etc. The final result is a compensated nut, compensated fret placement, and compensated saddle that together produces nearly perfect intonation across the entire fingerboard. No extra parts, costs or sweetened tunings required. If it weren’t for the compensated nut you would never know anything was different about the instrument.

I would be more than willing to supply compensation numbers to any builder who would like to try it out and even incorporate it into their production models. You can read more about my process on the Guild of American Luthier’s website (http://www.luth.org), Web Extra’s for American Lutherie magazine Issue #116.

Gary Magliari

Interesting paper, I have done nut and saddle compensation as described Gore\Gilet books. I cheated, and used the same string set, base scale length and setup parameters as his example compensation. Both of your processes look for an optimal solution of scale lengths per string to minimise error with fixed fret positions. I do get much better intonation, especially in steel string guitars.

There was a PDF Tension-Compensated Fret Intervals that looks like a calculator. Is that available anywhere. I was looking to build an excel based calculator, I thought it might be possible to use Excel solver for the optimization routine, but have never gone through the effort.

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http://www.Harvestmoonguitars.com

Hopefully its OK to post a more direct link to the article for those interested:

The #116 Web extras page
http://www.luth.org/web_extras/al116/al116_extras.html

I need to read up, as I hope to employ nut compensation on my next build - a modern OM, redwood on Brazilain Tiger wood, with a Florentine. So many firsts.

I do have a comprehensive spreadsheet for my process but I'm not ready to hand it out yet. I spent many years perfecting it, however I would be willing to give you any compensation numbers you need if you want to use it yourself (free). I'm new to this forum and not sure how contacting each other works. Perhaps you can help me out in that area and I'll supply you with some compensation numbers.

Welcome to the forum Gary. I'm looking forward to reading about your process.

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Steve Smith
"Music is what feelings sound like"

Thanks Steve...

So if I change the set up (string height at nut, relief, action) or string set, I need a new nut, saddle AND fretboard? That would be a Catch 22.

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

Hi Trevor, I am aware of your work thru a friend of mine, TJ Turner. I salute you for the work you have done on intonation and lutherie.

Here is the point I was trying to make. It is a known fact that fretted instruments have a problem with inconsistent intonation throughout the fingerboard. The frets towards the nut end tend to sound sharp. This area being most noticeable because this is an area that is most commonly played. Through my work in the area of intonation and compensation I came to the realization that the root cause of this problem was a by-product of traditional compensation. This is the catch 22. A luthier in an attempt to compensate an instrument in the traditional manner fixes one problem but inadvertently creates a second problem.

Stand back a little and visualize it in this manner. Take a fret pattern calculated based on a 25.00" scale length. Now take that fret pattern and put it on a COMPENSATED string (nut to saddle) that is 26.00" long. A longer scale length string on a shorter scale length fret pattern. Move the fret pattern as a group until it plays in tune at the 12th fret. What happens at the remaining frets? Because the 25.00" fret pattern is shorter than it should be for the 26.00" string, the frets north of the 12th fret are too close to the saddle so they play sharp. The frets south of the 12th fret are too far away from the saddle so they play flat. My point being that the by-product of traditional compensation creates a MISMATCH between the open strings and the fret pattern and that's the catch 22.

Thanks, Gary.

To me, any compensation method that relies on moving frets around has severe limitations, for a number of reasons:

1) The fretboard is dedicated to one set up
2) Fret spacings have to be recalculated for each different job
3) It's an "averaged" solution across all strings if you want to use straight frets, so not a per-string solution
4) Standard fretting templates cannot be used
5) Fret slots have to be hand cut (more susceptible to error) or CNC'd
6) It's unnecessary

It's unnecessary, because, if you think about it, there are two ways to match scale length (for fretboard division purposes) to actual string length. Change the scale (and hence the fret spacing, as you advocate) or change the string length. With nut and saddle compensation, the string length gets changed anyway, and it can be arranged so that each individual string length can be matched to a standard scale length and give pretty well optimal results (given the usual limitations of frequency shifting due to sympathetic resonances etc.) This is how I advocate doing it. An objective function is written using the usual string equations and then a numerical optimiser is used to minimise the tuning errors for each fret, each string, essentially by "nudging" the nut and saddle positions about on a per-string basis until all the errors go away (are minimised). It retains standard scale lengths, straight frets and is done on a per-string basis. Seems to work pretty well...



(That's max ~0.5 cents error anywhere, optimised nut and saddle compensation, typical steel string set-up, as per "the book").

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

Trevor the only thing missing from this discussion is a couple of pints!

All points well taken. My data supports much of what you are saying. In my process the averaging or recalculating of scale lengths for each string to get two distant frets to match, such as the 3rd and 20th frets, ends up bringing all remaining fret positions (except for fret 1) within a few mils (0.001" = 1 mil) of each other before each fret position is averaged. Of course the first fret 1 ends up with all strings varying within a window of 0.030" which your chart seems to support also.

I've done my process a couple of ways. First I can create a completely optimized set of numbers for a specific setup. That includes compensated nut, fret pattern, & saddle. Secondly I have created a set of numbers for just a compensated nut & saddle while using a standard calculated fret pattern as you do. And I would agree with you that my fret positions from 2 on up the neck come very close to standard fret patterns. Of course the nut end of the fingerboard is always cut short a bit and the first fret would like to be repositioned.

1/2 a cent error! How is a person suppose to sleep at night?

I'm being admittedly quite hypocritical with all the charts I've made showing similar lines seeming to portray some precise results, but if we're to be honest with ourselves we should really be using range bars. Of course then a chart with a range of +/-5¢ would be mostly just a solid color.

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Eschew obfuscation, espouse elucidation.

While I am all for accuracy, I have to ask, what difference does it all really make when MOST human hearing has trouble distinguishing + or - 5 cents (or just slightly over 1 hertz difference)?

I use a compensated saddle (saddle canted ~ 2-3*, and top of saddle compensated) on my builds and just checked 2 of them and the most any note was off was 2 cents (most were nearly dead on). Now mind you, some people can hear that difference but in my experience not many. So, while it is good to know how to fine tune it I would save it for the most discriminating players and in turn save myself for the extra time and effort that goes into it.

Just my 2 (or in the case of my hearing - 5) cents:)

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"I've been had again"
Tim Benware
Creedmoor, NC

Sure, it's not for everyone, but I get plenty of people asking me to fix the intonation on their guitars. The issue is not so much the absolute pitch accuracy. If every one played single notes I doubt there would be much demand for this sort of stuff. The issue is really about the accuracy of intervals, where playing e.g. thirds in equal temperament is already pushing the boundaries of dissonance and another few cents error the wrong way tips it over the edge. For people who hear this stuff, every bit of extra precision helps. A guitar that plays equal temperament accurately is still "out of tune" (to Just temperament) but sounds a lot better than a guitar that won't play equal temperament accurately. Some guitars/players can get sufficiently accurate equal temperament without nut compensation (or fretboard compensation) and, of course, wonder why others bang on about this stuff. They're the lucky ones!

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

I have no absolute pitch recognition at all, but as Trevor says, it's the intervals where you hear the dissonance. I really only hear it on the B string, but it's enough to make me fiddle the tuning back and forth all the time trying to find the best compromise. I can't hear any problems on my compensated nut guitars, so I guess I just need one more notch of "close enough" than most people

DENNIS, you're hearing that "B" string more, assuming it's from a medium set, because it has very high stiffness. It has roughly the same stiffness as the bass "A" string. Higher stiffness means it has a higher increase in tension when fretted, therefore higher sharpening effect.

TREVOR, I tend to agree with you about customizing the fret pattern and whether it's worth the trouble or not. But when you (we) create a compensated nut and saddle you are optimizing it to a specific string set or type and specific setup conditions. So if the player switches to a different string set gauge that would certainly degrade the intonation and require resetting the nut & saddle offsets.

TO ALL, I'd like to go on record and define a couple of terms I coined during my work on intonation and compensation and use frequently to describe my work:

TENSION LENGTH - The entire string length under tension from tuner to tailpiece. Necessary for determining string stiffness.
OVERPRESS - The vertical distance pressed (centered between two frets) beyond or after a string first touches a fret. Important for determining deflection geometry.

Yes, in theory, when you use Trevor's method you'll need to change out the nut and saddle if you change the strings or action height. In practice it's much less of an issue, since you'd need to make fairly large changes in the strings or action to alter the intonation appreciably. If you are using custom fretting to get the intonation right then you'd need to swap out the fretboard to correct things when you use different strings, or action height, or tuning. Nuts and saddles are cheaper.

A few years ago I set up a 12-string for a customer. Since he normally plays in open G tuning I set up the nut and saddle compensation to match. In 'standard' tuning there was no fretted note on it that was more than three cents out, either the main strings or the octaves. I was worried that it might be too 'smooth', since it didn't have the 12-string 'crunch', but he liked it.

Hi Alan, The fretting is not as affected as you might think from string gauge changes. It's mainly the first fret that would like to move a bit, but that would be the same for any type of compensation system which is usually left in place as is. It's more the nut and saddle that are affected and their relationship to the fingerboard.

Thank you for your interesting work Magliari.

I might use your method although I really don't have a problem with intonation that is not fully compensated for by the players (buyers) who purchase my guitars. I make electric guitars mostly but also acoustic and not fretted instruments. I have a good ear for intonation and sometimes I am cursed by it.

I have an interesting approach when constructing a guitar. I don't do much repair but some restoration.
I copy an old fret scale exactly, usually the old Gibson scale. I measured it and put the frets there. For compensation, I set the bridge saddles where the strings ring equally in pitch with an open string and a fretted 12 fret. I don't use the 12 root method to calculate placement.

Some notes might play out of tune for some players when fretted at places on the fretboard but there seems to be a system of compensation by most professional and more expert players to "self-compensate". This system involves a little bending here and there of select strings to get chords into tune and to blend with other instruments. This is done by ear.

Having said that, I do believe there is a place for your system depending on the situation, whether in an ensemble, solo, steel string, nylon, electric, heavy gauge etc.

I think the real "catch 22" is the player's fingers. Some players disregard intonation by ear hoping that the fret placement is correct. I notice and hear this in a lot of beginners playing. I also hear some expert players who can compensate on the fly if a guitar is even badly or moderately in tune.

Again I value your work and don't want to discourage it in any way.

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Ken

Hi Ken. As a player I can relate to the concept of self compensating. When tuning I would do some averaging up the neck on the 3 highest strings. And sometimes I would adjust my tuning based on the song I was playing.

When I started this journey in 2004 I became very motivated to understand and try to find a solution for the problem of inconsistent intonation on the fingerboard. Having an engineering background I decided to approach the problem from that perspective. Instead of placing frets based on the 12th root of 2, I wanted the math to tell me where to place the frets based on action, string gauge, string deflection, desired pitch, etc.

In the end my work has taught me that the fret pattern created by my system is not that much different than traditional placement. The only exception is the first fret which varies quite a bit along with its relationship to the nut end of the fingerboard which likes to be shorter than normal. The biggest change from tradition is the requirement of a compensated nut. Along with the saddle the compensation at the nut follows the saddle to some degree. The higher the stiffness of the string the more the pair of offsets shift towards the saddle end just like in traditional compensation. Also when shifting the nut and saddle offsets together you more or less maintain the string's scale length which is important in my view.

However it sounds like you have a placement that works for you. And I would agree with you that the player can adjust or throw off the intonation depending on how they play.

Trevor I have your books and I've both read Gary Magliari's GAL article and talked to him in person after his GAL convention presentation on this topic, and I can guarantee you that the two of you have an absolutely huge amount of thinking in common about this.

I enthousiastically encourage you to read Gary's AL article.


If you can tolerate an enthusiastic amateur, I can offer the following reactions to your 6 points listed above:

1) Aren't all fretboard layouts affected by setup? I think both Gary's and yours should outperform standard intonation and fret layout practices even when the setup varies from the one the instrument was originally optimized for.

2) If the optimized fret spacings increase the accuracy of fretted notes then might it not be worthwhile?

3) Gary's method of using straight frets yet still calculating a per-string optimization is particularly brilliant in my opinion. To the best of my understanding it involves mathematical optimization of small scale length adjustments so that the error differences across each straight fret are still present in theory, but reduced below the threshold of hearing - say less than 1 or 2 cents. I'm hope Gary will correct or clarify if I'm getting any of this wrong.

4) An optimized fretting template can just as easily be made as a non-optimized one. If your intended point is that they can't be purchased from suppliers then yes that's a valid point, but certainly any machinist could cut one.

5) Sure but again, anyone serious about implementing Gary's method could have a template made. Perhaps someone wishing to simply try out his fret spacing could have an inexpensive plastic template laser cut for them.

6) Sure it's unnecessary, as no doubt many would argue about the compensation system presented in your books, but if it helps then why not try it?



I also have a question for Gary: Trevor and David Collins had a very interesting discussion about this topic in another thread and they seemed to differ only in their models of the end conditions of guitar strings: where David's model is based on a clamped end condition, and Trevor's is based on a pinned end condition. Do either of those conditions match with your mathematical model process?



I absolutely love seeing these discussions here from people who I really look up to and feel inspired by their pushing the boundaries of intonation performance on guitars.


I would love to see Gary, Trevor, David, and Mark French discuss intonation together. Mark has an article in AL #125 called "A Simple Modification to Reduce Frequency Errors in Guitars" where he presents a template for improved intonation using bridge and nut compensation along with a prescribed shifting of the first two frets. I'm sure all four of you have a huge amount of common understanding, but would have very interesting things to say to each other about your preferred methods!

Hi Durero (Leo). Thanks for your compliments and words of support. Your interpretation of point " 3) " is correct. Each string is calculated separately and then the scale lengths of each are adjusted until two distant fret positions line up (ex. fret 3 & fret 14). All remaining frets fall within a few mils (0.001's) of each other except for the first fret which has a broader tolerance window.

As far as point " 5) " there are a few builders that use my system. One had a template made for the StewMac fret slotting jig and the other two use CNC to cut the slots. One very important point I would like to make here is that regardless of what system luthiers use to place their frets, for good intonation it is imperative that they be placed using accurate means. Either with a CNC type setup or machine made template, etc. The idea of using a handheld square as a saw guide against the edge of a fingerboard will almost always introduce noticeable errors.

Your question about "END CONDITIONS", I base my wave model on pinned ends. However string stiffness is based on tension length, from tuner to tailpiece. I see that you mentioned Mark French. I had the pleasure of meeting him at my GAL presentation. He featured some of my work in his book, "Technology of the Guitar".

You also mentioned Trevor, Mark, David, & myself maybe having a discussion about this subject. There are others out there doing similar work like Greg Byers, etc. I might have something better. Wouldn't it be great fun to have a FRIENDLY show-down or competition amongst all the Masters of Intonation, LOL. Maybe it could be sponsored by GAL or some other luthier magazine or organization. A set of rules would have to be established along with a standard model for testing. A comparison to traditional methods could also be apart of this. I think it would generate great interest for builders, musicians, enthusiasts alike.

I can understand if some builders would shy away from something like this. Their reputations (and sales) might be based on their intonation PROWESS.

Thanks, Leo, for investing in the books. And yes, I've read the AL articles by Gary and Mark French. As Mark acknowledges in his AL article, I alerted him (in a discussion at the Seattle conference of the Acoustical Society of America, where we met in 2011) to the idea of half the compensation at the nut and half the compensation at the saddle as a simplified method of implementing nut compensation, and it's written up in the book in Design Section 4.7.5 where it first appeared. In the AL paper, Mark erroneously said that my method involves moving the frets around, which is incorrect and he apologised for that error when I informed him of it. Another point where Mark and I differ is in the intonation error he computes for frets 1 and 2 where he advocates shifting those two frets to reduce the error. My modelling doesn't show any need for fret shifting (less than +/- 1 cent error, all strings all frets, Fig. 4.7-21 in the book), but Mark hasn't published details of his modelling, so I can't check it.

Regarding Gary's work, I think you may have misinterpreted my remarks about fretboard layout as a means of improving intonation. The scale length (and so the fret spacing) can be matched to the string length by changing one or other of them. Gary changes the scale length, I change the string length and use an optimiser to seek the least intonation errors over all the frets on a per-string basis by moving the nut and saddle around (i.e. nut and saddle compensation and matching string length to scale length happens all together, on a per string basis). This means a standard fretboard (and it's associated fretting template) can be used rather than having to make a new template (or write a different CNC routine) every time a different set-up or string set is used. As a builder, having just a small number of standard fretting templates (and not having to re-compute fret spacings for every different guitar set-up) is a major benefit. That is what I meant about fret spacing adjustments not being necessary. The same effect can be achieved in a different way, retaining standard scale lengths, straight frets and an optimised solution for each string. So, to me, changing the fret layout is just an unnecessary complication which doesn't appear to yield any better or even as good (modeled) results. Check Gary's Fig. 9 in his AL article vs. my chart in this thread (Fig. 4.7-17 in the Design book. Note the different chart scaling!)

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Trevor Gore, Luthier. Australian hand made acoustic guitars, classical guitars; custom guitar design and build; guitar design instruction.

http://www.goreguitars.com.au

We need to keep in mind in all of this that 'perfect is the enemy of the good', and it's probably not possible to achieve 'perfect' intonation on a guitar in any case. As has been pointed out, all you need to do is be a bit less than perfectly consistent with left hand pressure and things will be out, often by a couple of cents or more. And, again, we're usually dealing with Equal Tempered Tuning, which is out as compared with 'just' tuning anyway, and sounds it. On top of all of that, most of my customers use more than one tuning anyway. There's no way you can alter the guitar every time they change the tuning, and I can't convince them to buy a different instrument for each tuning and carry it around, so...

A final point; if you look into it, a guitar with a reasonably well done nut and saddle compensation job will play at least as well 'in tune' as most other instruments. Pianos can be terrible, and most instruments have issues of one sort or another.

Hi Alan. In traditional compensation there is quite a bit of error built into an instrument right out of the box. My primary motivation was to try and design-out the inherent error in traditional construction and compensation before the instrument is picked up. Of course at some point we have to account for the human interface. That can vary quite a bit. So your point is well taken about how perfect is perfect.

As far as various tunings, I’m no more concerned about different tunings than I would be with traditional compensation. However string gauge changes will change the intonation profile like they would for any type of compensation.

And for your final point, I do agree that both nut and saddle compensation by itself goes a long way in producing well compensated instruments far beyond tradition. If anything our work has proved to us is that a compensated nut should be AS integral to modern instrument construction as a compensated saddle.

Gary Magliari