How to compensate for inaccuracy of equal temperament?

It is a widely recognized fact that equal temperament gives inaccurate intervals (particularly, the major third). People usually ignore it, but some people try to actually fix this.

I have heard that one easy way to improve your sound is flattening your B-string - this is claimed to improve how the "cowboy" chords (Am, C, D, E, G) sound.

Why does it work?

What other adjustments to string tension could I do? How does it depend on the key(s) I am going to play in?

(I am talking about the regular 6-string guitar with 12 frets per octave, without altering the frets themselves in any way).

• Mathematicians (and some musician) have been trying to square this circle for at least 1000 years already. Reading about the history is interesting, but you are unlikely to invent a "magic solution" that is better that those that are already known!
– user19146
May 10, 2017 at 17:41
• So, there are known solutions, which are in a sense optimal. This would be a good place to describe them. I know one solution (probably too weak to be of use), and I'd be happy to know why it works, and how it can be improved. May 11, 2017 at 8:09
• "Squaring the circle" is an imperfect analogy, since it isn't susceptible to compromise, whereas the practical application of tuning systems to the performance of harmonic music is. Oct 25, 2021 at 6:43

It's not exactly equal temperment that is to blame. It's the fact that a modern steel string guitar is the result of a design history that is not completely compatible with equal temperment.

There are two main factors at work. First, the way a guitar works is based on the idea that one fret position can create the right fundamental frequency for six different strings. Now this is true enough that people make and enjoy all kinds of music with guitars all the time, and have done so for over 100 years. But it's not actually 100% accurate. As our ears develop as musicians we can start to detect the inaccuracies of the fret positions and it can annoy us more and more, no matter how much we adjust the intonation set at the bridge or play around with different string gauges.

The second factor is that a long time ago, acoustic guitars were strung with gut, not steel, and jigs and templates for necks, bridge position, and most importantly fret positions were created for gut strings. Then when steel string guitars were developed, those jigs and templates were not always adjusted to match the radically different string thickness and stiffness of steel strings. Some of those fret position calculations, jigs, and templates have persisted to this day. That means a lot of nuts are positioned too far from the first fret, even on guitars from major manufacturers.

Combine that with the desire to make sure a new guitar doesn't buzz no matter how hard it's played, which means you need to cut the nut slots pretty shallow, and you create major intonation problems for the open chords (including the cowboy chords) on many steel string guitars that are fresh from the factory.

Some "quick" fixes are to cut the nut slots down a bit (at least for players who have a more appropriate touch) and even to move the nut about 1/32" to 1/16" closer to the first fret, which has the heart-stopping requirement of permanently removing a sliver of the fingerboard.

A more sophisticated fix is to have a compensated nut (e.g., Earvana, Buzz Feiten, etc.) professionally installed.

All of these things help (and there are many other options out there instead of or in addition to these!), but there is no solution to the problem. The guitar will always be an approximation of equal temperment. One thing that I believe makes the guitar a compelling instrument for people to play and hear is the very fact that it's "wrong". It has a subtle rebellious quality. It sometimes sounds a little close to just intonation, other times it's an approximation of equal temperment, and most of the time it's just a bit inharmonic everywhere. I think people have learned to like it that way.

• Yet another reason to make guitars with a zero fret. Although most of the major manufacturers don't.
– Tim
May 10, 2017 at 14:45
• @Tim Also probably part of the reason why Ian Anderson from Jethro Tull pretty much always used a capo. May 10, 2017 at 14:48
• I shall now look upon players who insist on capoing, in a different light!
– Tim
May 10, 2017 at 15:06
• Is this part of the reason why you can't just tune a guitar by open/5th[4th] fret? I absolutely cannot get happy tuning that way. I've discovered I do it best by simply playing open strings from the top E down, one second or so apart. I can get it right pretty much first time like that. May 10, 2017 at 16:44
• @Tetsujin if you tune your strings acoustically, the fourths will be two cents narrower than equal and the major third will be 14 cents narrower than equal. Assuming the frets give equal half steps, then any intervals between the G and B strings will also be 14 cents narrower than equal. Since the pure fifth is 2 cents wider than equal, a fifth between the open G string and the third-fret B string would be 16 cents narrow. Ouch. A fifth between any other two adjacent strings would be four cents narrow, which is not nearly as painful, but in higher registers even these will beat noticeably. Dec 9, 2019 at 19:35

There is no easy fix (other than compensated nut or "true" temperament frets). You can flatten B or (alternatively G string), which makes G open chord shape sound more natural, but it will result in bad sounding C or E chord (or others).

You can also try to add an overdrive (even small amount), which will cover the the imperfection (of course it is not a perfect solution). There are a lot of players, which heavily rely on open G even they use equal temperament using crunch or overdrive.

As to the flattening the B string, if you do that you will have to avoid a lot of chord shapes (especially typical open C).

• "You can flatten B..., which makes G open chord shape sound more natural, but it will result in bad sounding C or E chord (or others)": how much does one flatten it? Yes, flattening it to match the corresponding harmonic of the G string will make it sound very sour with the E strings, but what about flattening it just a little bit? This might improve the G chord without overly detracting from the E chord. As to C chords, however, flattening the B will widen the major third between the first-fret C and the open E string, which is probably not what you want for your C chord. Oct 25, 2021 at 6:52

If I am understanding this correctly (starting off from Todd Wilcox explaining that the problem is specific to the guitar's design, not equal temperament per se, that guitars especially have marked intonation problems), this here looks like a solution:

True Temperament guitars, aka "squiggly frets"

Those guitars do not have straight frets, but rather oddly shaped ones, to provide a perfect(1) position, per string, to produce the correct note. Higher up the neck it looks somewhat like "fan frets", but on the lower frets they are no straight "rulers" anymore but go back and forth.

Apparently this is fairly new and only expensive custom made guitars are available. Here are electric guitars, which are most known for this: strandberg

I also had a few google hits on acoustic ones with such a neck, and the true-temperament company seems to sell them for luthiers to fit them on guitars. But the google hits for specific guitar makers and products were only in the cache, the sites apparently don't have them anymore.

But it is apparently viable. The guy from the first video, Mattias Eklundh, plays his electric models with those frets to this day, and while his eclectic style may not agree with as many as some vanilla acoustic playing, you can see/hear the guy can play:

Freak Kitchen, fusion oriented(?), ~ 2015

Fusion, "Math Metal"

Overall, Eklundh makes comparably heavy use of "harmonics" (in those E-guitar playing styles, meaning that the string is struck in a way that makes an overtone predominant in the timbre, as opposed to the fundamental), which works so well and sounds so right, as he explains in some of his videos, exactly because the usual intonation issues that would prevent this from sounding good are not present in his guitar.

(1) "perfect" with regards to reaching e.g. equal temperament, which is not achieved on every position of a regular guitar

• oh, I only now saw that Mat already mentioned true temperament frets (however briefly) - should I delete my reply as it's not as new as I thought then? Dec 8, 2019 at 22:24
• I think your answer is fine. +1 to offset the downvote. My guess is that it looked like an advertisement to the downvoter. I don't agree with that, but I can partly see how one could get that impression. Dec 9, 2019 at 2:55
• "a perfect position, per string, to produce the correct note": this is impossible. If a given note is in the perfect position for one chord, there will be another chord for which it is in the wrong position. Dec 9, 2019 at 4:52
• "there will be another chord for which it is in the wrong position" - well, the guy demoing it claims that's no more the case. It could be that I picked the wrong video, I remember him demoing both, how things sound wrong with a regular guitar for some chords while others sound right, and then how everything sounds right on his special guitar. If that's not both in the 1st video, I'll try to find the right one later. But perhaps you mean "wrong" in as much equal temperament is wrong - but that's a lot less bad than a guitar is usually wrong, from my current understanding. Dec 9, 2019 at 8:35
• Well it depends I suppose on what you mean by "perfect." It's certainly true that if your fifths F-C-G-D-A are all acoustically pure that the F-major chord will have a very high third, not an acoustically pure third. Perhaps this "true" temperament is a meantone temperament that favors chords frequently used in guitar literature. Whatever temperament it is, though, there will be some intervals that are more out of tune than equal temperament, as a compromise to allow some to be less out of tune. Perhaps they're hidden in rarely used chords. Dec 9, 2019 at 19:46