I’ve two steel-string guitars, a nylon-string, and an electrical. All are good manufacturers, properly maintained, with good strings, and have been arrange by my native luthier. I merely can not get them to play in tune all the best way up and down the neck. Irrespective of how rigorously I attempt, among the notes all the time simply sound out, particularly if I play with different devices. What am I doing mistaken? 
—Susan Richardson 

I really feel your ache. So many components of a guitar’s tone and vibe are subjective. Maintain, responsiveness, readability—these are intangibles that make one guitar a favourite and one other a dud. It appears as if tuning must be one factor we may merely measure—we have now digital tuners, in any case! Sadly, actuality is extra advanced, and fewer satisfying. 

I’ve had purchasers at each ability stage with this grievance. Whether or not seasoned gamers or golden-eared rookies, they got here in assuming that their tuners might be trusted to make every observe good, and in each case, I’ve needed to clarify utilizing an intermediate physics lesson.

Earlier than we get deep into the difficulty, let’s handle the commonest perpetrator: intonation and compensation. Totally different thicknesses of strings require totally different deviations from the theoretical bridge location. Many guitars have saddle areas which might be primarily a best-fit compromise that will get inside an allowable margin of error for the producer, for a mean set of strings. Since ultimate changes must be made for every particular participant’s most popular setup, it’s applicable for luthiers or techs to do that last-mile work after the guitar is bought.

If bridge changes can not appropriate the intonation, significantly on particular strings, it’s value wanting on the nut. First, confirm that the nut is appropriately positioned by capoing the guitar on the second or third fret and tuning these notes as carefully as doable. Then, take away the capo and see if the open strings are nonetheless in tune. If they’re persistently flat or sharp, the nut might not be positioned appropriately—an issue I’ve encountered on guitars at each worth level.

If the open strings don’t all agree, the guitar would possibly profit from nut-end compensation, through which the nut is adjusted on a per-string foundation. This can be a comparatively fashionable thought, and the physics have been confirmed mathematically by Trevor Gore. Empirically, the outcomes have been adequate to encourage aftermarket compensated nuts (such because the Earvana) and manufacturer-installed compensated nuts, akin to these on MusicMan guitars. Many luthiers haven’t but warmed as much as this method—if a straight first fret can intonate appropriately, it appears logical {that a} straight nut must also. I’ll confess to having been on this camp, although I not too long ago confronted a state of affairs that would solely be corrected with a single-string nut compensation and should admit that it was efficient.

Now, it’s time to face the elephant within the room: there’s a compromise constructed into
the tuning of each fretted instrument. All are constructed utilizing a system known as 12-tone equal temperament. On this system, the octave is split into 1,200 steps, known as cents. Every half step corresponds to 100 cents, and the connection between any two adjoining notes is a mathematically constant proportion (for many who are questioning, it’s the twelfth root of two, or 1.0594630943…). Multiplying the frequency of any pitch by that quantity ends in the frequency of the pitch one half step larger. Do that course of 12 occasions, and you’ve got gone up one octave. Guitar makers use the identical math to calculate fret-to-fret spacing on fingerboards. Appears completely cheap, proper?

Sadly, arithmetic has an objection, within the type of a unique system known as simply intonation. This method is mathematically elegant, primarily based on easy fractions and the equally swish physics of vibrating strings. This method traces again to Pythagoras and even prior, and is seductively easy. The octave is double the frequency of the foundation observe, and the string size is half. These easy fractions (2/1 and 1/2) repeat for the fifth (3/2 or 2/3), then the fourth (4/3 or 3/4), then the key third (5/4 or 4/5), and so forth. We’ve all encountered this once we play harmonics on an open string. The harmonic on the seventh fret (the fifth of the dimensions) divides the string into thirds, or three equal components. The fifth-fret harmonic (the fourth of the dimensions) divides the string into 4 equal components.

As with equal temperament, you’ll be able to apply these fractions each to the bodily string and to the observe frequencies, and that is the place the difficulty begins. If we begin from the identical observe and apply the mathematics of each methods, the ensuing notes don’t line up as they need to. For instance, if we start with a observe of 100Hz, then the simply fifth is discovered at 150Hz. If that instrument had a scale size of 25 inches, then the place for the seventh fret can be at 16.66666 inches.

Let’s examine that to the identical observe/fret with equal temperament. Ranging from 100Hz, we find yourself with a fifth observe at 100 occasions (1.05946 to the seventh energy), or 158.74. This can be a disagreement of almost 6 p.c. The fret location can be at 16.685 inches—a distinction of about half a fret width on a classic guitar. Among the notes within the scale disagree much more. The foremost third is the worst offender, with a disagreement of over 13 p.c between the 2 methods, equal to a distinction in fret placement of over 1/8 inch!

This has brought about issues for composers as properly. Early polyphonic devices had been tuned to simply intervals, which sound most harmonious in any given key, because the overtone sequence for the dimensions tones are likely to align mathematically. Nevertheless, that is problematic for any piece of music that features a key change or modulation—the E pure in the important thing of C main just isn’t fairly the identical because the E pure that’s the sixth (5/3) of G main. If a composer needed to make an easy modulation from G main to C main, the harpsichord would instantly be barely out of tune within the new key, audibly sufficient to trigger issues.

This age-old downside was well-known to the Pythagoreans, who wrestled with it with out success. Nevertheless, it remained largely theoretical for hundreds of years, as most devices had been both monophonic (flutes, for instance), fretless (ouds, violins, and so forth.), or capable of be fine-tuned (e.g., lutes, with their tied-on frets). Traditionally, string supplies additionally weren’t exact sufficient to persistently ship actual intonation each day.

With the arrival of contemporary polyphonic devices such because the harpsichord (and later, the piano), together with rising sophistication in composition, an alternative choice to simply intonation was vital. The answer: the equal-tempered system that’s now constructed into virtually each guitar we play. Put merely, the idea was to distribute all of the errors and disagreements evenly all through the dimensions, making every observe equally and minimally mistaken. The mathematical irrationality of the numbers concerned was offensive to many at first, and there was appreciable resistance to the concept that each observe in a scale might be one way or the other barely mistaken, and nonetheless would produce harmonious music.

J.S. Bach composed probably probably the most well-known piece of music to handle this conundrum. His The Nicely-Tempered Clavier from 1722 was meant to show the viability of the 12-tone equal-tempered system. It consists of 24 items, masking each key in each main and minor variations, and was meant to show {that a} single instrument was able to taking part in every bit with out retuning, and will produce a musically satisfying end in every key.

So what does this imply for you and your guitar? What I believe, and what I’ve discovered to be the case for a lot of of my purchasers, is that you’re each blessed and cursed with an ear able to discerning the compromises inherent in 12-tone equal temperament. It might be that you just use mixtures of open strings and harmonics in your music, or whenever you tune your guitar. If that’s the case, then you’ll by no means keep away from the disagreement between the naturally just-tuned harmonics and the equal-tempered fretted notes—the physics are merely towards us.

I’ve encountered just a few instances the place the mixture of the instrument and the selection of strings emphasised overtones within the notes, significantly within the low vary. Strings with thicker coatings are particularly vulnerable to this. In a single case, the open A contained a particularly distinguished and audible main third harmonic, which clashed badly with the fretted main thirds within the association. Once more, that interval is the worst offender, with roughly a 13 p.c disagreement between methods.

One of the best factor I can advocate is that this: get the instrument arrange as finest as doable, with a concentrate on intonation. This might embrace changes at each the saddle and nut ends. Experiment with string gauges, to fine-tune any remaining points particular to particular person strings. Subsequent, attempt to establish which notes sound mistaken, and the way. You could be listening to issues that may be resolved with adjustment and string choice. Nevertheless, armed with a bit of information about intonation methods, you might also uncover that you’re going through the identical downside that has plagued composers, mathematicians, and luthiers for over 2,000 years. Sadly, there isn’t any actual answer out there there.

In closing, I’ll provide this: these little idiosyncrasies of pitch are a part of what makes guitars sound like guitars. Each recording we’ve ever heard of a guitar included them, and they’re one of many colours within the general image of what the instrument is. A guitar that would break the legal guidelines of physics and sound precisely good at each observe, on a regular basis, could be good, however it may additionally not sound like a guitar anymore. Moreover, the nuances of pitch and vibrato that all of us incorporate into our taking part in, and which give every of us our particular person voice, are sometimes far more substantial than these little cut up variations. I get as shut as I can to make my purchasers’ guitars play in tune, however theoretical perfection is neither doable nor vital for making good music.