A guitar cannot be in tune

Six essays of this site’s longest ladder are about a discrepancy that exists in arithmetic and has to be put somewhere. This one is about what happens when that arithmetic meets a physical object with frets in it, and the answer is that two more errors arrive and the correction available is smaller than the problem.

The first error is the temperament, and it is the largest #

A guitar’s frets are placed at the twelfth root of two — equal temperament, built into the object. That is one of the four classical answers to where the comma goes, and its price is stated: every major third is 13.7 cents sharp of pure, every minor third 15.6 cents flat, and every fifth 2 cents narrow.

Those are not small against what a listener can hear. The difference limen is about five cents, so a guitar’s thirds are nearly three times it away from just. They are, however, inside the width of the category — a third seventeen cents off is still a major third — which is the whole reason equal temperament works at all.

Every fretted instrument in twelve-tone equal temperament carries this and it is not a defect of guitars. It is the floor the rest of the essay sits on.

The second error is that pressing a string down sharpens it #

Fretting is not simply choosing a length. The string is above the frets by half a millimetre at the nut and two or three at the twelfth, and pressing it down to the fret makes it travel further than a straight line between its ends.

A longer path at the same stretched length means more tension, and more tension means a higher pitch. The size of it turns on a quantity that is easy to get wrong: a fractional stretch is not a fractional frequency change. A string is stiff in extension, so the tension rises by roughly a hundred times the fractional stretch, and the frequency by half of that.

Taking that ratio as one, which is what happens if the tension is assumed constant, gives 0.05 cents at the twelfth fret — a hundredth of the real figure, and small enough to conclude that compensation is unnecessary. It is not; guitars are compensated, visibly, and the compensation is millimetres.

The third error is that there is one correction per string #

A luthier’s remedy is to move the saddle back, lengthening the string slightly so that fretted notes come down to where they should be. That is one adjustment per string, and it has to serve every fret on that string and every chord that uses it.

A fixed length added at the bridge is a different fraction of every fret’s sounding length — a small correction at the first fret, where the string is long, and a large one at the twelfth, where it is half as long. So a compensation set to be right at one fret is wrong at the others, in a direction that is entirely predictable and cannot be avoided.

This is the same structure as a wind instrument’s tuning slide, where a fixed length is added to a tube whose sounding length shrinks as the notes rise. Two instruments, no shared mechanism, the same shape of residual — and in both cases the remedy is a length and the problem is a frequency.

What the best possible setup leaves #

The interesting question is not how badly a particular guitar is set up. It is what the best achievable setup leaves behind, and that can be searched for.

Take eight common open chords. For each, compute what every string actually sounds — nominal fret position, plus the fretting sharpness, minus the flattening from whatever compensation is set — and measure the worst departure of any interval from the just interval the chord’s name implies. Then search over per-string compensations to minimise the worst case across all eight.

The search returns zero on every string. It cannot find a compensation that improves the worst case at all, and the worst case it is stuck with is 15.6 cents — on the E minor shape, whose minor third is 15.6 cents flat of a pure 6:5 by construction. Uncompensated, the eight shapes run from 13.5 to 15.6 cents. Compensated as well as a search can manage, they run from 13.5 to 15.6 cents.

That is the essay’s result and it is worth stating flatly. The saddle has nothing to work with, because the quantity being measured is almost entirely equal temperament’s own thirds, and a saddle moves a string’s length rather than a temperament’s intervals. The thing guitarists blame for their instrument being out of tune — the setup — is worth about a cent of the fifteen. The rest is a decision made about a comma several hundred years before the instrument was built.

It is worth being precise about what this does not say. Compensation is not pointless: it is what makes a fretted octave agree with an open string, and an uncompensated guitar is audibly wrong at the twelfth fret in a way a compensated one is not. What the search shows is that the residual error in a chord is not the setup’s to fix. Two different measurements, two different answers, and conflating them is why the subject generates so much argument.

What the search finds when the objective changes #

The worst case is one objective among several, and the caveat at the foot of this essay says so without saying what the others give. They give different answers, and one of them is worth the whole section.

The three-cent gain looks like a refutation of the headline and is not. Look at where it comes from: of the eight shapes, six barely move and two collapse — E major from 14.45 to 2.46 and the barred F from 13.52 to 2.13, while A minor gets slightly worse. Both of those shapes put their major third on the G string, and the setting the search chose is 4.25 mm on that string alone.

Four and a quarter millimetres is not a compensation. On a 648 mm scale it flattens the G string by 12.0 cents at the first fret, 12.7 at the second and 22.6 at the twelfth — which is very nearly the syntonic comma at the bottom of the neck and a quarter-tone at the top. What the optimiser has done is not set up an instrument. It has detuned one string by a comma so that the two chords whose thirds live on that string come out pure, and left it a comma flat everywhere else.

Which is exactly the fourth remedy this essay lists, arrived at by a search that was not told about it. Flattening the G string a couple of cents to favour some keys over others is what guitarists do by ear, and the search rediscovers it the moment the objective stops being the worst case. It also prices it: the useful amount is twelve or thirteen cents rather than a couple, and the cost is a G string that is unusable above the fifth fret.

So the headline survives in the form it was stated. A saddle setting that serves the whole neck buys nothing in a chord. A saddle setting that buys something in a chord is not serving the whole neck — it is choosing where to hide a comma, and the choice has to be remade for the next song.

Those are small numbers beside the ones below, and they are worth having separately because they are the part no fret placement can fix: the sharpening happens after the string length is decided, so it is added to whatever the fret geometry gives.

The measurement that made the bug visible #

A note about how this result was arrived at, because it nearly was not.

The first run of the search reported a worst case of 1,200 cents on two of the eight shapes — G major and the barred F. Twelve hundred cents is an octave, and an octave is not an error; what had happened was that both shapes have a note exactly an octave above their bass, the reduction into the octave produced 1,199.98 rather than 0, and the comparison against a unison’s 0 reported the whole octave as a discrepancy.

It was visible only because the number was absurd. Had the shapes been arranged so that the artefact came out at 40 cents instead of 1,200, the figure would have printed it, the essay would have explained it, and it would have been a finding about guitars that was in fact a modulo.

A number large enough to be obviously wrong is a gift. The dangerous version of the same bug is the one that lands inside the range a reader would believe, and there is no reason to expect this one to have been the only instance — which is why the search’s output is reported here to two decimals and why the per-shape values are on the figure rather than summarised.

What the players who do fix it are actually fixing #

Several partial remedies exist, and each attacks a different one of the three errors.

Nut compensation — moving the nut, or filing its slots — attacks the fretting sharpening at the low frets, where it is proportionally largest. It buys a cent or two in first position, which is where most open chords are played, and it is the fix with the best ratio of benefit to cost.

Per-string saddle compensation is standard on electric guitars, where each string has its own adjustable saddle. It attacks the octave: set the twelfth-fret harmonic and the fretted twelfth to agree, per string. That is a real and useful adjustment and, as the search above shows, it does not help the chords much.

True Temperament fretting replaces the straight frets with wavy ones, giving each string its own fret position at each semitone. That attacks the temperament itself — it can put a different tuning under each string — and it is the only remedy that reaches the largest of the three errors. It requires a different fingerboard.

And a player’s own adjustment: tuning the guitar slightly off equal temperament to suit the key of the piece, which is what most guitarists do without describing it that way. Flattening the G string a couple of cents makes open E and open G chords sound better and open D worse, and that is choosing where to hide a comma, performed by ear, on a schedule of one decision per song.

The low string is wound, and why that is here #

One further piece of guitar arithmetic belongs to this field even though it is not about tuning, because it is the same kind of forced decision.

A guitar’s bottom E is 82 Hz on a 65 cm string. A plain steel wire of ordinary gauge at ordinary tension would need to be several metres long to sound that note; making it thicker instead works for the frequency and ruins the sound, because inharmonicity grows as the fourth power of diameter and a thick plain wire’s partials go sharp fast enough to stop being one note.

Winding also affects this essay’s subject directly. A wound string has a different stiffness-to-tension ratio from a plain one, so it sharpens by a different amount when fretted — which is why compensation is set per string, and why the largest compensations on an electric guitar are on the wound strings.

Why a violinist does not have this problem, and a singer has it less #

The comparison is the sharpest way to see what a fret costs.

A violin has no frets. A player’s finger can go anywhere on the string, so the intervals available are continuous and a violinist adjusts them by ear toward whatever the passage wants — pure thirds in a sustained chord, sharpened leading notes in a melodic line, and neither of them equal temperament. The bow’s full harmonic spectrum gives them beat-based feedback precise to a couple of cents, which is far finer than the errors this essay is about.

A singer has the same freedom and the same feedback, which is why a cappella ensembles drift: given the freedom to tune every chord purely, a progression in pure ratios does not come home, and the drift is a comma per circuit. That is the cost of not having frets, and it is the reason fixed-pitch instruments exist.

So the instruments divide into two families with opposite problems. A fretted or keyed instrument commits to a temperament and is therefore always a fixed amount wrong. A continuous-pitch instrument can be exactly right at every moment and cannot stay in one key. Six essays of this ladder are about the first family’s choices; this one is about what happens when the choice is fixed in wood, and the drift essay is about the second family’s.

The other family has a version of the same problem. A progression tuned in pure ratios at every step arrives a syntonic comma below where it started, and a guitar cannot do this — its frets will not let it — which is the one respect in which a fixed-pitch instrument is better off than a flexible one.

What the picture cannot show #

The model has two free parameters and they are not measurements of any guitar. The action taper and the stiffness-to-tension ratio are typical values, stated in the figures. A high action or a heavy gauge makes the fretting sharpening several times larger, and a well-set-up classical guitar with nylon strings has a quite different ratio again.

The search is coarse, and it makes no difference. It steps each string’s compensation through seventeen values over four millimetres and does two passes. Re-run at 241 values over six millimetres with four passes it returns the same answer — all six strings at zero, worst case 15.641 — so the coarseness is not what is producing the null result. The objective is dominated by a term the compensation cannot reach, and the resolution of the grid has nothing to do with it.

The decomposition by interval class is the cleanest statement of that. Over the eight shapes at zero compensation, the unisons and octaves are wrong by a mean of 0.12 cents, the fifths by −1.83, the major thirds by +13.76 and the minor thirds by −15.26. Equal temperament’s own figures are 0, −1.96, +13.69 and −15.64. The fretting sharpening’s entire contribution is the difference between those two lists, which is under six tenths of a cent.

The chord set is a choice. Eight open chords in first position is what a guitar mostly plays and it is not everything a guitar plays. Barre chords, which are all the same shape transposed, have a different and more uniform error profile.

And the worst-case measure is one choice among several. Minimising the worst interval is not the same as minimising the average, and a luthier tuning by ear is optimising something else again — probably a weighted sum with the thirds counting for more than the fifths, because a rough third is more noticeable.

Whose instruments, and when #

Fretted instruments have been argued about since fretting existed. Vincenzo Galilei advocated equal temperament for the lute in 1581 partly because a fretted instrument makes any other temperament awkward — a fret crosses every string, so a fret position is a commitment for all of them at once, which is the constraint this whole essay is about, four hundred years earlier.

The compensated saddle is nineteenth-century on guitars and older on other instruments. Per-string adjustable saddles arrive with the electric guitar in the 1950s. True Temperament fretting is from the 1990s.

The claim that most of a guitar’s error is the temperament is a claim about a fretted instrument in twelve-tone equal temperament playing triadic music. It is not a claim about fretted instruments generally: a sitar’s movable frets, a saz’s microtonal fretting and a fretless oud are three traditions that made different commitments and do not have this problem in this form.

Where this goes #

This is the seventh rung of the comma ladder and the first to be about an object rather than a system. What it establishes is that the four classical answers are not the end of the question — an instrument that adopts one of them still has to be built, and the building introduces errors of its own that the answer did not anticipate.

The instruments field’s last essay leaves the instrument entirely. What is radiated has still to reach somebody, and it does not go in all directions equally — so the spectrum a listener receives depends on where they are sitting, which makes a recording a choice of seat rather than a neutral record.