The Tuner Is Right. So Is the Chord.

Imagine the small betrayal: all six open strings have earned their green lights. You put down a major chord, let it hang, and something still seems to rub. Back to the tuner. Still green.

There are at least two different things to investigate here. One is whether the guitar produces the notes it is supposed to produce when you fret it. The other is what “supposed to” means in the first place.

I wanted to separate those questions before reaching for either a screwdriver or a sweeping claim that guitars can never be in tune. The interesting answer lives between those reactions. A well-adjusted instrument can faithfully deliver a compromise. A badly adjusted one can add errors of its own.

What the green light promises

Assume a conventional chromatic tuner set to twelve-tone equal temperament, with A4 at 440 Hz. That qualification matters: tuners can use other references and presets. We are examining one particular promise, not every box with a needle.

In this system, going up an octave doubles frequency. The twelve semitone steps inside that octave divide the doubling into equal ratios. Each step multiplies frequency by the twelfth root of two. Four steps—a major third—multiply it by about 1.259921.

A simple-ratio major third uses 5:4, or 1.25. Close, but not identical. Measured in cents, where a semitone spans 100, the equal-tempered third is 400 cents and the 5:4 third is about 386.31. The equal-tempered version is roughly 13.69 cents wider. These are the interval relationships explained in Joe Wolfe's UNSW introduction to temperament, not evidence that your tuner has lost its mind.

The display is answering “Does this pitch match my reference?” It is not answering “Will every part of this chord line up into one unruffled vibration?”

A small disagreement at 1,100 hertz

Take an A at 220 Hz and a C-sharp above it. This is an idealized two-note example, not a report of a listening session.

At a 5:4 ratio, the C-sharp is 275 Hz. In equal temperament, it is about 277.183 Hz. The two possible C-sharp fundamentals differ by only about 2.183 Hz, but fundamentals are not the whole sound of a guitar string.

An ideal vibrating string has partials at whole-number multiples of its fundamental. As UNSW's explanation of strings and harmonics shows, the string can vibrate in several modes at once. Our 220-Hz A therefore has a fifth partial at 1,100 Hz. The 275-Hz C-sharp has a fourth partial at exactly the same frequency: 4 × 275 = 1,100.

Raise the C-sharp to its equal-tempered target, and that fourth partial moves to about 1,108.73 Hz. Now the two partials miss each other by about 8.73 Hz. If both are sufficiently present, their interference can produce amplitude fluctuations at that rate: beating.

That calculation explains a possible source of movement inside the interval. It does not tell us how conspicuous it will be on a particular guitar. Partial strength and decay matter; real strings also depart from the ideal harmonic model. Nor is this the same phenomenon as a Tartini difference tone, the auditory/nonlinear effect Wolfe uses to introduce his essay. Related territory, different mechanism.

The point is narrower and more useful than “this chord sounds bad.” Both fundamentals can land exactly where the tuner wants them, while some of their partials still refuse to coincide.

Why not just flatten the third?

For this pair of notes, we can specify the narrower third without difficulty. The trouble begins when we want the same small inventory of pitches to serve everywhere.

Stack three pure major thirds. Their ratios multiply:

5/4 × 5/4 × 5/4 = 125/64 = 1.953125.

But an octave is 2. The stack falls short. Three equal-tempered major thirds, by contrast, multiply to 2 exactly. Wolfe uses this same closure problem to show why temperament is a compromise rather than a failure to find the clever enough adjustment.

A fixed, octave-repeating twelve-note grid cannot make every major third 5:4 and also make those endpoints coincide. Other tuning systems and performers who adjust pitches in context have other options. This is not an argument that just intonation is impossible. It is an explanation of what this particular grid gives up to make the same interval ratios available across keys.

Flattening a third for one relationship is a choice about that relationship. It is not a proof that the rest of the instrument has become universally correct.

Then your finger arrives

None of that gets the physical guitar off the hook.

The neat fret-spacing calculation assumes that shortening the vibrating length does not also change the string's tension or mass per unit length. On a real guitar, pressing a string down does more than choose a shorter speaking portion. It deflects and stretches the string, changing its tension. String stiffness introduces additional departures from the simplest model.

Varieschi and Gower's study of intonation and compensation, published in the American Journal of Physics, develops this distinction mathematically and tests compensation one string at a time on a monochord sonometer fitted with part of a classical-guitar fingerboard. A spring-loaded device supplies the fretting action. That is actual laboratory work, though not a survey of every guitar or a test of listener preference. Their practical nut-and-saddle compensation is an approximation to corrections that otherwise vary by string and fret.

Compensation aims to bring the real instrument closer to its intended pitch grid. It does not remove the grid's compromise.

This gives a useful way to think about an octave check. In the ideal spacing, the twelfth fret halves the speaking length and doubles frequency. Equal temperament and a pure octave agree about that doubling. If the fretted octave is sharp against its target, blaming the major-third compromise explains nothing: this is an octave, not a third.

But one successful octave check is not a certificate for the whole neck. It samples one position under one touch. UNSW also notes that stiffness can put a real string's second partial slightly above an exact doubling, so even the familiar octave harmonic is not an infallible reference. Other frets and changes in touch can reveal more; no single observation identifies a unique repair.

Two questions instead of one accusation

My starting assumption was too simple: get the notes right and the chord should settle. Now I would ask two questions. Is the instrument meeting its intended pitches? And what relationships do those pitches choose?

The first can expose something worth correcting. The second can explain why correction does not mean abolishing every beat. Neither makes the other irrelevant. “It's all a compromise” can be as lazy as blaming every restless chord on the guitar.

And beatlessness is not the sole purpose of music. I have calculated an ideal interval here, not ranked real voicings or listened my way to a preferred tuning. The next question I want to test is what happens when the same major third moves to another register or another voicing: how much of the apparent problem belongs not to the notes we chose, but to which parts of their sound we bring forward?