A guitar tuned by harmonics hides a comma

Every instrument with open strings has to decide how they are tuned against each other, and three essays about the string quartet worked out what its decision costs. The tuning a string quartet cannot change found that tuning in pure fifths puts the open strings on a Pythagorean chain; the cello cannot hear its own tempering found that the chain cannot be tempered by ear at its lowest string; and an open string pulls the quartet flat found the chain dragging an unaccompanied quartet a few cents below the pitch it was given.

In all three the quartet had one freedom the arithmetic kept returning to: its stopped notes are not fixed. A violinist puts a stopped note wherever the ear wants it, so however the open strings are tuned, every other note can move round them.

A guitar does not have that freedom. Its stopped notes are wherever its frets are, and the frets are cut into the neck at an equal temperament. So a guitar’s open strings, tuned by ear against each other, have to agree with a temperament that is already fixed — and the tuning a guitarist can hear most easily is not that temperament.

Four fourths and a third close short by a comma #

Standard tuning is E, A, D, G, B, E: a fourth, a fourth, a fourth, a major third and a fourth, spanning two octaves. The arithmetic of tuning it pure is one line. Four pure fourths are $(4/3)^4 = 256/81$, a pure major third is $5/4$, and their product is

$$\frac{256}{81}\cdot\frac{5}{4} = \frac{320}{81} = 4\cdot\frac{80}{81}.$$

Two octaves is four. So four pure fourths and a pure third fall short of two octaves by exactly 81 to 80, a syntonic comma, 21.51 cents. No guitar tuned by pure intervals can put its high E on its low E’s double octave and its B a pure third above its G at the same time; one of the two has to give.

That is the same comma that somebody has to pay the comma found a just progression losing per cycle, and the same one the tuning a string quartet cannot change found between every third two open strings make and a just third. It turns up on the guitar because a guitar’s tuning is a cycle too — four intervals of one kind and one of another, required to close on a double octave.

Three ways to tune by ear, and where each puts the comma #

The quickest way to tune a guitar by ear is by harmonics. Touching a string at its fifth fret sounds its fourth partial, two octaves above the open note; touching the next string at its seventh fret sounds its third partial. When the fourth between the two strings is pure those two harmonics are the same pitch, so nulling their beat tunes a pure fourth — and the method is taught because a slow beat between two clear harmonics is easy to hear. Three variants cover what guitarists actually do.

Fretted unisons. Each string is matched to the string below it stopped at the fifth fret, and the B to the G stopped at the fourth. Every comparison is between an open string and a fretted note, so every open string ends exactly where the frets put it. This is equal temperament by construction, and it hides the comma by spreading it evenly, the way any equal temperament does.

Harmonics with a pure third. The four fourths are tuned by harmonics, and the B against the G’s fourth-fret harmonic — the G’s fifth partial, which is the same pitch as the open B’s fourth partial when the third between the strings is pure. Every interval is pure. The A lands 1.96 cents flat of its frets, the D 3.91, the G 5.87, and then the pure third takes the B another 13.69 below its fret, to 19.55 flat, and the last pure fourth takes the high E to 21.51 cents flat of its double octave. The whole comma is on the top two strings.

Harmonics with the B from a twelfth. The four fourths are tuned by harmonics, and then the B is matched to the low E’s seventh-fret harmonic, its third partial, which is a pure twelfth above the low E; the high E is tuned a pure fourth above the B. The high E comes out exactly on its double octave and the B 1.96 cents sharp. The fourths and the twelfth are pure, and the comma has gone into the one interval nobody tuned: the third between the G and the B, which is now a Pythagorean third, 21.51 cents wider than just.

The three are the whole choice. A comma cannot be tuned away by pure intervals; it can be spread across every interval, or put in the octaves, or put in the third.

Each open string against its own frets #

The most direct test a guitarist can make is the one the fretted method is built from: play an open string and the same note stopped on the string below. Tuned by fretted unisons the two agree by construction. Tuned by harmonics they disagree, and by the amounts in the figure.

The fourths contribute 1.96 cents each, the difference between a pure fourth and a tempered one, and at every string but the B that is all there is. A difference of two cents between two notes of the same name is at the edge of what a careful listener can hear, well inside the five cents how small a difference is audible put at the middle of the range, and a guitarist comparing an open string against its fretted unison would not notice it.

The B is a different matter. With a pure third it is 13.69 cents flat of the G string’s fourth fret, and with the B from the twelfth it is 7.82 sharp. The open B string is the one open string whose disagreement with its own fretted note is audible, in either method, and it is the string guitarists most often report retuning after tuning by harmonics.

The open chords are where it is heard #

A guitar is played in chords, and the open chords of the first position mix open strings with fretted notes on every beat. That is where the comma a tuning left becomes an interval.

With a pure third on G–B, the octaves are what suffer. In the open E major chord the E on the D string’s second fret and the open high E are 17.60 cents short of an octave, and so are the open B and the B on the A string’s second fret; in G major the open G and the G on the high E string’s third fret are 15.64 short. An octave 17 cents narrow is not a subtle interval. It beats visibly on a sustained chord, and it is exactly the complaint players make about guitars tuned carefully by harmonics: the chords sound worse than the tuning felt.

With the B from the twelfth the octaves are nearly clean — 3.9 to 5.9 cents wide — and the damage is in the one third that uses two open strings. G major’s open G and open B are 21.51 cents wider than a just third, a Pythagorean third, seven cents wider than even the fretted third that equal temperament gives every other third on the neck.

And with fretted unisons every octave is exact, every fifth two cents narrow, and every major third 13.69 cents wide — the familiar sound of a guitar in equal temperament, whose thirds are wide everywhere and nowhere worse than anywhere else.

What the ear hears is beats #

Cents describe an error; a guitarist hears a beat, and the beat rate is what decides whether an error is noticed. Each of the intervals above beats between the two partials that should coincide — an octave between the lower note’s second partial and the upper note’s first, a major third between the lower note’s fifth and the upper note’s fourth — at the difference of their frequencies.

With a pure third on G–B, the open chords’ octaves beat. E3 to the open E4 beats 3.33 times a second, the open G to the high E’s third-fret G 3.51, the A string’s B to the open B 2.49. An octave is the interval every listener expects to be still, and a strummed chord sustaining with a slow three-a-second wobble in its outer voices is exactly what “out of tune” means to a guitarist. The G–B third, the one interval the method made pure, does not beat at all — a gain in the one interval nobody listens to for beats.

With the B from the twelfth, the octaves beat 0.56 to 1.33 times a second, slowly enough to pass in a strummed chord, and the G–B third beats 12.21 times a second. The same third on an equal-tempered guitar beats 7.78 times. A third beating eight times a second is part of how a guitar’s chords sound; one beating twelve times is rough.

Fretted unisons give octaves that do not beat and thirds that beat 6.5 to 7.8 times a second, which is the reason the method is the one guitarists settle on however they begin. The unison is the coarsest thing in the room found an octave or unison the least sensitive interval to set by ear, and the arithmetic here is the other face of that: an octave is hard to tune exactly and easy to hear badly, and a method that puts a comma into it is the worst choice on the neck.

The equal-tempered answer is itself a way of dividing a comma. A fraction of a comma describes the meantone tunings that narrowed each fifth by a quarter of a syntonic comma so the thirds would be pure. Equal temperament narrows each fifth by a twelfth of a Pythagorean comma and leaves every third about two thirds of a syntonic comma wide — which on a guitar is the 13.7 cents in every third of the chords above.

Why a quartet never meets this #

The quartet’s open strings carry the same comma, and it shows up in exactly the same interval.

A quartet’s open C–E is a Pythagorean third, 21.5 cents wide of just, for the same reason the guitar’s twelfth-tuned G–B is: a chain of pure fifths or fourths produces Pythagorean thirds, and a Pythagorean third is a syntonic comma wider than a just one. The quartet’s cello C and violin E sit at the two ends of the chain, and the chain itself puts them 21.5 cents further apart than a just third and two octaves.

The difference is what each instrument can do about it. A quartet playing a C major chord does not have to use the open E. The violinist can play the E stopped on the A string, and put it wherever the chord wants it — a just third above the cello’s C if the players listen for it, or sharp because of where it goes if the line wants that. The comma in the open strings is a comma between two pitches the players can decline to sound together.

A guitarist cannot decline. The open chords are open because the open strings are in them, and the fretted notes around them are fixed. The guitar’s comma is a comma between pitches the instrument makes the player sound together.

That is why the quartet’s repair and the guitar’s are the same repair with different costs. Narrowing the quartet’s fifths to meet a keyboard makes its open thirds the keyboard’s, and the cello cannot hear its own tempering found that impossible to do by ear at the bottom of the chain. Tuning a guitar by fretted unisons makes its open strings the frets’, and it is easy — because the fret supplies the tempered note and the ear only has to match a unison, which a tuner counts beats with the fastest beat available. A guitar can be tempered by ear because it carries its temperament in its neck.

The intervals the arithmetic uses #

Every interval above is computed from small whole numbers and nothing else: a pure fourth is 4:3, 498.04 cents; a pure major third 5:4, 386.31; a pure twelfth 3:1, 1901.96. A fretted interval is a hundred cents a fret. Each method’s strings are built from the low E by the intervals that method matches, and each chord interval is the difference between the two notes’ pitches, open strings from the method and stopped notes from their string plus a hundred cents a fret.

What the frets and strings add #

Frets are not exact. A real neck’s frets are placed for equal temperament and then compromised: pressing a string stretches it and raises a stopped note by a few cents, which a compensated saddle offsets on average and not string by string. So the fretted unisons method matches each open string to a note that is itself a few cents sharp, and a guitar tuned that way is a few cents off its own ideal in a direction the arithmetic here does not model.

Harmonics are not exact either. A real string is stiff, and a stiff string’s partials are stretched sharp, more for thicker strings and higher partials. The fourth partial at a fifth-fret harmonic and the third at a seventh-fret one are stretched by different amounts, so a fourth tuned by harmonics is very slightly wider than pure, and each method’s numbers above move by a fraction of a cent to a cent or two.

Guitarists do not tune by one method. Many tune by harmonics and then adjust the B string to a chord they are about to play, which is a fourth method: choosing by ear, chord by chord, where the comma goes. The arithmetic says what each pure choice costs; it does not describe the compromise a player settles on.

What a table of cents cannot say #

That any of these is wrong. An equal-tempered guitar has thirds 13.7 cents wide in every chord and it is how the instrument sounds to most listeners. A guitar tuned with a pure G–B third has a sweeter G major chord and a sourer E major one. Which is better depends on what is played, and a table of cents does not know.

How loud each error is. A narrow octave between an open high E and a fretted E beats at a rate set by their frequency; a wide third between open G and open B beats more slowly and among many other partials. The beat rates are in the arithmetic and their salience in a strummed chord is not.

Whose guitars #

The six-string guitar in standard tuning is the case, and the arithmetic applies to every fretted instrument with a third in its tuning among fourths: the lute and the viol in their Renaissance tunings, which had the third between the third and fourth courses rather than the second and third, and the modern bass guitar, which has no third and so no comma — four fourths on a bass are required to close on nothing, and a bass tuned by harmonics drifts from its frets by only 1.96 cents with each string, 5.87 at its highest. The viol is the historical case worth naming, because its frets are tied gut and can be moved: a viol player could slide a fret to take the comma where the music needed it, which is the freedom a quartet has and a modern guitar has given up.

Still open: what an open tuning does with the comma #

Standard tuning is one choice of intervals among many. Open tunings — D A D G A D, D G D G B D, and the rest — replace the fourths-and-a-third with fifths, octaves and fourths chosen so that the open strings make a chord. Their open chords are the tuning’s own intervals, so a tuning by pure intervals makes those chords pure; and their fretted chords are still on an equal-tempered neck. Working out, for each common open tuning, which intervals a pure tuning leaves where on the frets, and how far its most-played fretted chords then sit from the open chord it was tuned to be, would say whether an open tuning moves the comma into the notes a player seldom frets or into the chords a player reaches for most.