Tune a guitar for the shape of its key's open chord

Tuned for one shape, a cent is worth five frets took a guitar tuned so that the E-shape barre chord is exactly just wherever it is moved, and searched every way of fingering a I–IV–V ground in E. The route that keeps every chord in the E shape is exactly in tune and moves the hand twenty frets; any other fingering is between twelve and a half and fifteen cents out; and a player should keep to the shape if a cent of the piece’s mean error is worth more to them than about five frets of movement.

That essay took the E shape as given, because the earlier searches had found it for a piece in E. But a guitar can be tuned for any of the five moveable shapes — the open C, A, G, E and D chords, moved up the neck — and each choice makes a different fingering of every chord just and puts it somewhere else on the neck. The open question was which shape a tuning should serve in a given key, and whether the answer is a fact about the key, the progression, or neither.

It is a fact about the key, and a simple one. In every key whose tonic is an open chord, the shape to tune for is that chord’s own, and the reason turns out to be the one thing the moveable shapes cannot do: an open chord needs no barre.

Seven keys, five shapes #

The piece is the ground of the essay before — I, IV, I, V, IV, I, for sixteen, eight, eight, four, four and eight beats — played in seven major keys. For each key and each of the five shapes, tune the guitar to make that shape exactly just, and find the route that keeps every chord exactly in tune, which on a tuning for one shape means playing every chord in that shape, choosing only where on the neck.

The pattern is clean. In C major the C-shape tuning moves the hand 20 frets through the piece and every other shape moves it 24 to 28. In D major the D shape moves it 16, against 20 to 28. In G major the G shape, 16. In A major the A shape, 16. In E major the E shape and the A shape tie at 20. In each of the five keys with an open tonic, the shape of that open chord moves the hand least.

The two keys without an open tonic confirm it from the other side. F major and B♭ major have no open chord for their tonic in any shape, and there the best any tuning can do is 24 frets. Their shortest just routes are four to eight frets longer than those of the open keys.

Why the open chord wins #

The obvious explanation would be that some shapes lie more conveniently for a I–IV–V than others — that the D shape’s IV and V fall closer to its I, say. They do not. Count the hand’s position as the fret of the barre, so that the open chord’s position is the nut itself, and every shape in every key comes out at 24 frets or more: I at some fret, IV five frets above, V seven above, and the ground’s shifts are 5, 5, 7, 2 and 5 frets whatever the starting point. With the barre as the measure, the shapes tie.

The difference is entirely in where the hand actually sits for an open chord. An open G chord is fretted at the second and third frets; the hand is two frets above the nut, not at it. An open A chord’s fingers are all at the second fret. So the hand that plays the open tonic is already partway to the IV chord’s barre at the fifth fret, and each of the ground’s four moves to or from the tonic is two frets shorter. In G major that is 2, 5, 2, 7, 5, 2 — sixteen frets — where the E shape, with no open chord in that key, goes 3, 8, 3, 10, 8, 3 for twenty-four.

The nut does the index finger’s job. An open chord is the one fingering on the guitar in which the lowest fret of the shape is not stopped by the hand, and that makes it the one fingering whose hand position is above its root. A tuning that makes the key’s open chord exact puts the piece’s most-played chord at the one place on the neck where the hand is not where the chord’s root is. Every other shape plays the tonic as a barre, and a barre puts the hand at the root.

Dragging the key shows the same thing in each. In D major the D shape’s path is identical to the G shape’s in G — 2, 5, 2, 7, 5, 2 — and the E shape, which has no open chord in D, has to start at the tenth fret and wrap down to the third for the IV chord. In E major the E and A shapes tie: the E shape starts from the open E, whose lowest stopped fret is the first, and the A shape plays the tonic barred at the seventh but gets the IV chord as an open A, at the second. Each has one open chord in the piece; they save the same amount.

What each key’s best tuning is worth #

The tuning’s other job is to make staying in the shape worth the movement, and the exchange rate at which it does so depends on the key.

Tuned for the best shape, the just route is worth taking at between 3.9 frets a cent, in A major, and 8.2, in G major. The spread comes from the second-best route in each key — the route a player would take if they valued movement a little more — and how much movement it saves. In A major the next route plays the V and the second IV in other shapes, saving ten frets for 2.55 cents, so a cent has only to be worth 3.9 frets before the just route wins. In G major the next route plays only the V chord out of shape, as an A shape at the fifth fret, and saves just four frets for half a cent; a cent must be worth 8.2 frets before those four frets are worth spending. On the frets the same decision needs a cent to be worth between 57.5 and 115 frets in every key. Tuning for the key’s best shape makes staying in it worth between seven and thirteen times less movement than it would be on the frets.

The frontier in G major #

The break-even is the last corner of a frontier, and G major’s frontier shows why its number is the largest.

Tuned for the G shape, five routes are cheapest at some rate. The route that never moves plays everything at the tenth fret and is 8.60 cents out. Four frets of movement buy most of the way to just — the open G, the C as an A shape at the third fret, the D as an open D shape — at 2.66 cents. Then each further four frets buys less: 1.47 cents for eight, 0.49 for twelve, and the last four frets, which bring the V chord into the G shape at the seventh fret, remove the last half cent. That last step is expensive because it is small, and it is the step the break-even measures. A player who values a cent at anything between four and eight frets would play G major tuned for the G shape with every chord in the shape except the V, and be within half a cent of just.

The corners are the same kind of object a tuning is right for some chords and wrong for the rest found on the cost surface of the tuning itself: points where one more unit of one quantity buys a fixed amount of the other, joined by straight segments. There the quantities were cents on different chords; here they are cents against distance, and the frontier’s convexity is what makes the rate at each corner well defined.

Measured along the string #

A fret is not a fixed distance. The frets close up as they climb the neck — the first is thirty-six millimetres from the nut on a 650-millimetre string, the twelfth is nineteen from the eleventh — so a shift of five frets near the nut is a longer movement of the arm than a shift of five frets at the ninth.

Measured in millimetres the result survives with one exception. The D, G and A shapes in their own keys all move the hand 475 millimetres, the C shape in C 613, and each is the least of its row. In E major the tie breaks: the A shape moves the hand 508 millimetres and the E shape 577. The A shape plays its tonic at the seventh fret, where the frets are shorter, and gets its one open chord on the IV; the E shape plays its open tonic at the nut and makes its long shifts in the widest part of the neck. So in E the shape to tune for depends on whether the hand’s effort is in frets or in distance, and for every other open key it does not.

A tuning is a choice of key #

The sharper way to put the finding is that tuning a guitar for a shape is a choice of key. The tuning for the G shape is the best tuning for G major and a poor one for everything else — in C major it moves the hand 26 frets where the C shape’s tuning moves it 20, in A major 24 where the A shape’s moves it 16. Nothing about the tuning itself names a key: the six offsets that make the G shape just are defined by intervals, not pitches. The key enters only through the one chord that can be played open, and it enters decisively. A keyboard’s unequal temperaments favoured some keys by tuning their thirds, and what a temperament cannot do is favour all of them; a guitar tuned for a shape favours a key by a different route entirely, through the hand rather than the pitches.

That is a new reason for a very old habit. Players who retune a guitar nearly always retune it for a key — often by harmonics, which a guitar tuned by harmonics hides a comma found decides where the comma goes — an open G tuning for music in G, D A D G A D for music in D — and one tuning has no comma to place found that those open tunings are exactly the ones in which a single barred shape carries the music. The standard-tuning version of the same habit is the one here: a player who tunes for the shape of the key’s open chord gets the tonic at the nut and every other chord in the same shape, which is the arrangement that moves the hand least and keeps every chord just. The chord a tuning gives up is a fingering found that a tuning serves one family; what the key does is decide which family is worth serving.

How the table is computed #

The five shape templates are the open chords of standard tuning — C as x 3 2 0 1 0, A as x 0 2 2 2 0, G as 3 2 0 0 0 3, E as 0 2 2 1 0 0 and D as x x 0 2 3 2 — each moved by the frets that bring its root to the chord wanted, and again an octave higher if it fits below the fourteenth fret. A shape’s tuning is the one that makes that shape alone exactly just, with the lowest string pinned at zero, as counting beats moves the price of a chord, not the tuning pinned it; a string the shape does not sound is left where the frets put it. For each key and tuning the cheapest route is found exactly, by dynamic programming over the six chords, at exchange rates swept over six orders of magnitude; the route at the top of the sweep is the one that keeps every chord just, and its movement and break-even rate are what the figures report.

The hand’s position for a fingering is its lowest stopped fret, so an open chord sits where its fingers are rather than at the nut. Millimetres are measured along a 650-millimetre string, with the nth fret at 650 times one minus two to the minus n over twelve.

What the table does not weigh #

How hard each shape is to play. The G shape barred up the neck is a four-fret stretch that few players use, and the C shape barred is not much easier; the E and A shapes are the barre chords every method teaches. The table weighs distance, and a player of G major tuned for the G shape would be playing its IV chord as a barred G shape at the fifth fret, which is an awkward chord. A search that weighted difficulty would favour the E and A shapes everywhere.

Other progressions. The ground here has three chords a fourth and a fifth apart, the commonest harmonic ground in the repertoire the shapes are taught for. A progression with a chord on the second or sixth degree, or with minor chords, would move the hand differently, and minor chords are not in the five shapes at all.

Key as a player chooses it. A guitarist picks a key for a singer, a band or a song, not for a tuning, and the table says only what follows once a key is fixed.

The string’s own length. The millimetre figures assume a 650-millimetre scale; a shorter scale scales every distance and changes no ranking.

Whose tunings #

The keys with open tonics — C, D, E, G and A — are the keys of folk and blues guitar in standard tuning, and they are its keys for the same reason the table finds: the open chord is the easiest one to play and rings on its open strings. What the table adds is that they are also the keys in which a retuning for one shape costs the hand least, so a player in one of them who wanted the whole piece exactly in tune has a single obvious tuning to choose. In F and B♭ that player has no good choice: every shape leaves the tonic as a barre, and the just route is a third longer. The equal-tempered frets, which serve every key equally badly, are the one arrangement under which a guitarist in F loses nothing to a guitarist in G, and that is why a guitar cannot be in tune seen from the side of the keys.

Still open: the minor chord no shape has #

Every chord in this essay is major, and the five shapes are the five major-chord families. A piece with a minor chord in it — the sixth-degree chord of the commonest pop ground, I–vi–IV–V — has to leave them. The minor shapes are families of their own, and a minor shape and its major twin differ in exactly the note that matters: the open E minor chord is the open E chord with the third lowered, so the pair of strings that carries that note is asked for a major third by one shape and a minor third by the other.

The earlier essay measured that disagreement as 29.33 cents, the largest between any two open shapes. On a guitar tuned for the E shape, then, the E-minor shape should be the worst fingering on the neck, and a minor chord in the piece has to be played in some other minor shape and paid for. The computation is the route search here with the minor shapes added as families of their own, and it would say which minor fingering a tuning for one major shape can best afford, and whether a single minor chord in a piece is enough to move the best tuning off the key’s own shape.