Tuned for one shape, a cent is worth five frets

The chord a tuning gives up is a fingering found that what a guitar’s six string offsets fix, from the music’s point of view, is fifteen differences between pairs of strings, and that a chord shape asks each pair for a particular difference. A tuning can meet every demand of one shape exactly, and a shape moved about the neck without changing its fingering asks for exactly the same differences, so it stays exactly just wherever it goes. Every open chord shape is a family of one; a tuning serves one family and no more.

That left the choice a player actually makes. Given a tuning set for one family, every chord still has several fingerings, and only one of them is in the family. The player can stay inside it — at the cost of moving the hand wherever the family’s version of each chord happens to sit — or leave it for a fingering nearer to hand and pay in intonation. The prediction was that the cheapest route through a piece stays in the family far longer than a player choosing by comfort would, and the question was a number that belongs to guitar teaching rather than to tuning: how many frets of hand movement a cent of intonation is worth.

The number is 4.8. Above it, the cheapest route through a I–IV–V ground keeps every chord in the tuned shape; below it, the route starts to leave. On equal-tempered frets the same decision needs a cent to be worth fifty-seven frets, which is to say that nobody playing on the frets should ever make it.

Five shapes, and one of them tuned #

Every major chord on a guitar in standard tuning can be played in the shape of any of five open chords moved up the neck — the open C, A, G, E and D shapes, each with the nut replaced by a barre. Teaching books call it the CAGED system, and it is exactly the set of families the earlier essay had in mind: five fingerings per chord, each one a transposition of an open chord.

Tune the guitar to make the E shape exactly just. That is the tuning the earlier searches found for a song built on the E shape — the A string and the B string raised 1.96 cents to make their fifths and fourths pure, the G string lowered 13.68 cents to make the E chord’s third a just 5 : 4, the D string and the top string where the frets put them. Then every fingering of every chord in a ground on E major can be scored.

The E-shape fingerings are exact: the open E chord, the A chord barred at the fifth fret, the B chord barred at the seventh, the E chord again at the twelfth. Every other fingering is between 12.5 and 15 cents out, and every one of them is worse than it would be on plain frets — the A shape by six cents more, the D shape by seven. A tuning that serves one family perfectly serves every other family worse than no tuning at all.

The lower bars carry a small surprise of their own. Even on equal-tempered frets the fingerings are not equal: the E shape is the most in tune at 5.60 cents and the C shape the least at 8.99. The frets distribute the error evenly among intervals, not among shapes, and different shapes put different intervals on the strings.

What leaving the shape costs #

The earlier essay predicted that the cost of leaving the tuned family would be thirteen cents or more every time, and the figure prices that exactly. The cheapest way out of the E shape is the A shape, at 12.51 cents; the most expensive is the D shape, at 14.99. Nothing costs less than twelve and a half, because every other shape asks at least one pair of strings for an interval the E shape’s tuning has deliberately put somewhere else — most often the third, which the E shape puts on the G string over the bottom string and which the A shape, for instance, puts on the B string over the A string, a pair the E tuning leaves exactly as the frets have it, nearly fourteen cents from where a just third would need it.

That makes the decision at each chord binary in effect. A fingering is either exact or at least twelve and a half cents out; there is no fingering that is nearly in tune. Somebody has to pay the comma in exactly the keyboard sense, and here the somebody is a fingering: the tuning has concentrated all the error on the shapes it does not serve. The route search is therefore not weighing small improvements against small movements. It is deciding, chord by chord, whether to pay a large fixed price in intonation or a variable price in distance, and that is why only four routes ever win.

The search, and its answer #

A route through the piece is a choice of fingering for each chord. Its cost is two numbers: the piece’s mean departure from just, weighting each chord by how long it sounds, and how far the hand moves, counted as the change in the fret at which it sits — the lowest stopped fret of each fingering, or the first fret for an open chord. Combining them needs an exchange rate, and the exchange rate is the thing being asked for, so it is swept: at each rate the cheapest route is found exactly, by working through the chords one at a time and keeping, for each fingering of each chord, the cheapest way of arriving at it.

Tuned for the E shape, only four routes are ever the cheapest. If movement is nearly free, the route that stays in the E shape throughout is exactly just and moves the hand twenty frets over the piece. If a cent is worth less than 4.8 frets, a route that plays the B chord as an A shape at the second fret and the last A chord as the open A shape costs 2.09 cents and ten frets. Below 3.6 frets a cent, a route that moves the whole piece up to the twelfth fret costs 4.30 cents and two frets. And if movement is precious beyond a third of a fret a cent, the hand stays at the seventh fret for the whole piece and pays 12.09 cents — the price of playing every chord in a shape the tuning was not set for.

On the frets the same search is almost indifferent. Its cheapest route in intonation is also the E shape throughout, at 5.60 cents, and the route that never moves is 6.85. Every cent between them costs from three to fifty-seven frets of movement, and a player who values their hand at all has no reason to follow the shape. Tuning for one shape does not only make that shape just; it makes staying in it worth twelve times as much.

Where the hand goes #

The four routes are not a gradual series; they are four different ways of holding a guitar.

The just route is the one a student of barre chords knows: open E, the E shape barred at the fifth fret for A, back to open E, barred at the seventh for B, down to the fifth, home. The hand travels four or six frets at every change. The ten-fret route is the same until the V chord, which it plays as an A shape at the second fret and follows with the open A — the route of a player who does not want to reach the seventh fret for four beats. The two-fret route and the unmoving one are position playing: find a place on the neck where every chord of the piece lies under the hand, at the twelfth fret or the seventh, and fit whichever shape is there.

Position playing is exactly what a tuned guitar punishes. Its whole purpose is to change shape so as not to move, and a tuning set for one shape charges every change of shape between twelve and fifteen cents.

The rate, turned #

The route that is cheapest changes at three exchange rates, and the dial moves across them.

Each route is the cheapest over a whole range of rates, and the ranges are wide except in one place. The just route wins from 4.8 frets a cent upward, and the route that never moves from a third of a fret downward; the two intermediate routes share the interval between, and the ten-fret route is the cheapest only between 3.6 and 4.8. So a player’s valuation does not have to be known precisely. What matters is which side of about five frets a cent it falls on.

That valuation has a plausible order of magnitude, which is the reason to think it matters. A shift of four or five frets on a guitar neck at a chord change is routine for a player of barre chords and awkward at speed; fourteen cents on a chord’s third is about the gap between a just third and an equal-tempered one, which is audible on a sustained chord. A player who thinks a fourteen-cent third is worse than a five-fret shift — which is to say most players who would bother to retune — is above the threshold and should play every chord in the tuned shape.

A leap is not a distance #

Counting movement in frets treats a seven-fret shift as seven times a one-fret shift, and a guitarist would not. The difficulty of a position change is largely in making it at all.

Counted in leaps, the frontier has the same ends and different middles. The route that never moves is still 12.09 cents, and the only just route is still the one that keeps every chord in the E shape, now counted as five leaps rather than twenty frets. The intermediate routes are different: one leap, from the twelfth-fret position down to the seventh and staying there, for 5.42 cents; two leaps for 3.13. The last step, from 3.13 cents to exactly just, costs three more leaps, and the exchange rate for it is 0.96 leaps a cent — about one leap a cent. The answer to the question survives the change of units: a player should stay in the tuned shape if a cent of the piece’s mean error is worth more to them than about one shift of hand position.

A small surprise about the frets #

The comparison with equal temperament has an implication beyond the tuned guitar. On the frets, the route that keeps every chord in the E shape is the most in tune of all the routes, at 5.60 cents, and the route that never moves is the least, at 6.85. A guitarist who has never retuned a string and plays every chord of a ground in E as an E-shape barre chord is, by this count, playing 1.25 cents closer to just than one who plays the same chords in position at the seventh fret. That is not much, and it takes twenty frets of movement to get, but it is the same direction as the tuned result and it says something the earlier essays did not: equal temperament has a favourite shape, and it is the one whose intervals — fifths, octaves and a single major third — the frets approximate best. The same fact is behind a guitar cannot be in tune: the frets are a compromise among intervals, and each shape samples the intervals differently. A player who changes shape to stay in one place is choosing, without knowing it, among five different compromises. What a temperament cannot do is make every chord equally good; what it can do, on a guitar, is make the shapes nearly equally bad, and the spread from 5.60 to 8.99 cents is how nearly.

How the search is made #

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 number of frets that brings its root to the chord wanted, and again twelve frets higher if it still fits below the fourteenth fret. The tuning for a shape is the one the earlier searches find for that shape alone, with the lowest string pinned at zero, as counting beats moves the price of a chord, not the tuning pinned it. A fingering’s error is the mean, over its sounding pairs of strings, of the departure of each interval from the five-limit just ratio its span names, exactly as every earlier essay on the guitar’s tuning counted it; a route’s error weights each chord by its beats.

The route is found by dynamic programming over the six chords, which is exact: for each exchange rate the cheapest route is the true minimum over every combination of fingerings, not a search that might stall. The rates are swept over four orders of magnitude, and because the frontier of error against movement is convex, the sweep finds every corner of it. The rate at which one corner gives way to the next is the movement difference divided by the error difference between them.

What the search leaves out #

Reach. A fingering’s difficulty is not only where the hand is but what it has to do there. The G shape and the C shape moved up the neck are awkward stretches; the E and A shapes are the barre chords every player learns. A search that weighted fingerings by difficulty would push further toward the E shape, which is the direction the result already points.

Speed. Four beats of a chord leave time for a shift that a quaver does not. The piece here is slow and the movement cost is the same at every change; a fast piece would raise the cost of every shift and move the threshold up.

The sound of the shape. A barre chord and an open chord do not sound alike: open strings ring, and a barre stops them. The error counts intervals, not timbre, and a player choosing the open E over the barred E at the twelfth fret is choosing a sound as well as a position.

Bending. A guitarist can pull a stopped note a few cents by pressure. That is a way to fix a non-family fingering on the fly, and it is not available on the open strings, which in several of the non-E shapes carry the offending note.

Whose guitar playing #

The piece is a I–IV–V ground of the kind that runs through blues, folk and rock guitar, stated rather than transcribed, and the shapes are the moveable major chords every guitar method teaches. Players who retune a guitar for a piece usually do so to an open tuning that makes one chord ring on the open strings — often by the harmonics, which a guitar tuned by harmonics hides a comma found decides where the comma goes — and one tuning has no comma to place found those tunings are exactly the ones in which a single barred shape carries the music. The arithmetic here says why that pairing is stable: once a tuning is set for one shape, the player’s cheapest route is to keep the shape, and the player’s habit of barring one shape up and down the neck is the route the search finds. A tuning is right for some chords and wrong for the rest posed the problem from the tuning’s side; this is the same problem from the hand’s.

Still open: which shape the tuning should serve #

Everything above takes the family as given — the E shape, because the earlier searches found it for a piece in E. But the tuning is the player’s choice too, and a guitar can be tuned for any of the five shapes. Each choice makes a different fingering of each chord just and puts it at a different place on the neck.

The question that follows is which shape a tuning should serve for a piece in a given key, when the player will then follow the cheapest route. The prediction the table above makes is that the answer depends on where each family puts the piece’s chords: a family whose tonic chord is an open shape starts the hand at the nut and may keep it lower on the neck. The search is the one here run five times — once for each family’s tuning — and it would say whether the shape to tune for is a fact about the key, about the progression, or about neither.