A tuned guitar has no minor chord in tune

Tune a guitar for the shape of its key’s open chord found that in every key whose tonic is an open chord, the moveable major shape a guitar should be tuned for is that open chord’s own: a player who then keeps every chord in the shape plays exactly in tune, and the open tonic lets the hand sit above the nut and travel least. Every chord in that essay was major, because the five moveable shapes are the five major-chord families — the open C, A, G, E and D chords, moved up the neck.

It ended on the chord those shapes cannot supply. The commonest ground in popular song is I–vi–IV–V, and its second chord is minor. A minor chord has shapes of its own — the open E minor, A minor and D minor chords, moved up the neck — and each is its major twin with the third lowered, which the chord a tuning gives up is a fingering measured as the largest disagreement between any two shapes on the neck. So a guitar tuned to make a major shape exact has to play its minor chord in some minor shape and pay for it. The questions were which minor shape it can best afford, and whether a single minor chord is enough to move the best tuning off the key’s own shape.

No tuning for a major shape can play any minor chord within eight and a half cents of just, and one minor chord in a quarter of the piece is enough to make the E shape’s tuning the most in tune in every key — but only for a player who values a cent more highly than the essays before this one assumed.

Every minor shape on every tuned guitar #

Tune the guitar for each of the five major shapes in turn, and score each of the three minor shapes under each tuning by the mean departure from just of its sounding pairs of strings, the same measure every earlier essay on the guitar’s tuning has used.

Every one of the fifteen combinations is worse than the frets. The E minor shape, which on equal-tempered frets is 5.74 cents from just, is between 9.78 and 10.95 on every tuned guitar. The A minor shape runs from 8.60, on a guitar tuned for the E shape, to 14.47 on one tuned for the G shape. The D minor shape is worst of all on the C shape’s tuning, at 16.94. There is no tuning among the five on which a minor chord is as good as it would have been on the frets, and the best combination on the whole table is still more than eight and a half cents out.

That is a sharper form of what tuned for one shape, a cent is worth five frets found about major fingerings outside the tuned family, which sat between twelve and a half and fifteen cents out. The minor shapes are slightly less bad than the wrong major shapes, because a minor shape shares some intervals with any major one; but they are never good, and they are never as good as untuned frets.

Why no minor shape can agree with a major one #

A tuning fixes the differences between the six strings’ offsets, and a shape asks each pair of strings it sounds for a particular difference. Two shapes can share a tuning exactly only if they agree on every pair both of them sound.

No major shape and minor shape agree everywhere. The nearest pair is the C shape and the E minor shape, which disagree by at most 11.73 cents and agree on four of their ten shared pairs; next come the D shape against E minor at 13.69, and the E shape against A minor and the A shape against D minor at 15.64. Each major shape and its own minor twin — E and E minor, A and A minor, D and D minor — disagree by 29.33 cents, the largest figure on the table, even though the E pair agrees on ten of fifteen pairs: the five pairs that carry the third are asked for a just major third by one shape and a just minor third by the other, and the two just thirds sit on opposite sides of the fret, 13.69 cents below it and 15.64 above.

The twins are the case that looks most compatible and is least. A guitarist moving from E to E minor lifts one finger from the G string; the tuning that made the first exact has to be most wrong for the second on exactly that string.

The pop ground in five keys #

The piece is now I–vi–IV–V, twice round at four beats a chord, so the minor chord holds a quarter of the time. In G major that is G, E minor, C and D.

The most-in-tune route under a tuning plays every major chord in the tuned shape, which is exact, and the minor chord in whichever minor shape that tuning makes least bad. Its whole error is therefore the minor chord’s, carried for a quarter of the piece. On the E shape’s tuning the minor chord goes in the A minor shape at 8.60 cents, and the piece averages 2.15 cents — in every key, because the minor shape’s error does not depend on where on the neck it is played. On the A shape’s tuning the minor chord is best in the D minor shape, at 9.78 cents, and the piece is 2.44. The G, C and D shapes’ tunings do worse, at 2.51, 2.54 and 2.73.

So the tuning that is most in tune for the pop ground is the E shape’s, in every key, and the reason is not its major chords but its minor one. What the piece needs from the tuning is a host for the chord no tuning can make exact, and the E shape’s tuning, whose offsets are small on every string but the G, is the least hostile host.

It is worth putting against the frets. On equal-tempered frets the most-in-tune route through the ground is 5.64 cents from just. A guitar tuned for the E shape, with its minor chord paid for, is 2.15 — still less than half. One minor chord does not make retuning pointless; it makes the choice of tuning a question about the minor chord.

The minor third that was already there #

Why the A minor shape, and why on the E shape’s tuning, has an answer that can be read off the strings. The E shape’s tuning lowers the G string by 13.68 cents so that the open E chord’s third, G♯ on the first fret of that string, is a just major third above the bottom E. It raises the B string by 1.96 cents to make the chord’s fifth pure. Between the G string and the B string, then, the tuning has put a difference of 15.64 cents above the frets — which is exactly the amount by which a just minor third, 6 : 5, is wider than an equal-tempered one.

That is not an accident. The E chord’s G♯ and B, on those two strings, are the third and fifth of the triad, and the interval between a major triad’s third and its fifth is a minor third. Tuning the major triad just tunes its upper minor third just as well, and the A minor shape happens to put its own minor third — A on the G string’s second fret, C on the B string’s first — on exactly that pair. Of the A minor shape’s ten intervals, two are exact on the E shape’s tuning, and one of them is the chord’s defining minor third. Three notes at once is a triad of two thirds stacked; a tuning that makes one stack just has made both of its thirds just, and a minor chord that reuses the upper third inherits it.

The rest of the A minor shape is not so lucky. Its octave on the G string and its minor tenth on the B string are both 15.64 cents out, because the same two offsets that made the minor third exact sit badly against the A string. That is the 8.60 — an exact minor third and a set of other intervals the tuning was never asked about.

The key’s own shape, or the E shape #

The earlier essay’s result was that the key’s own shape moves the hand least. The E shape’s tuning is the most in tune. When those disagree, the answer depends on how much a cent is worth.

In G major the G shape’s tuning reaches 2.51 cents for fifteen frets of movement, and the E shape’s reaches 2.15 for twenty-one. The E shape’s tuning buys 0.36 cents for six frets, which is worth it only to a player who values a cent at more than 16.7 frets — three and a half times the value the major ground in E needed before its player should stay in the tuned shape at all. Below that valuation the G shape’s tuning is cheaper at every point on its frontier.

Across five keys the picture is a band. Below about three frets a cent the cheapest tuning is whatever lets the hand sit still, and it is rarely the key’s own. Between eight and twelve frets a cent, the key’s own shape is the tuning to choose in four keys of five — C, E, G and A — and D major is best tuned for the G shape. The D shape’s own tuning makes every minor shape at least 10.9 cents out; the G shape’s tuning puts the vi, B minor, in the E minor shape at the seventh fret beside the D chord in the G shape, and hosts it at 10.04. At twenty frets a cent C and G have moved to the E shape, and at fifty every key has. The earlier essay’s rule survives a minor chord for players of moderate fastidiousness, and gives way to a single universal tuning for the most fastidious.

One key, turned #

The dial takes G major across the whole range.

At five frets a cent, the value tuned for one shape, a cent is worth five frets found typical, G major is best tuned for the G shape and played with the open G, the open E minor shape, the G shape barred at the fifth fret for C, and the D first as an A shape at the fifth fret and the second time as the G shape at the seventh — 3.24 cents from just, eleven frets moved. At eight frets a cent the D chord moves into the G shape at the seventh fret and the piece is 2.51 cents. From twenty frets a cent the guitar should be retuned for the E shape, every major chord barred in it, and the E minor played as an A minor shape at the seventh fret: 2.15 cents, twenty-one frets.

Tuning for the minor chord instead #

There is an obvious alternative: tune for a minor shape, so that the minor chord is exact and the three major chords pay. The arithmetic rules it out quickly. Tuned for the E minor shape, the ground in G major is at best 5.28 cents from just; for the A minor shape, 5.67; for the D minor shape, 6.91. Three quarters of the piece is major, and every major fingering on a minor shape’s tuning is several cents worse than the minor chord was on a major one. The best of them is barely better than the frets’ 5.64. A tuning should serve the chords that hold the most time, and in a major-key ground those are major.

Where the numbers come from #

The three minor templates are the open chords E minor as 0 2 2 0 0 0, A minor as x 0 2 2 1 0 and D minor as x x 0 2 3 1, moved by the frets that bring their roots to the chord wanted, and again an octave up if they fit below the fourteenth fret. The five major templates and their tunings are the earlier essays’; each tuning makes its own shape exactly just, with the lowest string pinned at zero. A fingering’s error is the mean departure from the five-limit just ratio of each interval its sounding pairs of strings form, a minor third’s ratio being 6 : 5 and a minor sixth’s 8 : 5.

The ground is I, vi, IV, V, twice round at four beats a chord. For each key, tuning and exchange rate the cheapest route is found exactly by dynamic programming over the eight chords, with the hand’s position taken as the lowest stopped fret of each fingering; the most-in-tune route is the one found at a rate of ten thousand frets a cent. The disagreement between two shapes on a pair of strings is the difference between the offset differences each would need to make its interval on that pair just.

What a minor chord’s cost leaves out #

Other minor fingerings. Guitarists also play minor chords in the shapes of the open G minor and C minor, which are awkward and rarely taught as barre chords. Adding them would give each tuning more to choose from and could lower the minor chord’s cost, though not below the eleven cents of disagreement the closest major and minor shapes already show.

Seventh chords. A minor seventh chord and a dominant seventh add a seventh whose just ratio is itself a choice between 7 : 4 and 16 : 9, and the seventh has two places to be found that choice decides a chord’s character. None is in the ground here.

The minor chord’s place in the piece. Four beats in sixteen is the pop ground’s own proportion. A piece that dwelt on its minor chord — a verse that sits on vi — would raise the minor chord’s share of the error and move the choice toward the E shape at lower valuations.

Whether 8.60 cents is heard. Every minor chord on the table is further from just than on the frets, and on the frets a minor chord is already 5.74 to 8.15 cents out and heard as in tune. A comma under the threshold is the nearest the collection comes to saying when a departure becomes objectionable, and the answer there depended on context.

Whose music this is #

The I–vi–IV–V ground is the harmonic basis of a great deal of popular song from the 1950s on, often called the doo-wop progression, and it is played on guitar with exactly the open and barre shapes the table uses. The arithmetic says something specific about retuning for it: a player who wants such a song more in tune than the frets allow should, above a high enough valuation on intonation, tune for the E shape whatever the key, and accept that the minor chord will be the one chord in the song that is not just. Below that valuation, the rule of the essay before mostly holds: tune for the key’s own open chord, except in D major. A guitar cannot be in tune because its frets serve every key equally; a retuned guitar can serve one family exactly, and a minor chord is the proof that it can serve only one.

Still open: a tuning halfway between two families #

Every tuning in this essay is exact for one shape. The search that a tuning is right for some chords and wrong for the rest ran over six continuous offsets found that a piece’s best tuning is usually a corner — exact for some chords, abandoning others — and so far that has been assumed here rather than tested. A piece with a minor chord in it is the natural place to test it, because it has two families whose demands cannot both be met, and a tuning halfway between the E shape and the A minor shape might leave the major chords a few cents out and the minor chord much closer.

The computation is the continuous tuning search of the earlier essays run on the most-in-tune route here, alternating with the route search until neither changes. It would say whether the pop ground’s best tuning is one of the five family tunings or a compromise between two — and if it is a compromise, it would be the first tuning in these essays that is not exact for anything.