A silent string tunes the minor chord

A tuned guitar has no minor chord in tune tuned a guitar to make each of the five moveable major-chord shapes exact in turn, and then asked how badly each tuning played a minor chord. The answer was a table of fifteen numbers, every one of them worse than the frets, the best of them the A minor shape on a guitar tuned for the E shape at 8.60 cents. The I–vi–IV–V ground, which has one minor chord in four, was therefore most in tune tuned for the E shape, at 2.15 cents in every key.

It ended by questioning the one assumption that result shared with every earlier essay on the guitar’s tuning. A tuning is right for some chords and wrong for the rest had searched over six continuous offsets and found that a piece’s best tuning was usually a corner — exact for some chords, abandoning others. A piece with a minor chord in it is the natural place to test that, because its two families of chord cannot both be exact, and a tuning halfway between them might leave the major chords a little out and the minor chord much closer.

The search, run properly, says no to the compromise. At every share of the piece the best tuning is exact for one of the two chords. But the same search found something the question was not looking for, and it undoes the earlier table rather than confirming it.

No compromise at any share #

The cleanest version of the question takes two chords and nothing else: a major chord in one shape and a minor chord in another, the minor holding some share of the time. The search is the one the earlier essays built — each string’s offset moved in turn to its best value, then pairs of strings together, refined through four grids down to a two-hundredth of a cent — started here from forty-two places, because on this surface a dozen starts can miss the best corner.

With the minor chord holding anything up to about half the piece, the best tuning makes the major chord exact and leaves the minor chord 6.26 cents out. With it holding more, the best tuning makes the minor chord exact and leaves the major 7.04 out. The two corners swap in one step, and between them there is nothing: no share of the piece makes a compromise the best tuning.

The step is not at one half, which is worth a sentence because it looks as if it should be. The major-exact corner costs the minor chord’s 6.26 cents times the minor share; the minor-exact corner costs the major chord’s 7.04 times the major share. They tie where those are equal, at 7.04 ÷ (6.26 + 7.04) = 52.9 per cent. A piece needs to be slightly more than half minor before the minor chord should be the exact one, because a major chord is the more expensive of the two to leave out.

That the optimum is always a corner has a reason, and it is the reason the earlier search found corners too. The cost is a weighted sum of absolute departures — each pair of strings a chord sounds contributes how far its interval sits from just — and a sum of absolute values is smallest at a point where as many of its terms as possible are exactly zero. Moving a tuning off one chord’s exact point costs that chord at the full rate of every pair it sounds, and buys the other chord at most the same rate, weighted by a smaller share. So the tuning goes all the way to one side.

All fifteen pairings keep the major shape exact #

Five major shapes and three minor shapes make fifteen pairings, and the same search on each, with the minor chord holding the pop ground’s quarter of the time, gives the same answer fifteen times: the major shape exact to within a thousandth of a cent, the minor chord paying. The lead’s hypothesis — a tuning between two families — is not the best tuning for any of them.

But the minor chord’s bill is not the one the earlier table printed.

The C shape’s tuning, placed for an E minor chord, plays it 6.26 cents from just. The earlier table gave that combination as 10.30, and said no combination anywhere got below 8.60. The D shape’s tuning plays E minor at 6.91, where the table said 10.92. The A shape’s plays it at 9.39 rather than 10.95. And five of the fifteen cells — every cell in the E and G rows — do not move at all.

What separates the rows that move from the rows that do not is visible in the chord diagrams, and it has nothing to do with harmony.

What “tuned for a shape” did not fix #

The open C chord is played x 3 2 0 1 0: five strings sounded, the low E string not played. The open A chord is x 0 2 2 2 0, again without the low E. The open D chord is x x 0 2 3 2, without the low E or the A. The E and G chords sound all six.

A tuning that makes a shape exact fixes the differences between the strings that shape sounds, and nothing else. The low E string’s offset does not appear in any interval of a C chord, so a search told only to make the C chord just has no reason to put it anywhere in particular, and puts it wherever it happens to stop. The earlier essays’ “tuning for the C shape” left the low E string 7.82 cents above the A string. That value was arbitrary — any other would have made the C chord exactly as just — and every result that depended on it inherited the arbitrariness.

The E minor chord is played 0 2 2 0 0 0, and it sounds the low E string: it is the chord’s root. So on a guitar tuned for the C shape the E minor chord’s root was wherever a search had abandoned it, and the 10.30 cents in the earlier table was mostly the cost of that accident.

Swept, the low E string draws a V with a flat-bottomed floor. At 7.82 cents above the A string the E minor chord is 10.30 cents out; at 13.68 below, it is 6.26, and that is the best setting. The figure is not arbitrary. On a guitar tuned for the C shape, the notes E on the D string and the high E string are the C chord’s third, tuned 13.68 cents flat of the fret so that the third is 5 : 4. Tuned to the best place, the low E string is simply another E the same as those — an octave of the C chord’s own third. The E minor chord built on it then has its root agreeing with its own octaves on the D and high E strings, which is most of what a minor chord’s intervals need.

What is left, 6.26 cents, is the E minor chord’s G and B. On a C-shape tuning the G string is set as the C chord’s fifth, two cents sharp, and the B string as its octave; neither is where an E minor chord’s minor third and fifth want them. Those are strings the C chord does sound, and the C chord holds three times the minor chord’s share of the piece, so moving them costs more than it buys — the corner argument again.

The ground in five keys, again #

Put the placed tunings back into the pop ground and the earlier conclusion reverses.

The hero figure is the whole of it. With its low E string tuned to the chord’s third, the C shape’s tuning plays every major chord of I–vi–IV–V exactly and the minor chord as an E minor shape at 6.26 cents, for a quarter of the time: 1.57 cents over the piece, in every key, against the E shape’s 2.15. The D shape’s tuning, with two strings to place, is next at 1.73. The A shape’s improves a little, to 2.35. The E and G shapes’ tunings are unchanged at 2.15 and 2.51, because there was nothing in them to change.

The chord that makes the difference is not the one the earlier essay named. There, the E shape’s tuning won because the A minor shape happened to put its own minor third on the pair of strings the E chord had already tuned as a just minor third — an inherited interval. Here the C shape’s tuning wins because it has a spare string and the E minor shape wants that string for its root. The first is the fact three notes at once turns on — a major triad’s upper third is a minor third, so tuning one tunes the other — arriving by coincidence on a shared pair of strings; the second is a degree of freedom nobody had used.

What it does to the hand #

The earlier essays priced a tuning against how far the hand moves, and that exchange rate is where a player’s choice is actually made.

At five frets a cent — the valuation tuned for one shape, a cent is worth five frets found typical — G major is now best tuned for the C shape with its low string placed, at 1.57 cents and nineteen frets of movement. The earlier answer at that rate was the G shape’s tuning at 3.24 cents and eleven frets. From eight frets a cent the C shape’s placed tuning is the cheapest in four keys of five, and A major holds out on the D shape’s until fifty. Twenty-five of the forty cells in the earlier envelope name a different shape.

At the bottom of the range something unexpected appears: in C, D, E and A major, a player who values a cent at one or two frets is best served by the D shape’s placed tuning and never moves the hand at all, at 4.9 to 6.9 cents. Among the five tunings, the D shape’s with its two silent strings placed is the least bad for a player who plays every chord open — which tune a guitar for the shape of its key’s open chord had expected of the key’s own shape rather than of D in four keys out of five. It is still worse than the frets, 5.64 cents, and a player who will not move should not retune at all.

Two results that were not wrong, and one that was #

It is worth being exact about what this revises, because most of the earlier essay stands.

The minor pairings table was right about the strings it measured. The chord a tuning gives up is a fingering and the earlier essay’s disagreement table compared what two shapes ask of the strings they both sound, and no major shape and minor shape agree on those. That is untouched. It never said anything about a string only one of them sounds.

The corner result was right, and is now stronger. Every best tuning is exact for something. The earlier search found it on a handful of pieces; here it holds at every share and for every pairing.

The minor chord’s cost was wrong for the three shapes that leave strings silent, and with it the claim that no tuned guitar plays a minor chord within 8.60 cents, and the conclusion that the pop ground wants the E shape in every key. Each of those numbers was a property of where a search had abandoned a string it had no reason to place, reported as a property of the shape.

That is a failure of a particular kind, and it is worth naming because it would recur anywhere the same move is made. A tuning “for” a chord is underdetermined whenever the chord does not use every string, and a result that holds a tuning fixed while asking about a second chord is a result about the arbitrary part. The repair is not a better search. It is asking the second chord’s question at the same time as the first.

Where the numbers come from #

The five major templates are the open chords C (x 3 2 0 1 0), A (x 0 2 2 2 0), G (3 2 0 0 0 3), E (0 2 2 1 0 0) and D (x x 0 2 3 2); the three minor templates are E minor (0 2 2 0 0 0), A minor (x 0 2 2 1 0) and D minor (x x 0 2 3 1). A chord’s error is the mean departure from the five-limit just ratio of the intervals its sounding pairs of strings form, and a piece’s is the time-weighted mean of its chords’. The lowest string is pinned at zero, since moving all six strings together changes no interval.

A placed tuning is the result of the continuous search on the major shape and the minor shape together, the minor weighted by a quarter, which is the ground’s proportion. For each major shape the minor shape it hosts best is used, and in every key the ground is then routed exactly by dynamic programming over its eight chords, as before, with the hand at the lowest stopped fret of each fingering.

Where the model stops #

A string that is silent in the tuned chord is not silent in the piece. Tuning the low E string for the E minor shape changes every other chord that sounds it — any G shape, any E shape, any bass note played on it. Within the pop ground routed through C shapes that costs nothing, because no chord in the route uses the low string except the minor one; a piece whose route puts a G-shape chord on it would pay, and the search would then weigh that chord too.

The minor chord’s share is the ground’s. A verse that sits on vi would raise it; at 53 per cent minor the tuning should jump to making the minor chord exact, and a piece that spent that long on one minor chord would be better called a minor-key piece.

Only five-limit ratios. A minor chord’s third could be taken as 7 : 6 or a minor seventh as 7 : 4, and a comma under the threshold is the nearest this collection comes to saying when cents of this size matter to a listener. The difference between 6.26 and 8.60 is a difference in tuning, not a promise of audibility.

And a real guitar. String stiffness, fret placement and the pull of a pressed string all move a stopped note by amounts comparable to the differences above, and a guitar cannot be in tune is about the frets that make every one of these offsets a compromise before any tuning begins.

Still open: every string placed for every chord in the route #

The placing here was done for two chords, a major shape and the minor shape that shares the piece with it. The pop ground routed through C shapes happens to need nothing else. Most pieces are not so obliging: a route that uses two major shapes, or plays a bass note on a string one shape leaves silent and another sounds, puts more than one chord’s demands on each free string, and the question becomes which shapes the route should use given that the tuning can be set for all of them at once.

The computation is the alternating search the earlier essay proposed, done without the shortcut this one took: route the piece, search the tuning on every chord of the route at its own weight, reroute, and repeat until neither changes, from enough starting tunings to see whether the answer depends on where the search begins. Run on the pop ground from the five placed tunings and from random ones, a first trial already found tunings exact for the D shape reaching 1.73 cents from one start and 1.83 from another, which is the sign of a surface with more corners than the two-chord problem has. Whether a whole piece can be tuned better than its best single major shape allows — by giving each silent string to a different chord — is the question that would say whether the right unit of a guitar’s tuning is a chord shape at all, or the route.