A guitar tuned for a still hand is tuned for nothing

A silent string tunes the minor chord searched continuously over a guitar’s six strings for a piece of one major chord shape and one minor, and found no compromise: at every share of the piece the best tuning is exact for one chord and makes the other pay. It also found that the tunings earlier essays called exact for a shape were exact only on the strings the shape sounds, and that placing the C shape’s silent low string for the minor chord made the pop ground’s most-in-tune tuning the C shape’s, at 1.57 cents in every key.

Its closing section asked the question at the level of the route. A real piece is played along a route — each chord in some fingering, the hand moving between them — and a tuning could in principle serve the route rather than any one shape, giving each silent string to whichever chord needs it. The computation was the alternating search the earlier essays proposed: route the piece, retune for every chord on the route at its weight, reroute, and repeat.

At the most-in-tune end of the scale that search finds nothing new. Everywhere else it finds something the earlier essays had assumed away, and at the bottom of the scale it finds the compromise the search for one had given up on.

Routing and retuning in turn #

The objective is the one the earlier essays on the guitar used throughout: a route’s mean departure from just, over the sounding pairs of strings of each chord, weighted by how long the chord lasts, plus the hand’s movement in frets divided by the exchange rate — the number of frets of movement one cent of intonation is worth. At a high rate a player values intonation and will move; at a low one they will stay put and accept what the tuning gives.

For a given tuning the cheapest route is found exactly, as before. For a given route the best tuning is found by the continuous search over five string offsets, the lowest pinned. The two are alternated until the route stops changing, from seventeen starting tunings: the five placed shape tunings of the earlier essay, equal temperament, and eleven seeded random ones. Every reported tuning is the best of those seventeen by the route’s own objective, and the figures check that it is never worse than the best placed shape or than the frets.

The ground is I–vi–ii–V, which has two minor chords where the pop ground has one, so the minor chords now hold half of the piece. The pop ground’s figures are drawn alongside it where they differ.

At fifty frets a cent, the shape #

At the most-in-tune end the answer is the earlier essay’s. In every key, at fifty frets a cent, the best tuning for the route is the C shape’s with its low string placed, and it plays both minor chords of the ground in the E minor shape at 6.26 cents: 3.13 cents over the piece, since the minor chords hold half of it.

The route never gives the two minor chords two different shapes, though it could. It puts both into the one shape the silent string was tuned for, so one placement serves both, and the whole-route search, free to give each silent string to a different chord, finds nothing better. At the most-in-tune end the unit of a guitar’s tuning really is the chord shape.

At the bottom, nothing #

At one to three frets a cent, a player will not move the hand to gain intonation, and in all five keys of this ground the route the search settles on moves no frets at all. The tuning it finds for that still hand is exact for no chord on the route.

In G major, at two frets a cent, the route plays every chord at the seventh fret — the G as a C shape there, the E minor as an A minor shape, the A minor as a D minor shape, the D as a G shape — and the tuning lowers the A, D and B strings by two cents each. No chord on the route is in tune: they sit between 6.8 and 9.0 cents from just, 7.7 on average. But on the frets the same route, with nothing retuned, averages 7.9, and every placed shape tuning, forced to move the hand or to play its shapes badly, costs more. In C major the same thing happens at 6.6 cents; in D and A at 6.5.

This is the compromise tuning — the one a tuning is right for some chords and wrong for the rest found no sign of when it searched for a piece’s best tuning, and that the earlier essay found none of at any share of a piece. It exists, and the reason nobody found it is that every earlier search let the hand move freely. A tuning that is exact for some chord is worth having only to a player who will go and play that chord in its shape; a player who stays put plays four different shapes in one place, and the best a tuning can do for four shapes at once is split their disagreements.

The gain is small — about a fifth of a cent in G, D and A, about four tenths in C and E — and it is small for a reason: the offsets are two cents, because four shapes disagree with each other by more than any one of them disagrees with the frets. The honest description is that the best tuning for a still hand is equal temperament nudged, in the direction the route’s chords agree on.

In the middle, a minor shape #

Between the two ends the route’s tuning is exact for something, and what it is exact for is decided by the route rather than the ground.

In G major at five and eight frets a cent, the route that balances movement against intonation plays the G chord open in the G shape, both minor chords as open E minor shapes — the E minor open and the A minor at the fifth fret — and the D as a C shape at the second fret. The tuning the search finds for it is exact for the E minor shape, with the G string raised 15.6 cents, and the major chords pay 7 to 10 cents each.

That is the minor-exact corner the earlier essay found only when the minor chords held more than 53 per cent of the piece. Here they hold exactly half, and the minor corner wins anyway, because at these rates the route does not play the major chords in one shape: it plays them in whatever shape keeps the hand near the nut, and two different major shapes cannot both be made exact. Two minor chords in one shape can. The shape a tuning should serve is the one the route repeats, and a route with a moderate value on intonation repeats its minor shape.

In E major the same middle band goes the other way — exact for the A minor shape at twelve and twenty frets a cent — and in C, D and A it is exact for a major shape. Which chord wins is a fact about where each key’s chords fall on the neck.

The pop ground, which has one minor chord #

The pop ground behaves the same way with one difference. At the top of the range, every key’s best tuning is the C shape’s with its low string placed, at 1.57 cents — the earlier essay’s result, confirmed by a search that was free to do otherwise. At the bottom, C, D and E major settle on a still hand and a tuning exact for nothing, at 6.21 cents against 7.3 to 7.6 on the frets: here the compromise is worth a cent or more.

In C major the still-hand tuning is worth 1.4 cents over the frets and at least a quarter of a cent over any placed shape, which, with E major’s identical figure, is the largest margin over the frets the compromise has anywhere in the two grids. The reason is the pop ground’s single minor chord: with three major chords and one minor on a still hand, the four shapes the route plays disagree less with each other than the I–vi–ii–V ground’s two major and two minor shapes do, and the split between them can be finer. The compromise is most useful where the chords on the route are most alike — which is the opposite of the condition under which a placed shape is most useful, where one shape recurs.

G and A major are the exceptions, and they are the keys in which a placed shape’s open chords make the whole ground playable without moving: G major is best on the C shape’s tuning even at one fret a cent, with the hand moving four frets, and A major on the D shape’s with the hand not moving at all. When the ground can be played still and in one shape, the compromise has nothing to do.

What the three regimes mean for a player #

The earlier essays on the guitar measured a cent at about five frets for a typical player, which sits in the middle band. There the whole-route tuning is often a minor-shape corner, and it is always a little better than the best placed shape: in G major at five frets a cent, 6.51 against 6.77. At the edges it either coincides with the placed shapes (at the top) or beats them by a margin that is mostly about not moving (at the bottom).

The regimes are also a warning about how the earlier results on the guitar’s tuning should be read. Each earlier essay reported a best tuning and, where it could, the exchange rate at which a different one took over; the rates it reported were all in the upper half of this figure. The lower half was never computed because the question was always “which shape”, and a shape is the answer only once the player will follow it. A figure that fixes the rate at its high end is a figure about a particular kind of player, and a reader who plays the guitar near the nut and changes position rarely was never the player those figures described.

So the advice the earlier essays gave — tune for the key’s own open chord, or for the C shape with its low string placed — is advice for a player who values a cent at a dozen frets or more. A player who values it less should retune for the route they will actually play, and one who values it least should nudge equal temperament and keep still. Tune a guitar for the shape of its key’s open chord was right about its own valuation and silent about the others; the grid in the hero figure is what it looks like across all of them.

Why the shape was the unit only at one end #

It is worth saying why every earlier result on the guitar’s tuning came out as a shape. A tuned guitar has no minor chord in tune and its successor reported the most-in-tune route under each tuning — the route at an exchange rate of ten thousand frets a cent, which lies beyond the right-hand column of the hero grid. At that valuation a player will move anywhere for a hundredth of a cent, the route can put every chord of a family into the one shape the tuning makes exact, and the tuning’s job reduces to making one shape exact and placing its silent strings. Everything the chord a tuning gives up is a fingering found about shapes disagreeing follows from that reduction.

Lower down the reduction fails, because the route stops being free. A player who values a cent at two frets will not move seven frets to play a chord in the shape the tuning prefers; they play the shape that is under their hand, and the tuning has to serve whichever shapes those are. The shape was the unit of tuning only for the player who was willing to follow it around the neck.

How the search runs #

The five major templates and three minor templates are the earlier essays’: the open C, A, G, E and D chords and the open E minor, A minor and D minor, moved up the neck to reach each chord, within fourteen frets. A chord’s error is the mean departure from the five-limit just ratio over its sounding pairs of strings; the route’s is the time-weighted mean; the hand’s movement is the change in the lowest stopped fret. For a fixed tuning the cheapest route is found by dynamic programming over the eight chords; for a fixed route the tuning is found by coordinate descent over the five free offsets with coarse-to-fine grids and a diagonal pass, from the current tuning and from equal temperament. The alternation stops when the route repeats, and the best of seventeen starts is kept.

Where the search is thin #

Seventeen starts are seventeen starts. The objective is piecewise linear in the offsets and the route is discrete, so the alternation can stop at a corner that is not the best. The figures check that the result is never worse than any placed shape or the frets, which it could not be since those are among the starts; they cannot check that nothing better exists.

A still hand is not a player’s whole technique. The lowest rates model a player for whom moving costs more than a cent is worth. Real players move for reasons the objective does not have — voice leading, open strings for resonance, the next passage — and a tuning for a still hand is a tuning for one piece played one way.

The frets are exact. Every offset here is measured against equal-tempered frets, and a guitar cannot be in tune is about the ways a real neck is not — string stiffness, fret placement, the pull of a pressed string. A tuning two cents from the frets is inside those errors on most guitars, which is a reason to think of the still hand’s tuning as a direction rather than a setting.

Only five-limit ratios, and only the chord’s own pairs. As throughout: a minor chord’s third is 6 : 5, and whether a listener hears 7.7 cents as worse than 7.9 is not a question this arithmetic can answer; a comma under the threshold is the nearest the collection comes.

Still open: a tuning that knows the next piece #

Every tuning here serves one ground in one key. A guitarist does not retune between songs, and the tuning a player would actually choose is the one that serves a set — several grounds, several keys, played along the routes a player’s valuation implies.

The computation is the same alternation with a set of routes rather than one, each weighted by how much of the evening it takes. The prediction the grids make is that a set spanning several keys cannot share a placed shape’s corner — the C shape is best for every key only at the top of the range — and that at a player’s typical valuation the best tuning for a set is a compromise exact for nothing, like the still hand’s. If it is, then the one tuning every guitarist actually uses, equal temperament on the frets, is not a failure to tune for anything but the limit of tuning for everything, and the question becomes how far from it a set of songs can pull.