A guitar tuned for a set is tuned for the D shape

A guitar tuned for a still hand is tuned for nothing found the best tuning for one ground in one key by alternating two searches: route the piece, choosing a fingering for every chord, then retune the six strings for the chords on that route, and repeat until neither changes. The answer depended on how much hand movement the player will trade for a cent of intonation. At the most-in-tune end it was one major shape’s tuning with its silent string placed. In the middle it was sometimes a minor shape’s. For a player who will not move at all, it was a tuning exact for no chord whatever, every string within two cents of its fret.

That essay’s closing section pointed out that nobody retunes between songs. The tuning a player would actually choose is the one that serves a set — several songs in several keys, each played along the route the player’s valuation implies. It made a prediction as well. A set spanning several keys should not be able to share a single shape’s corner except at the most-in-tune end, so at a typical valuation the set’s best tuning should be a compromise exact for nothing, like the still hand’s. If so, equal temperament on the frets is the limit of tuning for everything, and the only question is how far from it a set can pull.

The prediction fails at the valuation it was made for. At five frets of movement to the cent, the figure tuned for one shape, a cent is worth five frets measured for a typical player, a set of five keys wants a tuning exact for the D shape. It pulls the high E string nearly twelve cents flat of its fret. A set shares a shape after all. It shares the one shape every key’s route happens to contain.

The same alternation, run on an evening #

The search is the earlier essay’s, extended to more than one song. Every song in the set is routed under the current tuning at the same exchange rate. The tuning is then searched on the union of all their chords: each chord is weighted by its share of its song and each song by its share of the evening, which here is equal for every song. The routes are recomputed, and the two steps repeat until no song’s route changes. The objective is the set’s mean of each song’s cost: its mean departure from just intonation over the sounding pairs of strings of each chord, plus the frets its hand moves divided by the exchange rate.

The search starts from the five placed shape tunings, from equal temperament and from two seeded random tunings. The earlier essay used seventeen starts for one song. This uses eight for a set, because each alternation is now a tuning search over as many as eighty chords, and every figure checks that the result is no worse than any placed shape or the frets.

Two sets are run. The first is the I–vi–IV–V ground twice round in C, G, D, A and E major, five songs. The second adds the I–vi–ii–V ground in the same five keys, ten songs, half of them with two minor chords instead of one. For every song the earlier essay’s single-song search supplies the comparison: what that song would cost if the guitar were retuned just for it.

At the top, the set changes nothing #

At fifty frets a cent, where a player will move anywhere for a hundredth of a cent, both sets settle on the C shape’s tuning with its low string placed — the tuning a silent string tunes the minor chord found best for the pop ground in every key, on its own. Every song in the set wants that tuning anyway, so tuning for the set costs exactly nothing over retuning for each song. At twelve frets a cent the answer is the same tuning, and the cost of not retuning is a tenth of a cent for the single ground and fourteen hundredths for the mixed set.

So the top of the range is the case the prediction allowed: every key can be played in the C shape, the tuning makes that shape exact, and the set is five copies of one song. The interesting part of the grid is below it.

In the middle, the shape every key contains #

At five frets a cent the set’s tuning for the single ground is exact for the D shape. Its offsets from the frets, with the low E pinned, are A +2.0, D +2.0, G +4.0, B +2.0 and high E −11.7 cents. Measured from the D string, the four strings the D shape sounds sit exactly where the D shape’s own placed tuning puts them: G two cents up, B level, the high E 13.7 cents down, which is a just major third below the equal-tempered one. The set pulls the guitar that far from equal temperament, and it does so on one string.

Why the D shape and not one of the others is visible in the routes.

The D shape moves up the neck without an open string in it, so it can be played at any fret, and at a moderate valuation the routes use it wherever it keeps the hand near the chords around it. In C major the tonic is a D shape at the tenth fret and the dominant a D shape at the fifth. In G major the tonic is a D shape at the fifth and the dominant an open D. In D, A and E major the tonic, the subdominant or the dominant is. Every song in the set plays two or more of its four chords in the D shape, and no other shape recurs in every key. The tuning that serves the whole set is the tuning for the fingering the whole set shares, just as the route’s own tuning served the shape a single route repeated.

The cost of not retuning is small because the routes adapt to the tuning. Each song gives up between a tenth and half a cent against its own tuning, C major the most (5.66 against 5.17) and A major the least (4.29 against 4.21), and the set as a whole gives up 0.22. Against the frets the set gains 2.1 cents. A player who tuned the high E twelve cents flat for a set of five pop songs in five keys would, on this arithmetic, be two cents closer to just than on the frets, and within a fifth of a cent of retuning for every song.

The silent strings are placed for the set #

The D shape leaves the low E and A strings silent, and a tuning that makes the D shape exact says nothing about them. So the placed D tuning of the earlier essays put them where the ground’s minor chord wanted them, at A +2.0 and D +8.5 measured from the low E. The set’s tuning puts them elsewhere: for the single ground, the A and D strings level with each other, two cents above the low E; for the mixed set, the A string back on its fret.

That placement is worth 0.20 cents over the placed D tuning for the single ground (5.36 against 5.57) and 0.06 for the mixed set. The silent strings are sounded by the set’s other chords: the minor chords, played as E minor and A minor shapes at around the fifth fret, and the subdominants played as C shapes. The set places them for all of those at once. A silent string tunes the minor chord found that a shape’s silent strings are free to serve another chord. Here they serve the set’s other chords, which within a single song is the same argument.

At the bottom, it depends on the set #

Below five frets a cent the two sets part, and they part exactly where the earlier essay’s prediction was aimed.

For the single ground, the D shape survives all the way down. At one and two frets a cent the set’s tuning is the same D-exact tuning, A +2.0, D +2.0, G +3.9, B +2.0, high E −11.7, and the routes move very little: four frets in G major at one fret a cent, seven at two, none in the other keys. A player who stays put can still play the dominant in a D shape in every one of these keys, because the D shape reaches every key’s dominant within a few frets of where that key’s other chords lie. Not retuning costs 0.19 and 0.18 cents, and the frets would cost 1.4 more.

For the mixed set, the prediction holds, and only there. At one and two frets a cent the set’s tuning is exact for nothing: the A, D, G and B strings are raised by two to four cents and both E strings stay on their frets. It is the still hand’s tuning again, equal temperament nudged in the direction the chords on the routes agree on. It gains only 0.44 cents over the frets and costs 0.48 over retuning for each song, the largest cost of the whole grid. The half of the set that is I–vi–ii–V has two minor chords where the pop ground has one. At a still hand those songs play their minor chords in the E minor and A minor shapes, whose own tunings ask something different of the high E from what the D shape asks; the search finds no placement of that string that serves both halves of the evening better than leaving it on its fret.

So the prediction was not wrong about the mechanism. A set can drive the best tuning to a compromise exact for nothing: it needs a player who will not move, and songs whose still-hand routes do not share a shape. It was wrong about how common that is. A typical player’s valuation is five frets a cent, and at five both sets share the D shape.

How far a set can pull from the frets #

The earlier essay asked how far from equal temperament a set of songs can pull a guitar. The answer depends on the valuation and not much on the set.

At a typical valuation the set pulls one string 11.7 cents flat and three others two to four cents sharp, gaining one to two cents over the frets. At the most-in-tune end it pulls three strings 13.7 to 15.6 cents sharp, the C shape’s tuning, and gains three to four cents. Only at the bottom, and only for a set whose songs disagree, does it stay within four cents of the frets and gain less than half a cent.

Equal temperament is therefore not the limit of tuning for everything, as the prediction proposed. It is the limit of tuning for everything played without moving, and even then only when the songs do not share a shape. A set of pop songs in the five guitar keys still shares a shape at two frets a cent.

Why the D shape and not the C #

It would be natural to expect the C shape, the most-in-tune corner at the top of the range, to win throughout. It does not, and the reason is the same one the earlier essay found in the middle of its range: a tuning serves the shape the route repeats, and the routes repeat different shapes at different valuations.

At fifty frets a cent a route will put every major chord of a key into the C shape, wherever on the neck that takes it, because intonation is worth any movement. The hand moves 18 to 49 frets a song to do it. At five frets a cent it will not: the C shape of a key’s dominant can lie several frets from its tonic, and the D shape, with its root on the fourth string and a compact grip, is usually closer to whatever came before. Among the five major shapes, the D shape is the only one that the moderate route uses in every key, so it is the only shape the whole set can be tuned for.

This is a fact about the shapes’ geometry on the neck, not about harmony, and it would change with the set. A set in flat keys, or of songs that live around the seventh fret, could share a different shape. What should carry over is the rule. A set is tuned for whichever shape its routes have in common, and the one it has in common is decided by the valuation.

What the grid says to a player #

Put as advice, the grid is shorter than the argument behind it, and it overturns less of the earlier advice than it seems to.

Tune a guitar for the shape of its key’s open chord gave the advice for one song and a player who follows the tuning around the neck. That advice survives for a set only at its own valuation: at twelve frets a cent or more, a set of any of these songs is tuned for the C shape with its low string placed, and retuning between songs buys a tenth of a cent at most. A player at that end of the scale loses nothing by tuning once.

A player in the middle of the scale, the one the earlier measurements describe, should tune for the D shape and let the silent strings sit where the set’s other chords want them. A tuned guitar has no minor chord in tune still applies: the set’s minor chords sit seven to ten cents from just under this tuning, as they would under any major shape’s. What the set changes is which major shape is worth making exact, and the chord a tuning gives up is a fingering, not a chord — here the fingerings given up are every major chord the routes play outside the D shape.

A player at the bottom of the scale, with a set that mixes grounds, should nudge equal temperament and stop there. That is the only case in which the earlier search’s instinct, that a tuning is right for some chords and wrong for the rest and so should be right for none in particular, turns out to describe the best tuning. Even then it is worth less than half a cent, which is under any threshold a comma under the threshold would count as audible.

What the search leaves out #

Equal weights. Every song here holds the same share of the evening. A set dominated by one key would drift toward that key’s own tuning, and the cost of not retuning, a fifth of a cent here, would fall for the dominant songs and rise for the others.

Two grounds, five keys. The sets are the two grounds the earlier essays used, in the five major keys a guitar favours. Minor keys, songs in flat keys, and songs with borrowed chords are all outside them, and each could change which shape the routes share.

Eight starts. The objective is piecewise linear and the routes are discrete, so the alternation can stop at a corner that is not the best. The figures check that it is never worse than any placed shape or the frets; they cannot check that nothing better exists. The single-song results it is compared against used seventeen starts.

The frets are exact, and so is five-limit. As throughout: every offset is against ideal equal-tempered frets, a real neck is off by a cent or two in places, and a chord’s error is its departure from five-limit just intonation over the pairs of strings it sounds. Whether a set tuned eleven cents flat on one string sounds better at a gig than one on the frets is a question the arithmetic cannot answer. A guitar cannot be in tune is the reminder that the frets were never exact.

Still open: the set a player chooses #

Everything above takes the set as given and asks for its tuning. A player chooses the set too. The D-shape tuning is two cents better than the frets for five pop songs in five keys; whether it stays better for songs outside those keys is not computed here. The natural next computation turns the question over: given one tuning, which songs does it serve best, and how many keys can a set span before its best tuning stops being any shape’s? That needs the routes of many more grounds than two, and it would say whether the old practice of retuning between sets rather than songs — one tuning per half of the evening — is about the strings stretching or about the keys changing.