A guitar set keeps its tuning until the third minor song

A guitarist tuning once for an evening of pop songs in every key ends up, on the arithmetic of the essays before this one, with the tuning that makes the D chord shape exactly just wherever it is played. One guitar tuning serves every key, and it is the D shape’s found that from four keys to twelve. Two minor songs retune a guitar set for their dominant then added minor-key songs to the twelve-key set and reported a clean threshold: one minor song is absorbed, and two move the tuning to the C shape’s, the shape the minor songs’ major dominant is fingered in.

It ended with a prediction about keys. A minor song’s dominant was fingered in the C shape because of where its key put it on the neck, so a key whose dominant fell in the D shape should leave the tuning alone however many such songs were added, while the others should tip it at two. The computation proposed was to add minor songs key by key and see whether the number that moves the tuning is set by how many of their dominants are fingered outside the D shape.

Doing that exposed something about the earlier result first. Adding A minor twice moves the tuning exactly as A and E minor do; adding C and F minor does not move it at all, although their dominants are fingered in the C shape too. Those two facts cannot both be right about the rule, and pricing the choice directly shows why: two minor songs do not pay for moving the tuning, and three do. The earlier threshold was a property of the search that found it.

The choice as a vote #

The set’s tuning is the one that makes the mean cost of its songs smallest, where a song’s cost is its mean mistuning in cents plus its hand movement in frets divided by five — tuned for one shape, a cent is worth five frets — each song played by the cheapest route through its chords. Between two candidate tunings that choice is simple arithmetic. Fix the two tunings, route every song under each, and add up.

The two candidates are the ground’s own tuning, the D shape’s, which the twelve I–vi–IV–V songs choose on their own, and the C shape’s tuning, the one the search reported for the ground plus minor songs on A and E. Under each, every song is a route solve that takes milliseconds.

Moving the twelve ground songs from the first to the second costs them 3.90 in total. Moving a minor-key song on A from the first to the second saves it 1.69, because the C shape’s tuning makes its E major exact. So the move pays when the minor songs added save more than 3.90 between them. Two save 3.39. Three save 5.08.

Drawn as the set grows, the two means cross between the second minor song and the third. With two added, keeping the ground’s tuning gives a mean of 5.934 and moving to the C shape’s gives 5.970. The search had reported 5.975 under the C shape’s tuning for that set — a worse answer than simply leaving the guitar as it was.

Why the search missed it #

The search starts from idealised tunings — each of the five shapes’ family tunings made exact, equal temperament, and two random tunings — and alternates between routing every song and re-tuning for the chords the routes use, until nothing changes. It never scores a starting tuning as it stands, and none of its starts is the tuning the ground had actually settled on. From the D family’s idealised tuning, with two minor songs’ chords in the mix, the alternation drifted to something worse than the ground’s own, and the C shape’s start won the comparison among what was left.

Adding the ground’s own tuning as a start, and scoring each start before alternating from it, fixes it. With that, the ground plus one minor song keeps the D shape at 5.767; plus two, at 5.938; plus three it moves to the C shape at 6.008. The corrected search and the two-tuning vote agree at every count.

This is worth stating plainly because the earlier essay’s figures were drawn with the search as it was, and its headline — that two minor songs move the tuning — should read three. What survives is everything about why the tuning moves: it moves for the minor songs’ dominant, to the shape that dominant is fingered in, because making the one major chord exact is the cheapest thing a major-shape tuning can do for a minor song. What changes is the count.

Every key votes alike, except at the nut #

The prediction about keys assumed that where a key sits on the neck decides which shape its dominant takes. On a guitar tuned by moving whole strings, that assumption fails almost everywhere.

A tuning here is a number of cents per string, and it is the same at every fret. So a moveable shape is exactly as in tune at the eighth fret as at the third, and a minor song’s cheapest route in one key is its route in another moved up the neck. Under the ground’s tuning, C minor is routed E-minor shape at the eighth fret, A-minor shape at the eighth, C shape at the seventh; A minor the same shapes at the fifth, fifth and fourth. Every one of ten keys costs exactly 8.16 under the ground’s tuning and 6.47 under the C shape’s.

The neck is not endless at the bottom, and that is where the keys differ. F minor’s route runs at the first fret with its dominant as the open C chord; it costs less movement, but it gains exactly as much from the C shape’s tuning as the others. B minor’s route under the ground’s tuning puts its dominant in the D shape at the fourth fret, and saves 1.61 from the move instead of 1.69. And G minor puts its dominant, D major, as the open D chord: the D shape itself, which the ground’s tuning makes exact. It gains only 0.33 from the move.

So the key does matter, as the earlier essay suggested, but not in the way it suggested. It matters only where the nut lets a dominant fall into the shape the ground already tunes, and that happens for one key’s dominant in twelve.

Why the guitar’s tuning is the same at every fret #

That ten keys cost exactly the same is not a coincidence of these progressions; it is the structure of the problem, and it is worth spelling out because it decides what “a key” can mean to a guitar’s tuning.

A string moved a few cents sharp is a few cents sharp at every fret, because a fret divides the string in a fixed ratio and the ratio does not care how the open string was tuned. So a chord shape — which frets on which strings — has one mistuning under a given tuning, wherever along the neck it is placed. A silent string tunes the minor chord used exactly that fact to count a shape’s error from its sounding pairs alone. Moving a whole progression up the neck by a fret therefore changes nothing a tuning can see: every chord is the same shape with the same error, and every change of hand position is the same distance.

The only things that are not the same everywhere are the two ends. At the bottom, a shape cannot go below the nut, so a key whose chords would need it has to use a different shape or open strings, and open strings are the one place a guitarist can finger a chord that costs no hand position at all. At the top, a shape cannot go past the fourteenth fret. Between them the neck is uniform, and a guitarist’s key is invisible to the tuning.

That is why the earlier essays’ single-song results depended on the key while this set’s minor songs mostly do not. Tune a guitar for the shape of its key’s open chord found a single song’s best tuning to be the shape its tonic is played in near the nut — a statement about the nut, and so about the key. A guitar tuned for a still hand is tuned for nothing found that when the hand is free to move, the route reaches whatever shape the tuning makes exact. Put together: in the middle of the neck any key’s route is every other key’s, and only the keys whose routes run into the nut get a vote of their own. G minor is one; its dominant is the open D chord, and the open D chord is the D shape at fret zero.

The chord a tuning gives up is a fingering said that what a tuning abandons is never a chord but a way of fingering it. The same holds for what a key contributes: not its chords, which are the same up the neck, but the fingerings the nut forces on it.

G minor, which hardly votes #

Add minor songs all in A minor and the C shape’s tuning becomes cheaper at the third. Add them all in G minor and it does not become cheaper until the twelfth: twelve songs saving 0.33 each come to 3.92, a hair over the ground’s 3.90. Before that, though, a third tuning the vote does not include takes over: the E-minor shape’s, which makes the minor songs’ own tonic exact. Correction: an earlier version of this paragraph said a set could hold eleven minor songs in G and keep the D shape’s tuning. With the E-minor shape’s tuning in the vote, eleven move the guitar to it and ten are the most the D shape’s tuning keeps; the comparison between two tunings was right about those two and could not see the third. The corrected search agrees where it has been run: with six songs in G minor added it keeps the D shape, at a mean of 5.977, while three songs in A minor are enough to move it. The difference between the two keys is one chord in each song — the dominant — and whether the nut lets it be the one open chord the ground’s tuning already makes exact.

That is the earlier prediction’s form, with its number corrected. A minor song whose dominant falls in the D shape barely pulls the tuning; one whose dominant falls in the C shape pulls it with a vote worth about half of what the ground can resist. The earlier essay’s direct test — G minor twice leaves the tuning alone, A minor twice moves it — was right in its first half and wrong in its second, for the reason above.

Mixed sets behave as the vote says. Songs in G and B minor alternating save 1.94 a pair, so four of them come to 3.87 and keep the D shape; six come to 5.81 and move it. C and F minor together save 3.39 and keep it. The old search, run on those sets before the defect was found, gave exactly those answers — it agrees with the vote everywhere except on the two-song sets that fall just short of the threshold, such as A and E minor or C♯ and E♭ minor, where it reported a move the sum does not support. The corrected search, run on the circle’s first four songs and on three songs in A minor, agrees with the vote at every count.

Who pays for the move #

The penalty the minor songs have to overcome is not spread evenly. The ground songs in C and G gain from the move — by 0.50 and 0.33 — because, as a guitar tuned for a set is tuned for the D shape found, their own best tuning is the C shape’s. The songs in E and E♭ barely notice. The song in A pays 1.78, more than any minor song gains; the others between 0.1 and 0.5.

That puts the earlier essay’s account of why two songs looked enough in its proper place. It described the tuning as a weighted vote in which the C and G songs already favoured the C shape and were outvoted by the other ten, and the minor songs’ dominants were extra voices for the C shape. The account is right. It was the tally that was off: the ten outvote the C and G songs by 3.90, and each minor song brings 1.69.

The model the numbers rest on #

A six-string guitar with each string moved by a free number of cents, the low E pinned; every chord fingered in one of the five moveable major shapes or three moveable minor shapes within the fourteenth fret, open strings included as fret zero; a fingering’s mistuning as the mean departure from just of its sounding pairs; each song routed by the cheapest path through its chords at five frets of movement to a cent; songs weighted equally. The ground is I–vi–IV–V in all twelve keys and the minor song i–iv–V–i, each played twice round. The two fixed tunings are the one the search finds for the ground alone and the one it found for the ground plus minor songs on A and E. The corrected search is the earlier one with the ground’s own tuning added as a start and every start also scored unaltered.

What the model leaves out #

Only two candidates. The vote compares the ground’s tuning with the C shape’s. A tuning that served the minor songs’ minor chords rather than their dominant — the E-minor shape’s, which the earlier essay found taking over with twelve minor songs — is not in the comparison, and at large numbers of minor songs it wins. The corrected search, which has every family’s tuning among its starts, is the check that nothing else intervenes at two or three songs; it agrees.

Local search. The corrected search can still miss a tuning none of its starts leads to. It now cannot report a tuning worse than one it started from, which is the defect that produced the two-song result.

The same progressions and the same rate. At other exchange rates the ground’s own tuning changes: at eight frets a cent and above it is already the C shape’s, and a minor song adds nothing to move. At two or three frets a cent, one minor song was enough, in the uncorrected search, to take the tuning off the D shape. Those rates have not been re-run with the correction.

What the picture cannot show #

Whether any guitarist hears the difference. The margins here are a few hundredths of a cent-plus-frets in a set’s mean, and the tuning either side of the threshold differs by up to sixteen cents on three strings. The second is audible as beating in sustained chords; the first is a statement about which is cheaper by the model’s own measure, not about a listener’s preference.

And where the minor songs come in the evening. Every song counts equally and the order does not matter. A guitarist who tunes before the first song and meets the minor songs late might reasonably weigh the early songs more.

Still open: what the minor songs’ own chords would vote for #

The C shape’s tuning wins for the minor songs by making their one major chord exact. With enough minor songs the earlier essay found the E-minor shape’s tuning taking over, making a minor chord exact instead. Between the two there is a count of minor songs at which the vote passes from the dominant’s shape to the tonic’s, and it can be found exactly the same way as here: a third fixed tuning, three sums, and the counts at which each overtakes the others. What it would say is how many minor songs an evening needs before the guitar should be tuned for minor chords at all — which, on this arithmetic, is where the evening stops being a pop set with some minor songs in it and becomes something else.