A guitar set tunes for its minor chords only when nearly half its songs are minor

An evening of pop songs in all twelve keys, played on one guitar tuned once at the start, settles on the tuning that makes the D chord shape exactly just. One guitar tuning serves every key, and it is the D shape’s found that, and a guitar set keeps its tuning until the third minor song priced what happens when minor-key songs join the evening. Each such song saves 1.69 if the guitar moves to the tuning that makes the C shape exact, because its dominant chord is fingered in that shape; the twelve ground songs lose 3.90 between them by the same move. Two minor songs do not pay for it and three do.

That was a vote between two tunings, and it said so. A minor song has a tonic as well as a dominant, and its tonic is a minor chord. There is a tuning that makes the E-minor shape exact, which is the shape a moveable minor tonic is most often fingered in, and the earlier essay on adding songs from outside the ground found that tuning taking over once enough minor songs were added. Between the dominant’s tuning and the tonic’s there is a count at which the vote changes hands. Finding it needs a third fixed tuning, three sums and the counts at which each overtakes the others.

The count is ten minor songs in the generic key. Ten songs against twelve is forty-five per cent of the evening. Before that, the guitar is tuned for the minor songs’ one major chord rather than for their minor ones, however many minor songs there are.

Three tunings, and what each one makes exact #

The set’s tuning is still the one that makes the mean cost of its songs smallest. A song’s cost is its mean mistuning in cents plus its hand movement in frets divided by five, the exchange rate a cent is worth five frets established. Each song is played by the cheapest route through its chords under whatever tuning the guitar has. Fix the candidate tunings, route every song under each, add up, and the vote is arithmetic.

The three candidates are each a tuning the search has found for some set of songs, and each makes one chord shape exact:

• The ground’s, which the twelve I–vi–IV–V songs choose on their own. It makes the D shape exact.
• The C shape’s, which the ground plus a few minor songs chooses. It makes the minor songs’ dominant exact wherever it falls in that shape.
• The E-minor shape’s, the one family tuning in which a minor triad is exact. It makes the minor songs’ tonic exact wherever it falls in that shape.

Under the ground’s tuning, the twelve ground songs cost 66.81 between them. Under the C shape’s they cost 70.71, and under the E-minor shape’s 77.98. A minor song in a generic key costs 8.16, 6.47 and 5.69 under the three.

Read as prices and savings, the E-minor shape’s tuning serves every minor song best; it is cheaper than the C shape’s by 0.78 a song. It also costs the ground the most. Moving from the C shape’s tuning to it costs the twelve ground songs a further 7.28, so the move pays only when the minor songs added save more than that: nine save 7.1, ten save 7.8.

The second move costs twice as much and saves half as much #

The counts come out of two divisions, and the reason the second count is so much larger than the first is visible in the two numbers each division is made of.

The first move, from the ground’s tuning to the C shape’s, costs the ground 3.90 and saves each minor song 1.69: two and a third songs, so three. The second move costs the ground 7.28 and saves each minor song 0.78: nine and a third songs, so ten.

That the second saving is small is the surprise, and its cause is in what each tuning does to a minor song’s four chords. A generic minor song here is i–iv–V–i, twice round, and under all three tunings its cheapest route is the same: the tonic in the E-minor shape, the subdominant in the A-minor shape, the dominant in the C shape. The tonic fills half the song’s beats and the other two a quarter each.

Under the ground’s tuning nothing in that route is exact: the tonic fingering is 6.9 cents out, the subdominant 10.2 and the dominant 5.5. The C shape’s tuning makes the dominant exact and, by moving the A and B strings, brings the tonic in to 6.3. That is a large saving for one chord’s worth of change.

The E-minor shape’s tuning makes the tonic exact, which is half the song, and the tonic’s gain over the C shape’s tuning is 3.13. But the same two strings that make the E-minor shape exact throw the C shape out by 7.0 cents and the A-minor shape out by a further 2.3, and those cost 1.76 and 0.59. Three-quarters of what the tonic gains, the dominant and subdominant give back, and 0.78 is what is left.

That trade is the core of the result. The minor songs’ tonic is the chord heard most often in them, but in this cost there is no extra weight on the tonic; it counts by its beats like any other chord. The arithmetic does not care which chord is the home chord. It cares how many beats each chord occupies and how far out of tune each fingering is.

Between the two tunings there is no compromise #

The C shape’s tuning and the E-minor shape’s look like two separate answers, and on six strings they are almost one.

Both leave the low E, the D string and the high E at equal temperament and put the G string 15.6 cents sharp. They differ only on the A and B strings, which the C shape’s tuning puts 13.7 cents sharp and the E-minor shape’s 2.0. The reasons are the two chords. In the C chord the root C is on the A string and must lie a just major third below the E on the D string, which needs the A string 13.7 cents sharp. In the E-minor chord the A string carries a B a fifth above the low E, and a pure fifth needs it 2.0 sharp.

So every tuning between the two is one number: how sharp the A and B strings are. That turns the choice into something that can be drawn completely. Slide the two strings from one end to the other and price the whole set at every point.

The minor song’s cost rises steadily along the line from the E-minor end to the C end, from 5.69 to 6.47, because each fingering’s mistuning is an average of absolute departures and moves in straight segments with any string it contains. The ground’s cost does something else. It is 77.98 at the E-minor end, climbs to 80.7 with the strings six cents sharp, and falls to 70.71 at the C end.

That hump is itself a finding about routes. For any one fixed route through the twelve ground songs, the cost along the line is a sum of absolute values of straight lines, and a sum like that can only bow downward: it cannot be higher in the middle than at both ends. A cost that rises in the middle therefore means the ground’s cheapest routes on the two sides of it are different routes. The hand that plays the ground with the strings at two cents and the hand that plays it at thirteen are using different shapes for some of its chords, and in between neither set of shapes is well served.

Add the two and the set’s cost has its lowest value at one end or the other for every count of minor songs, never in between. The guitar does not drift from the dominant’s tuning to the tonic’s as minor songs accumulate. It jumps, between the ninth song in A minor and the tenth, by 11.7 cents on two strings at once.

A guitarist who has tuned for one shape and senses the evening turning would naturally nudge the two strings part of the way. On this arithmetic that is the worst thing to do: every intermediate tuning costs the ground more than either end does and saves the minor songs less than the E-minor end does.

Ten keys vote alike; two do not #

The two-tuning vote found that ten minor keys vote identically for the C shape’s tuning and that the exceptions sit at the nut. The same holds for the third tuning, but the exceptions are different keys.

The reason ten keys are identical is the one that held before. A tuning on a guitar is a number of cents per string, and it is the same at every fret, so a moveable shape has one mistuning wherever on the neck it is placed. A minor song’s cheapest route in one key is its route in another moved along the neck. In ten keys that route uses the same four shapes, and the song costs 5.69 under the E-minor shape’s tuning and 6.47 under the C shape’s.

E minor is different because its tonic, under the E-minor shape’s tuning, is the open E-minor chord — the nut deciding a key’s vote, as it did when a single song’s best tuning was the shape of its key’s open chord. The shape costs nothing to reach — the hand has no position to move from — and it is exact. E minor saves 1.56 over the C shape’s tuning rather than 0.78, and pays for the move twice as fast. F minor also routes near the nut, with its tonic at the first fret, and is cheaper under every tuning for the same reason; but it gains from the move exactly what the other keys gain, so it votes like them.

So songs all in E minor move the guitar to the C shape’s tuning at the third, exactly as the other keys do, and on to the E-minor shape’s at the fifth. Five songs saving 1.56 each come to 7.8 and pay the second move’s 7.28 alone.

G minor does something the two-tuning vote could not see. Its dominant, D major, is the open D chord under the ground’s tuning — the one chord shape the ground already makes exact — so G minor gains only 0.33 from the move to the C shape’s tuning, and the C shape’s tuning never becomes the cheapest. But G minor gains 1.11 from the move straight to the E-minor shape’s, and that move costs the ground 11.18. At eleven G-minor songs the E-minor shape’s tuning wins, without the C shape’s having been chosen on the way.

A correction to the two-tuning vote #

That last count corrects the two-tuning vote. It reported that a set could hold eleven minor songs in G minor and keep the D shape’s tuning, and that the tuning left both candidates for the E-minor shape’s only at twelve. With the E-minor shape’s tuning in the vote, eleven G-minor songs already move the guitar there: the set’s mean is 6.111 under the E-minor shape’s tuning against 6.155 under the ground’s, while at ten it is 6.130 against 6.126 and the ground’s tuning holds. The number the earlier essay should have given is ten.

The error is the same kind that essay itself corrected in the one before it, and it is worth naming as such. A vote between candidates is exactly as good as its list of candidates. The two-tuning vote was right about the two it compared — the C shape’s tuning really does not overtake the ground’s in G minor until twelve — and it was silent about a third that overtook both one song earlier. The note now in that essay records the corrected count.

The corrected search agrees with the three-tuning vote wherever it has been run. It is the alternating search of the earlier essays, started from every family’s tuning, the ground’s own and the two minor-song tunings, and scoring every start unaltered as well as after alternating from it. With the ground plus nine songs round the circle it returns the E-minor shape’s tuning at a mean of 6.076, the vote’s number; with nine songs in A minor it keeps the C shape’s at 6.140, and with ten it moves to the E-minor shape’s at 6.130; with ten in G minor it keeps the ground’s at 6.126, and with eleven moves to the E-minor shape’s at 6.111. No fourth tuning appears at any of those counts.

The evening the tuning describes #

The three counts — two, nine and ten, depending on the key mix — are statements about what an evening is.

Up to two minor songs, a pop set is a pop set and its guitar is tuned as if the minor songs were not there. From three to nine, the guitar is tuned for the minor songs’ dominant: their one major chord, the one that most resembles the ground’s own chords. That is a tuning in which the ground songs are slightly worse and the minor songs’ turnaround to the tonic is clean. Only from ten does the guitar tune for minor chords themselves, and by then the evening is forty-five per cent minor-key songs.

It is a familiar picture in another form. A tuned guitar has no minor chord in tune, on the arithmetic of the earlier essays, because every tuning that suits the major shapes leaves the minor shapes’ thirds rough — and a silent string tunes the minor chord found the minor shapes’ trouble concentrated in one pair of strings. Here, the major-shape tunings keep their hold on a guitar long after the minor songs arrive, because the minor songs themselves contain a major chord whose tuning is cheaper to serve than their own tonic’s. The evening’s minor songs vote, for most of their range, for a major chord. And what the tonic’s tuning finally gives up, at the tenth song, is not the dominant chord, which is still played, but a way of fingering it in tune: the C shape at the fourth fret, now 7.0 cents out.

The model the counts rest on #

A six-string guitar with each string moved by a free number of cents, the low E pinned. Every chord is 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 is the mean departure from just of its sounding pairs, and each song is routed by the cheapest path through its chords at five frets of movement to a cent. Songs are weighted equally. The ground is I–vi–IV–V in all twelve keys and the minor song is i–iv–V–i, each played twice round.

The three fixed tunings are the one the search finds for the ground alone, the one it found for the ground plus minor songs on A and E, and the E-minor family tuning, solved for that one shape alone. The line between the last two holds every string but the A and B where both put it and sweeps those two together from nought to fifteen cents sharp, with the two tunings’ own values on the grid exactly.

What the vote leaves out #

Only the shapes and the routes the model knows. A guitarist who fingers a minor tonic as a barre in the A-minor shape, or plays the dominant as an open E chord at the nut whatever the key, routes the songs differently, and the counts move with the routes. The counts here are what the cheapest route gives, not what any particular player does.

The tonic is not weighted. Every chord counts by its beats. A cost that charged a mistuned home chord more than a mistuned passing one would favour the E-minor shape’s tuning sooner; a cost that weighted the final chord of a song heavily would favour it sooner still. The model has no such weight, and nothing here says what it should be.

One exchange rate. At five frets a cent the ground prefers the D shape’s tuning. At other rates the ground’s own tuning is different, as the earlier essays found, and both counts would have to be found again.

What none of these figures can show #

Whether the tenth song is where a guitarist would hear the change. The vote compares sums over a whole evening, and the margins at the thresholds are a hundredth of a cent-plus-frets in the set’s mean. The tunings either side of the jump differ by 11.7 cents on two strings, which is audible in a sustained chord as beating. A player would hear the difference between the tunings long before any arithmetic of this kind said which was better, and might reasonably prefer whichever made the songs they care most about sound right.

And the order of the evening. Every song counts equally and the sequence does not matter. A set whose minor songs come together, late, is the same set to this vote as one that alternates.

Still open: whether a guitarist who retunes once does better than one who never does #

Every count here assumes the guitar is tuned once for the whole evening. A guitarist who can retune once — two strings by 11.7 cents, between songs — could play the major half of the evening in one tuning and the minor half in the other. On this arithmetic that is worth a great deal near the threshold, where neither tuning serves the whole set well. The computation is the same sums split at one point in the evening: the cost of each half under its own best tuning, against the set’s under one, and against what a retuning costs in time or in the risk of a string going out. It would say how far from even an evening’s mix has to be before one tuning for everything is the better choice, which is the version of this question a working guitarist actually faces.