A set free to change key becomes a set in two

A guitarist’s set has two reasons to care about its order, and they pull against each other. One is musical: neighbouring songs a fifth apart sit comfortably together. The other is mechanical: in a tuning adjusted for chords, a capo lets a song borrow a cheaper key’s open-string shapes, and each move of the capo between songs costs time on stage. A set steps by fifths where the capo moves searched all 41,304 orders of twelve minor songs, three each in E, A, D and B, and found the trade exact: with three capo changes, a set can have three fifths for nothing, and pays about seven points of the capo’s saving for each fifth beyond.

That essay treated each song’s key as given, and said so. Its closing question was the natural loosening. A singer will often move a song up or down a tone to suit the voice or the evening, and a song so moved might land on a fret where the capo already sits. Give each song a choice of keys, search the keys and the order together, and ask how much of the trade goes away, and whether a set that can transpose can have every neighbour a fifth apart and the capo’s whole saving at once.

The search says no to the second part, and says something unwelcome about the question’s own yardstick along the way. A count of neighbouring fifths has been standing in for flow, and when a set is free to change key it finds out how to satisfy that count without going anywhere.

How the search is done #

Twelve songs with three possible keys each and an order among them looks like a much larger problem than 41,304 orders, about twenty-two billion, and no exhaustive pass is possible. It does not need one, because nothing about a set depends on more than where it is now. The state is which songs remain, the last key played, the fifths and the moved songs so far, the capo changes spent and the fret the capo is on. A dynamic program through those states takes about a second in all four candidate tunings, and it is exact.

It has one built-in control. With no song free to move, it must return the exhaustive front from the earlier essay, and it returns it to nine decimal places at all twelve counts of fifths, so the two searches agree where they overlap. The price of a song is the same as before: what the best route through its chords costs in tuning error in that tuning at that capo fret, in the same units as the rest of this series, with a capo limited to the seventh fret. And the saving is measured against the same two end points, a set played at the nut in its own keys and a set in its own keys with a capo free to move for every song.

Note what that means for the score. A moved song can do better than its own key ever could, so a set could in principle exceed the old 100 per cent. In practice it does not at three capo changes, but the scale is a yardstick borrowed from the fixed keys, not a limit.

Moving songs does not dissolve the trade #

At three capo changes, moving songs adds between five and twenty points at any count of fifths. The shelf rises from 84 to 92 and is exactly as long as it was: three fifths for nothing, a fall at the fourth. Seven fifths go from 61 to 75, and eleven from 37 to 57. There is no count at which a free set reaches the whole saving, and the slope past the shelf is about the same as before, so the trade has been raised and not removed.

The earlier essay’s hoped-for mechanism, a song landing on a fret where the capo already is, turns out not to be the main thing going on. To see why, it helps to look at the keys themselves.

Every song in this tuning is trying to be in B minor, the key the tuning was raised for, which costs 4.53 at the nut. A capo is how the others get there: E minor at the fifth fret, D minor at the third, C minor at the first. A minor cannot. Reaching B minor’s shapes from A minor would take a capo at the tenth fret, past the limit of seven, so A minor sits at 5.09 with no way down. Three of the twelve songs in this set are in A minor.

So the first thing a free set does is obvious once it is printed. It moves an A minor song up a tone, into B minor itself, where it costs 4.53 at the nut and needs no capo at all. That is worth 0.56 per song, about eight points on the saving scale, and it is the whole of the rise in the shelf. The moved song did not land on a fret the capo was already on; it left the one key the capo could not help. The mechanism the question proposed, in which a transposition is a chance to fit a fret, does appear, but only at high counts of fifths, where the two A minor songs at the end of a set become G minor and sit at the third fret with the D minor songs around them.

The shelf stays three fifths long for the reason it had before. Each capo change buys a fifth because a fifth always crosses frets: the keys a fifth apart want different places for the capo, and a change is where that crossing is paid for. Moving a song does not alter this. B minor wants the nut and E minor wants the fifth fret, so a set that swaps its A songs for B and its D songs for E still has a fret to cross at every fifth it wants to take for nothing, and three changes still pay for three of them.

The best set on the raised shelf is worth writing out, since it shows what a moved song does in practice: B A B A B E B E D E D E, with the capo at the nut for the first five songs, at the fifth fret for the E minor songs and back for the B between them. Two songs have moved a tone, and the three capo changes are used where the set steps from B to E. Nothing about it would be hard to play, though the D and E songs at the end want the capo at different frets and share one by accepting a small loss on each. A singer willing to move only one song should move an A minor song to B minor, which gives the whole eight points at up to two fifths; keeping the third fifth as well takes a second move, of a D minor song to E. A guitarist who has been leaving the capo in place gains far more from the three changes than from the moved songs.

How many songs are worth moving #

A singer does not want to transpose half of a set, and the answer turns out to be that nothing asks for it.

Two moved songs buy most of what moving can buy, and six buy all of it. Three fifths reach 92 with two songs moved. Eleven fifths reach 49 with two and 57 with six. At seven fifths the steps go 61, 67, 67, 69, 75. Beyond six the numbers do not change, at any count of fifths and any number of capo changes tried. A whole set in which every song is free is worth no more than one in which half of them are.

The same arithmetic can be set against capo changes, since those are the other resource a guitarist spends.

Moving songs buys the most where fifths are many and capo changes are few. With one capo change and eleven fifths it nearly doubles the saving, 21 to 40. With six changes and three fifths it only adds the eight points that A minor gives it. A singer who will move songs is therefore most useful to a guitarist who will not move the capo, and of least interest to one who will.

What a set in two keys says about the yardstick #

The last figure is the one the search did not expect to produce. It prints the sets themselves.

At its limit, a set free to change key stops being in four keys and becomes a set in two, B minor and E minor in alternation. It gets eleven fifths, and 57 per cent of the saving, and it is plainly not what anybody meant.

It was not what the fixed-key set meant either, and that is the part worth reading back. The own-key set with eleven fifths reads B E B E B E A D A D A D. That is also a pendulum, one that changes its swing once. Eleven fifths in twelve songs must be a pendulum, and the reason is arithmetic rather than anything about the search. The four keys lie along a line on the circle of fifths, B–E–A–D, and a line has only three edges. A set that steps by fifth from key to key and never turns back can take only three such steps; the fourth has to reverse. With a tone’s freedom either way the chain lengthens to eight keys, C♯ to C, and a walk gets seven.

So the count on the horizontal axis of every figure in this essay and the last one measures something narrower than it seemed to. Three ways to measure how far a key is said that steps round the circle of fifths and notes in common are one measure, and it is a measure of distance between two keys. Summing it over a sequence says how many neighbours are close and nothing about whether the sequence goes anywhere. A rocking set scores perfectly. Tonal journey, which is what the key plan is the form describes for a classical movement, is a property of the whole path and is invisible to this count.

That raises the practical question of what a set that does walk the chain costs. It can be priced directly, by searching only the orders of the twelve songs that contain the run D–A–E–B or its reverse and have no other fifth. The best of them recovers 76 per cent, against the shelf’s 84. So a set which goes once through all four of its keys in order pays eight points for it.

The numbers depend on this set #

The set was chosen for the earlier essays’ reasons. Four minor keys in a chain, three songs each, with B minor among them, in a tuning raised for that key. The A-minor effect exists because the set has A minor in it and the limit sits at the seventh fret. Allow a capo at the tenth, which some guitarists would accept and few necks would enjoy, and the own-key shelf rises to 92 by itself, with no song moved, which is the check that A minor’s limit was the cause. Change the set to include G minor and the same argument applies with a larger effect, because G minor at 5.72 has further to fall.

A tone either way is not the whole of a singer’s freedom. Widening the choice to three semitones either way changes the shelf not at all and moves the middle of the front by at most eight points: the 57 at eleven fifths stays, and the only gain is at five and at nine fifths. The tone was never what was limiting anything.

Nothing is charged for moving a song. A moved song is simply priced as the key it lands in. A singer’s range, an arrangement already learned, a band that has rehearsed in the old key: every one of these has a cost, and none is on the scale. That is why the limit on the number moved is the useful dial rather than the unconstrained result. Two moved songs is plausible on most evenings, and six is a different evening.

The tuning’s prices are the model’s. Each comes from a route through a minor song’s chords, charged by tuning error at a stated rate, in a tuning fixed for the whole set. A capo is a transposition the tuning cannot see explains why capo fret and key are interchangeable to this cost, and one guitar tuning serves every key discusses the choice of tuning. None of these prices has been checked against a listener in a room, and a person playing at the nut in a tuning a few cents out may simply not care.

What a best set cannot say #

The search knows nothing of music beyond a distance between keys. Tempo, mood, the song the audience came for, the opening and the encore have no term. An order that is best on this front may be the worst possible evening. All the figures can do is say what a constraint costs. Nothing here recommends any set to anybody.

A key’s character is not in the model. Moving a song from A minor to B minor changes where the open strings fall, and hence its sound and what the fingers do, well beyond its cost in tuning error. For eighteenth-century keyboard tunings, keys that had characters measured a real difference in each key’s intervals. Here the tuning’s error is the only difference this cost can register. A guitarist may hear more in the move than is priced.

And whether a listener notices fifths is assumed. The earlier essays adopted key relation as a stated rule of set-building, and a set ordered by key relation moves the capo most priced it without asking how much of an audience hears it. After a song of three minutes, with applause in the gap, an unaccompanied memory of the last key is not obviously available. A set’s key relation might be wholly a matter for the player.

Whose set #

This is a solo or small-band guitarist in a tuning adjusted for chord shapes, with a capo, playing songs built on a few open chords, in a Western popular or folk tradition where a set is a sequence of separate songs. A set of the kind found in classical guitar recitals uses no capo and is often planned around a key scheme for its own sake, which the pendulum would badly fail. A band with a keyboard player is under none of these constraints, and a tuning that is not being adjusted has no prices for the search to work on.

Still open: a set that has to go somewhere #

The pendulum shows what the search needs. Fifth-neighbours is a local count, and what a set wants is a path: a limit on how often it may reverse along the chain, or a stated journey from one end of the chain to the other and back. The computation is this search with the direction of the last step added to its state, and a count of reversals beside the count of fifths. It would say what a set with no more than one or two turning points costs in capo changes, how much of the front’s upper reach survives once rocking is forbidden, and whether a set that climbs the chain and returns can recover anything like the shelf’s 84 per cent.