A set steps by fifths where the capo moves

A guitar tuned for its minor chords plays some minor keys better than others, and a capo lets a player borrow a better key’s shapes for a worse one. A capo is a transposition the tuning cannot see priced that borrowing song by song, and a capo left in place serves one key found that a capo which never moves recovers little of what a capo moved for every song would save. The space between is a capo moved a few times in an evening.

A set ordered by key relation moves the capo most put twelve minor songs, three each in E, A, D and B minor, into three orders and priced each with a fixed number of capo changes. Grouped by key, three changes recovered the whole of a moving capo’s saving. A different key each song, at random, recovered about half. And ordered by key relation — as many neighbouring songs a fifth apart as possible, the commonest principle for a set — three changes recovered thirty-five per cent: the worst of the three. The capo groups keys a tone apart, the essay found, and a set built on fifths keeps them furthest from each other.

It ended by asking for the order that serves both: the count of neighbours a fifth apart traded against the capo’s recovered saving, at a fixed number of changes. The suspicion was that a set could step mostly by fifths and occasionally by tones, with the tone steps placed exactly where the capo changes are spent.

Twelve songs in four keys, with no key twice running, have 41,304 orders. That is few enough to search every one.

The front, exactly #

The search prices each order the way the earlier essays priced a set: the cheapest way of playing every song in one of the four candidate tunings, with the capo at whatever fret each song is given, moving at most the stated number of times, reported as the share of a moving capo’s saving over the nut that it recovers. For each count of neighbours a fifth apart, from none to eleven, it keeps the best order.

With three capo changes the front is a plateau and then a slope. An order with no fifths, one, two or three recovers 84 per cent of the saving at best. From the fourth fifth on, each one costs about seven points: 78 per cent with four, 70 with five, 63 with six, 54 with eight, 46 with nine, 37 with eleven — every neighbouring pair a fifth apart.

Two things in the figure correct the earlier essay. Its fifths order, found by trying a few thousand random shuffles, had nine fifths and recovered 35 per cent; the best order with nine fifths recovers 46, and there are orders with ten and eleven fifths that recover more than it did. And its grouped order’s hundred per cent was an order that played each key three times running; with no key twice in a row, the ceiling is 84.

The dashed line is the typical order at each count, and it falls far faster than the best. A random order with three fifths recovers 62 per cent on average, against the best’s 84; with six, 46 against 63. Ordering a set for the capo is worth almost as much as the capo itself, whatever the count of fifths.

Why a fifth costs a capo change #

The plateau has a simple cause, and it is in which fret each key wants. The tuning the best orders use is the one the earlier essays raised for B minor, and in it B-minor shapes are the cheapest of all: a song in B minor costs 4.53 in the essays’ units, a song in E or D minor at the nut 5.72. The capo is how a key borrows those shapes. E minor with the capo at the fifth fret is played with B-minor shapes, and so is D minor at the third; both then cost 4.53. B minor needs no capo at all. A minor has no fret within reach that turns it into B minor, and it plays best open, at 5.09.

So the frets the four keys want are five, three, nothing and nothing, and the only two keys that share a fret are A and B — not because they are a tone apart as such, but because one of them is the tuning’s favourite key and the other cannot reach it. D and E, also a tone apart, want different frets, but each loses little at the other’s: D minor at the fifth fret costs 5.09, under half of what it gains at the third. A step a fifth apart, in this set, always joins a key that wants the capo off with one that wants it on: A to E, A to D, B to E. Either the capo moves, or one of the two songs sits at a fret that serves it badly.

A fifth step is therefore exactly as expensive as a capo change, and a set with three changes to spend can afford three fifth steps without any song played at the wrong fret. The tone steps cost nothing between A and B and a little between D and E, which is why the best orders are built from runs of those pairs.

That is the earlier essay’s observation that the capo groups keys a tone apart, stated as a mechanism — and corrected: what the capo groups is keys that reach the tuning’s cheap shapes at the same fret, which in this set happen to be a tone apart. It is also why the front is a plateau up to the number of changes allowed: each change buys one fifth step that costs nothing, and the fourth fifth has to be paid for with a song at the wrong fret.

Each change buys a fifth #

The same pattern holds for every number of changes, roughly. With one change the plateau runs to one fifth, at 76 per cent. With two it runs to two, at 84. With three, to three. With four and six the ceiling rises — 92 and then 100 per cent, at which point the capo can be at every song’s wanted fret — and the plateaus run to about two fifths at the top level and on to four and six at the next. Each extra fifth beyond the plateau costs a moving share of the saving, between five and fifteen points, and the most flowing order of all, eleven fifths, recovers 21 per cent with one change, 37 with three, 56 with six.

So the trade is not a curve a set list has to find by trial. It is a budget. Each capo change a player is willing to make buys roughly one neighbour a fifth apart at no cost, and the rest of the flow has to be paid for.

Where the best orders spend their changes #

The earlier essay’s suspicion was that the compromise would put its tone steps where its capo changes fell. The best orders do the opposite. Across the four orders drawn, the capo moves twelve times, and eleven of those moves fall on a step a fifth apart; the twelfth falls on a step a minor third apart. Every step between A and B minor, in every order, keeps the capo still.

The best order with three fifths makes the pattern plain: E D E D E, B A, D, A B A B. The opening run alternates E and D, a tone apart, and plays both at the fifth fret, where E is best and D gives up a little. Then a fifth to B and A, both with the capo off, alternating by tones. A fifth to D at the third fret, a fifth back to A, and a run of A and B to the end with the capo off. Three fifth steps, three capo changes, and the tones in between.

The order with eleven fifths has no tone steps at all — D A D A D A, E B E B E B — and spends its three changes on three of its eleven fifths, leaving eight fifth steps in which a song sits at a fret that is not its own. That is the thirty-seven per cent.

So a set that wants both flow and an easy capo should be built from runs of keys a tone apart, joined by fifths at the points where the capo moves. The tone runs are where nothing happens to the capo; the fifths are where the music moves and the capo moves with it.

Orders full of fifths are rare #

Most orders sit in the middle. Five and six fifths are the commonest counts, with 10,626 orders each; only 66 orders have ten and only two have eleven, and only two have none at all. A set list put together without attention to key relation will usually land at five or six fifths, and recover about half of the saving with three capo changes — 50 and 46 per cent on average — against 70 and 63 for the best orders at those counts.

The rarity at the top has a musical consequence the earlier essay’s search could not see. Its random search for the order with the most fifths found nine, because orders with ten or eleven are a few dozen in forty thousand. A musician building a set by key relation by hand would be in the same position: the orders that flow by fifths throughout are so few that they are found only by design, and they are the orders the capo likes least.

The tuning decides which steps are free #

The whole result rests on one tuning’s preferences, and it is worth seeing how much. The earlier essays chose among four tunings for whole evenings, and for this set the cheapest is the one raised for B minor. In it the capo’s work is to bring each key to B-minor shapes, and so the frets the keys want are set by their distances below B: five semitones for E, three for D, none for B. A minor, two semitones below B, would need the capo at fret ten to borrow the same shapes, which is beyond the reach the essays allow, and settles for the open string.

In another tuning the frets would be different and so would the free steps. The minor half’s own tuning belongs to one key found the best tuning for the minor songs alone helping only A, B and D minor; two minor songs retune a guitar set for their dominant found two minor songs enough to move a whole set’s tuning; a guitar set tunes for its minor chords only when nearly half its songs are minor found the choice of tuning itself moving as a set’s balance changed. Every such choice moves the wanted frets, and with them the pairs of keys that can stand next to each other without moving the capo. In this set they are the pairs a tone apart. In a set tuned for A minor they would be whichever keys sit the same distance below A.

What does not depend on the tuning is the shape of the trade. Each capo change is one opportunity to join two keys that want different frets, and a fifth step, in any tuning that favours one key, usually joins keys at different distances from it. A set list can spend its capo changes on the music’s motion, and a player who knows which keys share a fret in their tuning can build runs of those keys between the moves.

What a set list built this way sounds like #

The best order with three fifths reads, in its keys alone, like a set with three sections: a run alternating E and D minor, a run of A and B, and a return after a single D. Each section stays in a narrow band of two keys a tone apart, and each change of section is a move by a fifth. One retuning is worth what the pop songs pay for the minor tuning priced an evening’s single retuning, made between the pop songs and the minor songs; the capo’s three changes play the same part here at a finer grain, each one a seam between sections of a set rather than between the halves of an evening.

Whether a set of tone-paired runs sounds like a set with flow is a question about listeners. A tone step between minor keys is a strong move — a whole step up or down the scale’s floor — and a set that alternates E and D minor for five songs does not stay in one place the way a run in a single key would. The arithmetic says only that such runs are cheap for the capo and that the fifths can be kept for the joins.

What the arithmetic assumes #

The set is the earlier essay’s: twelve minor songs, three each in E, A, D and B minor, with each song’s cost the cheapest route through it in one of four candidate tunings at a given capo fret, and the whole set in the one tuning that makes it cheapest.

No key twice running. A set that plays two songs in one key back to back is a different musical choice and the earlier essay’s grouped order made it freely; the search here excludes it, as the earlier essay’s random and fifths orders did. With repeats allowed, the grouped order’s hundred per cent is available at two fifths, and the front would start higher.

Capo changes are free up to the limit and forbidden beyond it. A player’s real cost of a change — a few seconds of retuning by ear between songs — is not modelled. The fixed number of changes is the earlier essays’ device for pricing an evening.

A fifth is a fifth either way. Neighbours a fourth apart are counted as a fifth, as the circle of fifths counts them; the arithmetic of frets does not distinguish them.

What a search of orders cannot say #

What key relation is worth to an audience. The arithmetic counts neighbouring songs a fifth apart as the measure of flow because it is the commonest principle stated for building sets. Whether listeners hear a set that steps by fifths as flowing, or a set of tone runs joined by fifths as anything in particular, is not a question about frets.

What the songs are. The keys are the earlier essays’ choice. A set in four other minor keys would have different wanted frets, and the plateau’s length and height would move with them.

How a player moves a capo. Some players use partial capos, some retune instead, some play in open position with a different tuning for the second half; a guitar set keeps its tuning until the third minor song priced one of those alternatives. The search compares orders under one device only.

Still open: a set whose keys are not given #

The search fixes the twelve songs’ keys and orders them. A set list is often the other way round: the songs are chosen, but each can be played in more than one key, and a singer will move a song a tone up or down to suit the voice or the evening. Given a set whose songs each have two or three acceptable keys, the order and the keys could be chosen together, and every tone-apart transposition is a chance to land a song on a fret the capo already sits at. The computation is this search with each song’s key a choice among its acceptable ones, and it would say how much of the flow-against-capo trade disappears when the keys can move — whether a set that can transpose its songs can have every neighbour a fifth apart and the capo’s whole saving at once.