Pitch and tuning

A set ordered by key relation moves the capo most

Grouped by the fret each song wants, a set of twelve minor songs in E, A, D and B gets the whole of what a moving capo saves with three capo changes. A real set is not grouped by fret; it is ordered for the music, and the commonest rule for that order is key relation — neighbouring songs a fifth apart. That order is the worst the capo can face: no two neighbours want the same fret, and three changes recover thirty-five per cent of the saving against fifty-one for a random order. The capo groups keys a tone apart, and a set built on fifths is the order that keeps them furthest from each other.

Assumes: A capo left in place serves one key · A capo is a transposition the tuning cannot see

A guitar retuned for an evening of songs plays some keys much better than others, and a capo lets each song be played as one of the keys the tuning serves. A capo is a transposition the tuning cannot see found that moving the capo for every song lets nearly every minor key borrow the open-string fingerings of the tuning’s best one. A capo left in place serves one key then found that a capo left at one fret for a run of songs does something far weaker — it renames the keys rather than sending them all to one place — and that the useful question is how much moving is enough. For four songs in E, A, D and B minor, the commonest keys for guitar songs, three capo positions recovered everything, provided the songs could be regrouped so each position was one block.

That proviso was the essay’s open question. A set list has an order chosen for other reasons — the shape of an evening, tempo, and key relations between neighbouring songs — and a capo may move only between neighbours in that order. Whether an order that suits the music also suits the capo, or is exactly the order that makes the capo move most, was left to be computed.

It is the order that makes the capo move most.

The set and the three orders

The set is twelve minor-key songs, three each in E, A, D and B minor, each the same i–iv–V–i progression twice round, priced as the earlier essays priced them: the mean departure from just intonation of the chords as played, plus the frets the hand moves divided by five, which tuned for one shape, a cent is worth five frets set as the exchange rate. The guitar takes the best of the four tunings the evening essays chose among; for this set that is the tuning built for the minor song in A, in which A minor and B minor are cheap at the nut and E and D minor are cheapest with the capo at the fifth and third frets, where they play as B minor’s shapes.

The capo may change fret a stated number of times, between songs; the first song’s fret is free. For a fixed order, the cheapest assignment of frets is found exactly, and the result is reported as the share of what moving the capo for every song saves over playing everything at the nut.

Three orders are compared. Grouped by key: three E-minor songs, then three in A, three in D, three in B. At random: forty shuffles of the twelve songs with no key twice running, the usual rule for a set. By key relation: the order with as many neighbours a fifth apart as possible and no key twice running — B, E, B, D, A, E, B, E, D, A, D, A, nine of its eleven steps a fourth or fifth.

Three changes, three answers

Fig. 1 Twelve minor-key songs, three each in E, A, D and B minor, in three kinds of order, the capo allowed to change fret the number of times on the dial: the share of a moving capo’s saving recovered. The dots are the forty random orders. The buttons play the first song of the fifths order and of the grouped order as the capo plays them.

With three capo changes allowed, the grouped set recovers all of the moving capo’s saving: one block at the nut for A and B minor, one at the fifth fret for E, one at the third for D, and the order of the blocks does not matter. Random orders recover between 35 and 70 per cent, 51 on average. The order by key relation recovers 35 per cent — the worst of the forty-one orders on the chart, tied with the worst random one.

Turn the dial and the ranking holds. With one change, grouped 50, random 20, fifths 5. With two, 56, 37 and 25. With four, grouped is complete, random 60, fifths 42. With six, random 77 and fifths 59.

Fig. 2 The same twelve songs: the share of a moving capo’s saving recovered against the number of capo changes allowed between songs, grouped by key, in random orders with no key twice running (the mean of forty), and with neighbours a fifth apart.

Drawn against the number of changes, the three orders separate from the first change and never meet. The grouped set needs three; the random sets reach nine-tenths at eight; the fifths order reaches three-quarters at eight changes, two thirds of the way to moving the capo for every song. A guitarist who plays a set ordered by key relation and wants the capo’s whole benefit has to move it before nearly every song.

Why every neighbour wants a different fret

In a set ordered by fifths, every song wants a different capo fret from the song before it. Twelve minor-key songs ordered so that neighbours are a fifth apart, in the minor song's own tuning: the fret each song's own best capo position puts it at, and the fret the capo is at when it may change only three times. B minor wants 0, gets 0; E minor wants 5, gets 0; B minor wants 0, gets 0; D minor wants 3, gets 0; A minor wants 0, gets 0; E minor wants 5, gets 0; B minor wants 0, gets 0; E minor wants 5, gets 5; D minor wants 3, gets 3; A minor wants 0, gets 3; D minor wants 3, gets 3; A minor wants 0, gets 0. Recovered: 35% of the moving capo's saving. — Fig. 3 The fifths order song by song, in the minor song’s own tuning: the fret each song’s best capo position puts it at (dots), and the fret the capo is at when it may change only three times (bars). The button plays the first song as the capo plays it.

The song-by-song picture is a zigzag. B minor wants the nut, E minor the fifth fret, B the nut, D the third, A the nut, E the fifth, B the nut, E the fifth, D the third, A the nut, D the third, A the nut. No two neighbouring songs want the same fret. With three changes allowed, the cheapest plan leaves the capo at the nut for the first seven songs, then follows the last five as best it can, and almost every song is played somewhere other than where it wants to be.

The reason is arithmetic about the tuning, not about these songs. The tuning’s cheap keys are A minor and B minor, a tone apart, and the capo’s whole use is to send a song to one of them. A song in E minor reaches B minor with the capo at the fifth fret; D minor reaches B minor at the third; A and B need no capo. Two keys can share a fret only if one fret sends both to cheap keys, which means they must lie the same distance apart as two cheap keys do: the same key, or a tone apart. Two keys a fifth apart are seven semitones apart, and no fret sends both to A or B minor. A set built on fifths is a set in which no two neighbours can share a fret.

The cheap pair a tone apart may not be a quirk of this tuning, though only this tuning is priced here. The evening essays found their tunings serving particular open-chord shapes — one guitar tuning serves every key, and it is the D shapes traced one of them to the shapes the open strings favour — and a guitar whose tuning favours two shapes a tone apart gives the capo the same map. If that is the usual case, the capo’s map, on which a tone is the closest relation, belongs to the guitar rather than to one evening’s tuning, and the conflict with a set built on fifths would follow the guitar from tuning to tuning. Whether it is the usual case is a search over tunings this essay does not run.

Two minor keys a tone apart lose least by sharing a capo fret; every other interval loses nearly three times as much. In the minor song's own tuning, for every pair of minor keys an interval apart: how much dearer the two songs are played at the one capo fret that suits both best than each at its own best fret, averaged over the twelve pairs, and how many of the twelve pairs lose nothing. semitone: 0.80, 3 of 12 free; tone: 0.28, 6 of 12 free; minor third: 0.77, 4 of 12 free; major third: 0.77, 4 of 12 free; fourth or fifth: 0.77, 4 of 12 free; tritone: 0.77, 4 of 12 free. — Fig. 4 In the minor song’s own tuning, for every pair of minor keys an interval apart: how much dearer the two songs are when played at the one capo fret that suits both than each at its own best fret, averaged over the twelve pairs.

Over all twelve keys, the cost of making two songs share a capo fret depends on the interval between them in one way only. Two keys a tone apart lose 0.28 on average by sharing, and half of the twelve such pairs lose nothing. Every other interval — semitone, third, fourth or fifth, tritone — loses between 0.77 and 0.80, and only three or four pairs in twelve share for free. The capo groups keys a tone apart, and nothing else.

That is the same fact the earlier capo essay found from another direction, when it priced pairs of keys under a capo left in place and found that pairs a tone or a semitone apart recovered most. It explains why the fifths order is the worst: a set built on fifths never puts two keys a tone apart side by side, because the tone pairs among E, A, D and B — A and B, D and E — are not fifths. The music’s rule forbids exactly the neighbours the capo’s rule wants.

Not one bad order but a gradient

The fifths order is an extreme, chosen to maximise the rule. A set that only leans towards key relation pays a share of the cost, and the random orders show how the share grows.

Fig. 5 A hundred and twenty random orders of the same twelve songs, no key twice running: the share of a moving capo’s saving recovered with three changes, against the number of neighbouring songs a fifth apart, coloured by whether the order has two or more neighbouring songs a tone apart. The line joins the mean at each count.

Among random orders, those with three or four neighbouring pairs a fifth apart recover 62 and 57 per cent of the saving with three changes; five, 50; six, 47; seven, 42; eight, 38. The single random order with nine fifth neighbours recovers 33 per cent, beside the searched order’s 35. Each neighbouring pair a fifth apart costs the capo about four per cent of its saving. The dots with two or more tone neighbours sit high at every count, because a tone pair is a pair that can share a fret; the dots without them sit low.

So the arithmetic does not only condemn an extreme. A guitarist who builds a set by feel, preferring a fifth between songs where one is available, is moving along this gradient, and the capo pays in proportion. A set with half its steps a fifth apart is already giving away about a tenth of what three capo changes could recover, and the more consistently the set flows by fifths, the more it gives away.

There is a second reading of the gradient. The random orders’ spread at each count — twenty or thirty percentage points between the best and worst order with the same number of fifth neighbours — is the room an order has to place its other steps well. Two orders with six fifth neighbours each can recover 35 or 60 per cent, depending on whether their remaining steps are tones placed beside each other, where a single capo change can serve a block. That is the room the open question below would search.

Over all twelve keys the circle is not special

Fig. 6 Twelve minor-key songs, one in each key, in four orders — round the circle of fifths, chromatic, by whole tones, and at random (the mean of forty): the share of a moving capo’s saving recovered against the capo changes allowed between songs.

The result is sharpest for a set in a few keys repeated, which is what a guitarist’s set usually is. For a run of twelve songs in twelve keys, every order is costly and the differences are smaller. Round the circle of fifths, three changes recover 47 per cent and six 80; a random order gives 47 and 79; chromatic order 54 and 79. Ordered by whole tones — C, D, E, F♯, G♯, B♭, then the other six — the set recovers 53 per cent with three changes, 87 with six and all of it with eight, because each pair of neighbours a tone apart is a pair that can share a fret, and a run of whole tones is a run of such pairs.

So over the full circle the fifths order is merely average: with every key present once, no order can group much, and the circle’s alternation costs no more than chance. In a set drawn from a few keys, where grouping is possible, it is the order that throws the possibility away.

What the music’s order and the capo’s order are

The order by key relation is not arbitrary. Moving a fifth between songs keeps most of the notes and changes one, so a set built that way flows: each song arrives in a key that shares six of seven notes with the last. Keys are neighbours set out why a fifth is the closest relation between keys; the key plan is the form found the same closeness organising movements of longer pieces. A set list built for flow is using the same map.

The capo uses a different map. To a capo, a key’s neighbours are the keys that can be sent to the same cheap key by the same fret, and on a guitar tuned for A and B minor those are the keys a tone apart. The two maps agree only about the same key twice, which a set list avoids for variety. So on this guitar, the two good orders for a set — by flow and by capo — share almost nothing, and a guitarist has to choose which to serve. The arithmetic puts a price on the choice: about two thirds of the capo’s saving, with three moves, for an evening of the four commonest guitar keys.

One retuning is worth what the pop songs pay for the minor tuning found that a moving capo nearly doubled what retuning once in an evening was worth, and the earlier capo essay that a capo kept at one fret brought none of that. This essay finds the middle ground — a capo that moves a few times between songs in an order chosen for the music — and finds that the order chosen for the music is the one that collects the least of it.

A rule for guitarists, stated plainly

The capo groups songs whose keys are the same or a tone apart, on any guitar tuned to favour a pair of minor keys a tone apart, which is what the evening’s best minor tuning does. The practical rule that follows is short. Put songs in the same key together, and songs a tone apart beside them; spend the capo changes at the boundaries between those blocks; and expect a set ordered for smooth key relations to cost the capo most of its benefit. A guitarist who wants both can alternate: a block of songs grouped for the capo, then a move by a fifth into the next block, so that the music’s preferred step is taken where the capo would move anyway.

That rule is a consequence of two choices already made — tuning the guitar for the minor songs, and using the capo to send every song to the tuning’s cheap keys. A guitar set tunes for its minor chords only when nearly half its songs are minor set the first condition; a set with fewer minor songs, played in standard tuning, has no cheap pair of keys for the capo to aim at, and the order of the set would hardly matter to it.

The sums under the set

Songs, tunings and prices are the earlier essays’. A minor song is i–iv–V–i twice round, priced by mean departure from just and frets moved over five, on a fourteen-fret neck shortened by the capo; the guitar takes the best of four tunings for the whole set.

Capo changes are counted, not timed. Each change is one unit of the budget whatever its size, and costs nothing else. A real change takes a few seconds and a tuning check, which is why the budget is small, and a change from the nut to the fifth fret is no harder than one from the third to the fifth.

The fifths order is found, not written. It is the best of four thousand seeded shuffles of the twelve songs by the count of neighbours a fifth apart, with no key twice running. Other orders with nine such neighbours exist. In every one of them the nine fifth steps are steps between songs wanting different frets, since the argument above does not depend on the particular order; only the two remaining steps could join songs that share a fret.

What the sums cannot say

What a set is ordered by. Key relation is one rule among several; tempo, mood, the singer’s voice and the shape of an evening are others, and most sets answer to all of them. The figures price the key-relation rule in its purest form, and say only that to the degree a set follows it, the capo pays.

Whether the cents are heard. As before, the differences are fractions of a unit a song — a few cents on a held chord, a few frets of hand movement — and whether an audience notices a set that is two units dearer over twelve songs is a question about listening the sums cannot settle. A guitar cannot be in tune is the reminder of how small the margins are that the whole evening is optimising.

Still open: a set ordered for both

The two maps share little, but they are not opposite. A set can step by fifths and occasionally by tones, flowing most of the time and grouping where it can. The computation is an order searched for both at once — the count of fifth neighbours traded against the capo’s recovered saving at a fixed number of changes — and it would say how much flow a set gives up to recover most of the capo’s saving, and whether there is an order that keeps nearly all of both, with the tone steps placed exactly where the capo changes are spent.