A capo left in place serves one key

A capo is a transposition the tuning cannot see found that on equal-tempered frets a song played with the capo at fret c is, to the tuning, the same song c semitones lower played at the nut, on a neck c frets shorter. That let every minor key borrow the fingerings of whichever key the tuning serves best. In the tuning that raises the B string by 13.7 cents — the tuning searched for a minor song on A — the best key is B minor, at 4.53 in this collection’s units of cents plus frets over five, and with the capo up to the seventh fret eight of the twelve minor keys reached that price. A minor half spread across twelve keys, which had given a retuning nothing to work with, nearly doubled what one retuning between the pop half and the minor half could save.

Every one of those songs took its own best capo position. The capo moved between songs, often by several frets, and the essay ended by pointing out that this is not what most guitarists do. A capo is usually put on for a run of songs and left there. The question it left was whether one fixed position recovers most of what a moving one gives, or whether a minor half in scattered keys needs the capo to move to get anything at all.

The answer is closer to the second than to the first, and the reason is a symmetry that the moving capo hid.

What a fixed capo does to a run #

A moving capo chooses a different transposition for each song, and so can send every song to the same cheap key. A fixed capo applies one transposition to all of them. The song on C♯ minor at fret 2 is B minor’s song, as before — but the song on B minor at the same fret is now A minor’s, the song on A minor is G minor’s, and so on round the circle. The keys are not sent to one place. They are renamed.

With the capo at the fifth fret, E minor becomes B minor and costs 4.53, which is what a moving capo would have given it. A minor becomes E minor and costs 5.72, its price at the nut being 5.09; B minor becomes F♯ minor and pays 8.62 instead of 4.53; C minor and B♭ minor are sent to keys that need room above the capo the shortened neck no longer has. The twelve songs cost 78.0 with the capo fixed at the fifth fret, against 66.8 at the nut in the same tuning and 57.9 with the capo moved for each song.

Turning the dial makes the point fret by fret. Each position gives B minor’s price to one key and A minor’s to another — the two keys a tone apart that the raised-B tuning was built for — and takes something from every key that was already well placed at the nut. No position does better than the nut.

Over every key, a fixed capo is a reshuffle #

That is not a property of this tuning or of these songs. It follows from what a fixed capo is.

A run that contains every key once, played with the capo at fret c, is priced as a run that contains every key once played at the nut — the same twelve keys in a different order — except that every song now has c fewer frets to work in. The order does not matter to a total. The frets do, and they can only cost. So in every tuning, at every fret, the twelve-key run is at least as dear as at the nut, and the curves in the figure all rise from left to right.

In the raised-B tuning the total goes from 66.82 at the nut to 67.23 at the first fret, 70.13 at the second, 78.04 at the fifth and 91.25 at the seventh. In the E-minor shape’s tuning it is 66.68 at the nut and at the first fret — the one place in the figure where a capo costs nothing at all — and 97.0 at the seventh. The cheapest the run can be with one fixed capo is therefore the E-minor shape’s tuning at the nut, 66.68, which is exactly where the minor half’s own tuning belongs to one key left a mixed-key minor half before there was a capo.

So for a minor half in every key, a capo that stays in one place is worth exactly nothing. All of the 8.81 a moving capo saves comes from moving it. The symmetry is the same one one guitar tuning serves every key found in the moveable shapes: a transformation that treats every key alike cannot favour any of them, and a fixed capo is such a transformation applied to the keys rather than to the shapes.

Every key added takes the gain away #

A minor half in all twelve keys is the limiting case. A real run has a handful of keys, and the answer depends on how many.

For a run in one key a fixed capo is a moving one, and recovers everything. For two keys it recovers 48 per cent on average, and eleven of the sixty-six pairs get nothing from any fixed fret. For three keys it is 29 per cent, for four 19, for five 13.5, for six 9; by seven keys half of all possible runs get nothing at all, and at twelve the share is zero.

The curve falls faster than one over the number of keys, and the reason can be read off the hero figure. One fret puts at most two keys on the tuning’s two cheapest prices. Every further key in the run is renamed to wherever the fret sends it, which on average is no better than where it started and on a shortened neck somewhat worse. So the gain from the keys the fret serves is diluted by the keys it does not, and the dilution grows with the run.

The moving capo, by contrast, saves about seven tenths of a unit a song whatever the run’s size — 0.76 for one key, 0.68 for four, 0.73 for twelve. Its value is per song. A fixed capo’s value is per run, and a run has room for only one or two winners.

The pairs a fixed capo suits #

Which two keys a run contains matters as much as how many, and the pattern is specific enough to be useful.

Pairs a tone apart recover 66 per cent of the moving capo’s gain, and pairs a semitone apart 59. Pairs a minor third apart recover 34 per cent, a major third 38, a fourth 44 and a tritone 46.

The reason is that each tuning’s cheap keys come in a pattern, and a fixed capo moves the pattern without changing its shape. The raised-B tuning’s two best minor keys are B and A, a tone apart. The E-minor shape’s tuning’s two best are E at 4.91 and F at 4.89, a semitone apart. A pair of songs a tone apart can be landed on B and A minor at once by one fret, and a pair a semitone apart on E and F minor; the buttons play the first case, C♯ minor and B minor at the second fret, sounding as B minor’s and A minor’s fingerings. A pair a third apart matches neither pattern, and one fret can serve only one of its songs.

That turns a fixed capo into a question about a set list rather than about a guitar. A run of minor songs in keys a tone apart — B and C♯, A and B, E and F♯ — is the run that a capo left in one place serves best, and it serves it nearly as well as a moving capo would. A run in keys a third apart is the worst a fixed capo can be given.

The four keys guitarists use most #

Most of that is about runs a guitarist might choose. The run most guitarists actually have is different, and it gets nothing.

E, A, D and B minor are the minor keys whose chords lie under the open strings, which is why they dominate guitar music written at the nut. In the raised-B tuning the four cost 21.06 together at the nut: E minor 5.72, A minor 5.09, D minor 5.72 and B minor 4.53. Moved for each song, the capo saves 2.38, by putting E minor at the fifth fret and D minor at the third, where both become B minor’s song.

No single fret saves anything. At the fifth fret E minor becomes B minor’s 4.53, but B minor becomes F♯ minor and A minor becomes E minor, and the run costs 24.38. At the third fret D minor is served and the others are not, and the run costs 25.01. The best fixed position for E, A, D and B is the nut.

The reason is short. B and A minor are already the raised-B tuning’s two best keys at the nut, so any fret that improves E or D moves B and A off their prices. The four keys a guitarist reaches for at the nut are the keys the open strings serve, and a tuning built around those open strings has already given its best prices to two of them. A capo kept in one place can only trade among them, and the trade loses.

That is the surprising connection, and it runs against the usual reason for a capo. Guitarists generally reach for a capo to escape the open-string keys — to play a song in B♭ minor with A minor’s shapes. The arithmetic says that is exactly where a fixed capo pays, and that for the open-string keys themselves it is an obstacle rather than a tool.

How many places the capo needs #

If a fixed capo recovers little and a moving one recovers everything, the useful question is how much moving is enough.

For E, A, D and B minor, two positions recover 76 per cent — the nut for A, D and B and the fifth fret for E — and three recover all of it: the nut, the third fret for D and the fifth for E. If the songs can be reordered so that the D-minor song and the E-minor song each come at a moment when the capo can go on, that is two capo changes for the whole run.

For a run in every key the climb is long and nearly straight. Two frets recover 19 per cent, three 40, four 60, five 74, six 87, seven 94, and only all eight positions from the nut to the seventh fret give the full saving. The first few frets each add nearly two units to what the run saves, because each brings two more keys onto B minor’s and A minor’s prices; the last two add about half a unit each, because the keys they would serve are already served.

So the answer to the lead’s question has two parts. A minor half in scattered keys gets nothing from a capo left in one place; it does get most of what a moving capo gives from a capo that moves a few times, provided the songs can be grouped by the fret they want. A guitarist who is willing to move the capo three or four times in a twelve-song run, and to put the songs in an order that makes that possible, has three fifths of the moving capo’s saving. One who keeps it in one place has none of it.

What it does to a retuning #

One retuning is worth what the pop songs pay for the minor tuning priced an evening of pop songs followed by minor songs with one retuning allowed between them, and the capo essay found that a moving capo nearly doubled that retuning’s worth for a minor half in mixed keys, from 11.18 to 19.22, because it made the minor half one key.

With the capo fixed for the minor half, the minor half in every key is priced exactly as it is at the nut, since the nut is its best fixed fret. Nothing the moving capo added to the retuning’s worth comes from the minor half in that case. The doubling belonged entirely to moving the capo, and a guitarist who keeps the capo in one place for the minor half should expect what the retuning was worth before capos were counted at all. That is a statement about the minor half’s contribution; how the pop half is played is a separate choice, and the pop ground, being in every key, obeys the same symmetry.

What the arithmetic rests on #

The songs and tunings are the earlier essays’. A minor song is i–iv–V–i twice round, priced by the mean departure from just of the chords it is played on and the frets the hand moves divided by five — a cent is worth five frets — over the four tunings an evening chooses among, on a fourteen-fret neck. A capo at fret c leaves fourteen minus c, and that is the only thing a capo costs here.

Every run has each key at most once. The size figure counts runs of distinct keys. A run with three songs in A minor and one in E minor weighs A minor three times, and the best fixed fret for it is A minor’s own — which is the nut in the raised-B tuning. Repeated keys make a fixed capo more useful, not less, and the four-key run here is the case with no repetition.

The capo is perfectly in tune, and moving it is free. A real capo pulls each string slightly sharp, by an amount that depends on the string and the capo; a guitar cannot be in tune priced the same kind of error for a finger. Moving it between songs takes a few seconds and a re-check of the tuning, which is part of why guitarists leave it in place, and none of that is counted.

What the sums cannot say #

Which runs guitarists play. The claim that E, A, D and B minor dominate is a claim about guitar music written for the open strings — folk, singer-songwriter and much popular repertoire played at the nut. A run drawn from a repertoire written at the keyboard, or transposed to suit a singer, would have different keys, and the pair figure says what to expect of it.

Whether the capo’s voicings are heard as the same song. A song played as B minor’s shapes with the capo at the fifth fret sounds in E minor but voices every chord as the B-minor shapes give it. What a tablature keeps is a reminder that the same notes on different strings are not the same sound; the price counts cents and frets, not timbre.

Whether the cents are heard at all. The differences between runs here are fractions of a unit a song: a few cents of tuning on a held chord, or a few frets of hand movement. A tuned guitar has no minor chord in tune found that a few cents can beat audibly on a sustained minor chord; whether a listener notices a run that is two units dearer over four songs is not something a sum can settle.

Still open: a set list ordered for the capo #

The positions figure allowed the songs to be reordered freely, so that every song wanting one fret could be played together. A real set has an order chosen for other reasons — key relations between neighbouring songs, tempo, the shape of an evening — and a capo that may move only between songs in a given order is a constrained version of the same problem. The computation is the positions search with the order fixed, priced over a set list in which neighbouring songs are chosen by key relation, and it would say whether an order that suits the music also suits the capo: whether songs a tone apart tend to sit together anyway, as the pair figure would like, or whether the order a set wants for its own reasons is the one that makes the capo move most.