A capo is a transposition the tuning cannot see

The minor half’s own tuning belongs to one key searched for the best tuning for a run of minor-key songs on a guitar, with no pop songs to share it, and found that the answer depended on the key. Nine keys kept the tuning that makes the open E-minor shape exact. Three — A, B and D minor — found tunings of their own, each built around an open chord whose minor third it puts in tune: A minor’s and B minor’s raise the B string by 13.7 cents, D minor’s raises the high E. A minor half in one of those three keys wanted a tuning of its own; mix in a song in E minor and the advantage was gone.

The reason it gave was structural. In nine keys the minor chords cannot be fingered at the nut in the shapes allowed, so every tuning plays them up the neck on moveable shapes, where a shape costs the same wherever it sits and a tuning cannot favour one key over another. The essay ended by naming the device that changes that: a capo, which clamps every string at one fret and so moves the nut.

What a capo does to the tuning problem #

The price of a song in these essays is two things added together: the mean departure from just of the chords it is played on, in cents, and the frets the hand moves, divided by five — the rate at which a cent is worth five frets. Both depend only on the shapes a song is fingered in and the order they come in. On equal-tempered frets a chord shape’s intervals are the same wherever it is placed: moving a shape up five frets moves every note by five semitones and leaves every interval between strings, and so every departure from just, exactly as it was. It is why a guitar set keeps its tuning until the third minor song could compare shapes rather than keys, and it is the property one guitar tuning serves every key turned into a theorem about moveable shapes.

A capo at fret c applies the same move to the open strings. Every open string sounds c semitones higher, and every shape that used it now uses the capo instead. To the tuning, a song in a key k played with the capo at c is indistinguishable from the song in the key c semitones lower played at the nut — the same shapes, the same errors, the same hand movement — on a neck that ends c frets sooner. A capo is a transposition the tuning cannot see. The only thing it costs, in this arithmetic, is the frets it takes away above it.

Put the other way round, every tuning has a best key for each kind of song — the key whose open strings fall where its exact shapes want them — and without a capo a song gets that price only if it happens to be written in that key. The searches of the earlier essays were, in this light, searches for tunings whose best key was the key the songs were in. A capo dissolves the coincidence: the songs bring their own key, and the capo supplies the distance to the tuning’s.

That turns the question the earlier essay left into a simpler one. With a capo, a song does not have to be played in its own key; it can be played in whichever key its tuning serves best, provided that key is no more than a capo’s reach below it.

The figure is where the earlier essay left the minor keys, and it already contains the answer. The raised-B tuning is A minor’s own and very nearly B minor’s, and in it B minor costs 4.53: the lowest price any minor key reaches under any tuning the searches have found. A capo can bring any key down to B minor, so the question becomes which keys can bring B minor’s price with them.

Eight keys become B minor #

With a capo allowed up to the seventh fret, eight of the twelve minor keys cost 4.53 in the raised-B tuning: B minor itself at the nut, C minor with the capo at the first fret, C♯ at the second, D at the third, E♭ at the fourth, E at the fifth, F at the sixth and F♯ at the seventh. Each is B minor’s song moved up by the capo, and each pays B minor’s price to the hundredth. A minor and B♭ minor reach A minor’s 5.09, B♭ with the capo at the first fret. G and A♭ minor stay at 5.72, the price every key pays on moveable shapes in this tuning, because the capo positions that would make them B minor are the eighth and ninth frets. Slide the dial up to the ninth fret and they join the others; slide it down and the eight shrink to the keys within reach.

The E-minor shape’s tuning does well with a capo too, and the dots show how: eight keys reach 4.89, which is F minor’s price at the nut in that tuning, where the open C chord is the dominant — C minor with the capo at the seventh fret, F♯ to B minor at the first to the sixth. C♯, D and E♭ minor would need the eighth fret or higher, and E minor stays at its own 4.91. So the E-minor shape’s tuning with a capo also gives most keys a price no key had without one; but its best is still a third of a unit dearer a song than the raised-B tuning gives eight keys, and over a run of a dozen songs that is four units, about a third of what a single retuning saved on a mixed-key run of that length before there was a capo.

So the lead’s question has a clear answer and a slightly different mechanism from the one it proposed. A capo does admit the keys the open strings do not serve, and most of them; but it does so by turning them into B minor rather than A minor or D minor. The lead imagined a capo at the second fret turning the open D-minor chord into E minor; in the raised-B tuning that buys nothing, because D minor is not where that tuning is cheap — D minor’s own tuning raises a different string. The capo has to put the song in a key whose open chords the tuning in hand puts in tune, and for the raised-B tuning that key is B minor.

E minor at the fifth fret #

E minor is the clearest case, because it was the key that defeated the minor half’s own tuning in the earlier essay: one song in E minor took the best tuning for a mixed run straight back to the E-minor shape’s.

At the nut, in the tuning built for it, the E-minor song is the open E-minor chord, the same shape moved to the fifth fret for A minor, the G shape at the fourth fret for the dominant B — ten cents from just — and the open chord again. Its mean departure is 2.51 cents and its hand travels twelve frets: 4.91 in all.

With the capo at the fifth fret, in the raised-B tuning, the tonic is the A-minor shape two frets above the capo, exactly in tune; the subdominant is the open E-minor shape at the capo, 10.6 cents out; the dominant is the E shape two frets above the capo, 7.6 out. Its mean departure is higher, 4.53 cents, and the hand does not move at all: every chord sits within the same two frets. The song is cheaper by 0.38 because it has traded a little purity for a great deal of stillness, which is the exchange the chord a tuning gives up is a fingering found every tuning making in one direction or the other.

That changes what it means for E minor to have been the obstacle. Without a capo, E minor was the one key whose open chords the E-minor shape’s tuning served, and it pulled any mixed run back to that tuning. With one, E minor is just another key that can become B minor, and the pull is gone.

The pop songs are paid too #

A capo is not a minor-key device, and nothing in the argument above was about minor keys. The pop ground — I–vi–IV–V, in all twelve keys — is subject to the same transposition.

In its own tuning the pop ground costs between 4.29 and 5.78 a song at the nut, 66.81 for the twelve keys together. Its cheapest key is A major, at 4.29, where the open A and D chords are the tonic and subdominant and the D chord’s shape is the one the tuning makes exact. With a capo up to the seventh fret, five keys reach A major’s price outright — B♭, B, C and C♯ with the capo at the first to the fourth fret, and A itself — and the twelve together fall to 58.15. The keys that need a high capo to become A major do not get all of it: E with the capo at the seventh fret pays 4.86 rather than 4.29, because seven frets of capo leave only seven above it and A major’s cheapest route wants more room than that. That is the one place in this arithmetic where the capo is not free.

The search that found the raised-B tuning for a minor song can be run on a single pop song too, and for A major it finds a tuning that costs 4.25 — four hundredths less than the ground’s own. So for a pop half played with a capo, the ground’s tuning is already within a whisker of the best there is: the capo does the work a pop-specific tuning would have done.

What it does to one retuning #

One retuning is worth what the pop songs pay for the minor tuning priced an evening of the twelve pop songs followed by a run of minor songs, with one retuning allowed between the halves, against the cheapest evening in one tuning. With a capo both halves change, and so does the comparison.

For a minor half in mixed keys the capo nearly doubles the retuning’s worth. Without one, twelve minor songs round the circle save 11.18 — the ceiling of the earlier essay — because the best the minor half can do in mixed keys is the E-minor shape’s tuning, which a single-tuned guitar could have had anyway. With a capo, each song is moved to B minor in the raised-B tuning and the run is in effect one key: twelve songs save 19.22, and twenty save 25.21, still rising. The keys of the minor half have stopped mattering, as the earlier essay’s key-by-key ceiling stopped mattering when the right fourth tuning was offered.

For a minor half all in A minor, the capo makes the retuning worth less. Without a capo, twelve songs on A saved 18.34; with one, 15.23. The minor songs themselves are cheaper with a capo, at B minor’s price rather than A minor’s, but the cheapest single tuning has fallen further: with a capo every pop song and every minor song can move to whatever key the one tuning serves best, and the single-tuned evening is 14.20 cheaper than without. Some of what a retuning used to buy, the capo now buys on its own, and a guitarist with a capo and an all-A minor half has less reason to retune than one without.

Those two results look opposed and are the same fact read from two sides. A retuning is worth the difference between what each half pays in its own tuning and what both pay in one. A capo lowers both terms. Where the halves were already well served by distinct tunings — pop in its own, minor all in A — it mostly helps the single-tuned evening catch up. Where the minor half had no good tuning of its own because its keys were scattered, it gives it one, and the retuning gains.

How high the capo may go #

The seventh fret is a convention here, not a measurement. A capo high on the neck shortens the string to the length of a smaller instrument’s, and many players avoid going past the fifth or seventh fret for the sake of the sound.

With no capo the evening costs 144.66 in its cheapest tuning, the E-minor shape’s, and 133.49 retuned from the pop tuning to the same. Allow the capo to the second fret and the single-tuned evening’s best tuning changes to the C shape’s, at 143.05, while the retuned evening moves to the raised-B tuning for its minor half and falls to 127.08. At the fifth fret the two are 137.44 and 119.45; at the seventh, 135.24 and 116.02; at the ninth, 133.64 and 112.59. Over the first seven frets the single tuning’s evening falls by 9.42 and the retuned one by 17.47: the retuned evening gains nearly twice as much from the capo as the single-tuned one does, and the saving rises from 11.18 to 19.22.

Most of that comes early. The first two frets of capo are worth 4.79 to the retuning; the next five are worth 3.25 more. A player who will put a capo only at the second or third fret already has most of what a capo can give a retuned evening, because the keys a short reach brings to B minor — C, C♯ and D minor — are among the commonest.

What the arithmetic assumes #

The capo is perfectly in tune. A real capo presses every string onto the fret harder and from a different angle than a finger does, and pulls each string slightly sharp, by an amount that varies with the string’s gauge and the capo’s design; a guitar cannot be in tune priced the same kind of error for a fretting finger. A tuning set without the capo and played with it would carry that error on top of the ones counted here. A guitarist who retunes with the capo on would remove it.

Moving the capo is free, like retuning. Between every two songs in these evenings the capo may move to any fret. That takes a few seconds and is generally done, but it is not nothing, and a set that moved the capo for every song would be a fussier set than one that did not.

The shapes and the neck are the earlier essays’. Five moveable major shapes and three moveable minor shapes, fourteen frets, chords voiced in full. The minor song is i–iv–V–i twice round, the pop ground I–vi–IV–V; the chord a tuning gives up is the essay these routes come from.

What a capo does not change #

The keys a listener hears. Every price here is for the song in its own key, which is the point of a capo. What changes is the shapes the guitarist plays it in, and a guitarist with a capo at the fifth fret playing B-minor shapes is producing E minor with B minor’s voicings: the same pitches, but each chord voiced as the shape gives it, not as E minor’s own open shapes would. A listener may hear the difference in voicing more readily than the difference of a few cents.

The tone. A capo high on the neck gives the open strings a shorter vibrating length and a brighter, thinner sound, and the arithmetic gives no weight to that. What a tablature keeps is a reminder that the same notes on different strings and frets are not the same sound.

Whether the cents are heard. As a tuned guitar has no minor chord in tune found, a few cents on a held minor chord can beat audibly; whether a listener prefers E minor’s open chords at 2.51 cents and a moving hand to B minor’s shapes at 4.53 and a still one is a question about what the departures sound like, not about sums.

Still open: one capo for the whole minor half #

Every minor song here took its own best capo position, so the capo moves between songs, often by several frets. A guitarist who leaves the capo in one place for a whole run of minor songs — the commoner practice — plays every song in that run as the key the capo makes of it, and only the songs whose key falls on B minor or A minor in that position get the raised-B tuning’s best price. The question is then a joint choice of one capo position and one tuning for a run of minor songs in mixed keys, the same search as the earlier essay’s with the capo added as a seventh variable, and it would say whether a fixed capo 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.