The minor half's own tuning belongs to one key

One retuning is worth what the pop songs pay for the minor tuning let a guitarist retune once in an evening, between a run of pop songs in all twelve keys and a run of minor-key songs on A. The best evening played the pop songs in their own tuning, the one that makes the D chord shape exact, and the minor songs in the tuning that makes the E-minor shape exact. With ten minor songs or more the retuning saved 11.18, which was exactly what the twelve pop songs would have paid to be played in the minor tuning, and adding minor songs never raised it again.

That ceiling rested on a restriction the essay stated and did not lift. The retuning could choose only among three tunings, and all three had been found by searching for a tuning to serve a whole evening — the pop songs’ own, the C shape’s and the E-minor shape’s. Once the evening is cut in two, the second half is minor songs and nothing else, and nobody had asked what tuning a run of minor songs wants when it does not have to share. The E-minor shape’s tuning makes the minor song’s tonic exact, but the earlier essay on the minor songs found most of that gain handed back by the subdominant and the dominant. A tuning chosen for all four of the song’s chords might do better.

It does, and the way it does is not what the question expected. The minor half’s best tuning is not a better compromise among the song’s chords. It is a tuning for one key.

The same search, on one song #

Each song is priced as before: the mean departure from just of the chords it is played on, in cents, plus its hand movement in frets divided by five, the rate at which a cent is worth five frets. The minor song is i–iv–V–i, twice round. Its tuning is searched for the way the evening’s tunings were: from several starting tunings — each chord shape’s own, equal temperament, two random ones, and here also the three candidates of the earlier vote — the search alternates between routing the song through its cheapest fingerings and moving the strings to suit the chords those fingerings use, and every start is also scored as it is, so that a tuning already in hand can never be lost.

Run in each of the twelve keys, the search returns the E-minor shape’s tuning nine times. In C, G, E, F♯, C♯, A♭, E♭, B♭ and F minor no tuning anywhere does better for the song than the one the vote already had. In ten of the twelve keys that tuning costs the song the same 5.69, because the song is played up the neck on moveable shapes and the key only slides the hand; in E and F minor it costs a little less, 4.91 and 4.89, because the tonic or the dominant falls at the nut.

In three keys it does not return the vote’s tuning. D minor finds one that costs 5.35, saving 0.34. A minor finds one at 5.09, saving 0.60. B minor finds one at 4.53, saving 1.16. Those three are the keys whose minor chords can be fingered at the nut with open strings in them in the shapes a guitarist reaches for first — A minor and D minor as the open chords of those names, B minor as the A-minor shape two frets up with the open E-minor chord as its subdominant — and the search has found a way to put those open chords in tune.

Why most keys cannot be helped #

The nine keys that keep the E-minor shape’s tuning are not unlucky. They share a property that makes improvement impossible on these terms. Their songs are played entirely on moveable shapes, up the neck, and a moveable shape costs the same in cents wherever it is placed: the tuning’s error belongs to the shape, not the fret. Moving a song from C minor to G minor slides the hand and changes nothing else, which is why ten keys cost exactly 5.69 under the vote’s tuning and why one guitar tuning serves every key for the pop ground: on moveable shapes the keys are interchangeable, and a tuning cannot favour one over another.

For a song played that way, the question of its best tuning is the question of its best set of shapes, and that question was settled for the minor song when a guitar set keeps its tuning until the third minor song compared the shape tunings. The tonic’s shape exact is worth more than the dominant’s, and no tuning that tries to serve both does better than one that serves the tonic. The search confirms it from nine directions at once. What it adds is that this is also the best tuning when the pop songs are no longer there to be served: in those keys the minor half has nothing to gain from being alone.

The open strings break the symmetry. An open string is a fret the hand cannot move, so a chord containing one is fixed to a key, and a tuning can be shaped around it. E and F minor have open chords too — the open E-minor tonic, the open C as F minor’s dominant — but the E-minor shape’s tuning is already shaped around the first, so for them the vote’s tuning and their own coincide. A guitar tuned for a set is tuned for the D shape found the pop ground’s tuning drawn towards shapes that sit well on the neck; the minor half’s own tuning goes further and is drawn to a particular place on it.

What A minor’s tuning does #

The chord-by-chord picture says what has happened.

In the E-minor shape’s tuning the song is played around the fifth fret: the tonic as the E-minor shape at fret 5, exact; the subdominant, D minor, as the A-minor shape at fret 5, 12.5 cents from just; the dominant, E, as the C shape at fret 4, 7.0 cents off. Its mean departure is 4.89 cents and the hand moves four frets in the whole song.

In its own tuning the song is played at the nut and at the fifth fret in one shape. The tonic is the open A-minor chord; the subdominant is the same A-minor shape moved up five frets, which on a guitar is D minor; both are exactly in tune, because a tuning that makes a shape exact makes it exact wherever it is placed. The dominant is the open E chord, 7.6 cents from just, and it carries the song’s whole error. The mean departure falls to 1.89 cents — less than half — and the hand now jumps between the nut and the fifth fret four times a pass, sixteen frets in the song. At five frets to the cent the extra movement costs 2.4 of the 3.0 cents saved, and 0.60 is what is left.

So the minor half’s own tuning is a decision the earlier vote could not have made, because what a tuning gives up is a fingering rather than a chord. The vote’s tunings each made one shape exact and let the song choose the fingerings that suited it. This one makes the tonic and the subdominant the same shape, which is only possible in a key where that shape’s two placements are both convenient, and A minor, with its tonic at the nut and its subdominant five frets up, is such a key. The tuning is a property of the key as much as of the song.

String by string the difference is one string. The E-minor shape’s tuning raises the G string by 15.6 cents and the A and B by two. The A-minor song’s own tuning raises the B string by 13.7 cents and lowers the A and G by two. Both are the same move made on different strings: the minor third of the chord is fretted on that string in the shape the tuning serves, a minor third in equal temperament is 15.6 cents narrow of just, and raising the string by about that much puts the third in tune. In the open A-minor chord the third, C, is the first fret of the B string.

B minor’s tuning turns out to be almost the same one — the B string up 13.7, the A and G down 2.0 — because its tonic is the A-minor shape at the second fret, which the raised B string puts in tune wherever it sits; its subdominant, the open E-minor chord, is left 10.6 cents from just, and its dominant, the E shape at the second fret, 7.6. D minor’s raises the high E string instead, where the open D-minor chord frets its third. Three keys, two tunings, and each tuning is the one that puts a particular open chord’s minor third in tune.

What the pop songs pay for it #

A tuning built around open A-minor and D-minor chords is a poor tuning for anything else. Under it the twelve pop songs cost 95.62, against 66.81 in their own tuning and 77.98 in the E-minor shape’s. That is why no search for a whole evening ever found it: as a single tuning for an evening with even one pop song it loses badly, and it would only win an evening that was nothing but minor songs on A, B or D.

That makes it exactly the kind of tuning a retuning exists for. Where to hide the comma described keyboard temperaments as ways of sharing an error between keys a fixed instrument must play in turn; this is the opposite case, a tuning that shares nothing and is useless to every key but its own, and it is worth having only where it does not have to be shared.

The refund without a ceiling #

Give the retuned evening four tunings to choose among — the vote’s three and the A-minor song’s own — and the minor songs on A move to their own tuning as soon as there are any.

With one minor song the retuning saves 3.07 instead of 2.48; with three, 8.04 instead of 6.25; with nine, 16.31 instead of 10.94. At ten it saves 17.15 instead of 11.18, and from there it keeps rising, by the song’s own gain of 0.60 for every minor song added: 19.53 at fourteen, 23.12 at twenty. The earlier ceiling belonged to the three tunings, not to the evening. It held because the best the minor half could do was the tuning a one-tuning guitar would also reach, so once the single tuning was the E-minor shape’s the minor songs had nothing left to gain. With a tuning the single-tuning evening can never choose, every minor song gains something however many there are.

The single-tuning evenings are unchanged by the fourth candidate — at every count the cheapest single tuning is still the ground’s, the C shape’s or the E-minor shape’s — because the A-minor song’s tuning would have to carry the twelve pop songs at 95.62. Everything the fourth tuning buys, it buys through the retuning.

Only behind its own retuning #

The earlier essay found the running order deciding a retuning’s worth before the mix of songs did: ten minor songs in one run were worth 11.18 to one retuning, in two runs 6.70, alternated with the pop songs 2.30. The minor song’s own tuning sharpens that rule.

With the minor songs in two runs and one retuning allowed, the fourth tuning adds nothing: the saving is 6.70 either way. The single retuning has to leave the second half of the ground in whatever tuning the minor songs are in, and the A-minor song’s tuning would charge those six pop songs nearly eight a song where the E-minor shape’s charges them 6.5, so the retuning goes to the E-minor shape’s as before. With three retunings — into the first run, out of it, into the second — the guitarist can put both runs in their own tuning, and the saving jumps to 17.15, against 11.18 with three tunings. Three runs need five retunings for the same jump. The rule is the earlier essay’s count of retunings, one into each run and one out, less the last way out; what changes is that below that count the fourth tuning is worth exactly nothing.

So the minor song’s tuning is a specialist’s tool in the strict sense. It pays more than any of the others when the minor songs are together and have it to themselves, and it pays nothing at all the moment a single pop song has to be played in it.

Mix the keys and it is gone #

All of this assumed every minor song is on A. A real minor half is in several keys.

A run of A minor and B minor songs keeps a tuning of its own, since the two share one, and saves 0.88 a song — the average of their separate gains. A minor and D minor want different strings raised; a tuning searched for both saves only 0.28. A run of A, E, D and B minor saves 0.23. But one song in A minor and one in E minor take the best tuning straight back to the E-minor shape’s, with nothing saved, and so do five songs on A with five on E, and so does a run spread over all twelve keys. The E-minor shape’s tuning has its own open-string gift in E minor, where its tonic is the open chord, and a minor half that contains E minor prefers that gift to A’s.

The answer to the question the earlier essay left is therefore conditional, and the condition is the keys. If the minor half is in A, B or D minor and nothing else, it wants a tuning of its own, which none of the vote’s three is, and the refund a retuning buys has no ceiling. If it is in any other single key, or in a mixture that reaches beyond those three, the E-minor shape’s tuning is already the best a minor half can have and the earlier refund is all there is.

What the numbers rest on #

The guitar is the earlier essays’ six strings, each tuned by a free number of cents with the low E fixed, chords fingered in five moveable major shapes and three moveable minor shapes within fourteen frets, open strings counting as fret zero. The minor song is i–iv–V–i played twice; the ground is I–vi–IV–V in all twelve keys, in the circle-of-fifths order. Each song takes its cheapest route at five frets of movement to a cent. The search for a song’s own tuning is a local search from nine starts, alternating routing with string moves; it can miss a better tuning that none of its starts leads to, and the claim that nine keys keep the E-minor shape’s tuning is a claim about what that search finds.

The open-chord tunings depend on the shapes allowed. A guitarist who plays A minor up the neck rather than at the nut has no use for them, and one who uses a wider vocabulary of open voicings might find tunings for more keys than three.

What the sums leave out #

The cost of the retuning itself, which is zero here, as before. The minor song’s tuning moves the B string by 11.7 cents up from the pop songs’ tuning and the high E string by 11.7 back up from its lowered place, so the retuning is at least as large as the one the earlier essay priced.

Whether a player would accept the jumps. At five frets to a cent, sixteen frets of movement are a price the song pays and still comes out ahead. A player who values staying in one position more than that exchange rate does would never choose the open-chord fingerings and would never want the tuning. The exchange rate is a stated convention, and this result moves with it.

What these figures cannot show #

Whether the gain is heard. The A-minor song’s own tuning halves its mean departure from just, from 4.89 cents to 1.89, by putting the tonic and subdominant in tune and leaving the dominant 7.6 cents out — a dominant that, as a tuned guitar has no minor chord in tune found of minor chords, a listener may hear beating if it is held. Whether an audience would prefer two pure chords and an impure one to three moderately impure ones is a question about listeners, not about sums.

Still open: the minor half in a key the open strings do not serve #

Nine keys found no tuning better than the E-minor shape’s, and the reason looks structural: their minor chords cannot be fingered at the nut in the shapes allowed, so every tuning plays them up the neck on moveable shapes, where one shape’s tuning serves them all alike. A capo changes that. Clamped at the second fret, it turns the open A-minor chord into B minor and the open D-minor chord into E minor, and a minor half in E minor played with a capo could use the A-minor song’s own tuning — the same raised B string — and gain what A gains. The computation is this one with the capo’s fret added to every open string and its shapes, and it would say which of the nine keys a capo can admit to a tuning of their own, and whether a guitarist with a capo and one retuning can give every minor key the refund that only three have now.