One retuning is worth what the pop songs pay for the minor tuning

The essays on tuning a guitar for a whole evening, from the first one that tuned for a set on, have all made one assumption that no working guitarist makes: that the guitar is tuned once, before the first song, and left. A guitar set tunes for its minor chords only when nearly half its songs are minor followed that assumption to its end. An evening of pop songs in all twelve keys keeps the tuning that makes the D chord shape exact, the one that serves every key, until it has two minor-key songs in it; from three to nine it takes the tuning that makes the minor songs’ dominant chord exact, the C shape’s; from ten it jumps to the tuning that makes their tonic exact, the E-minor shape’s. Between those tunings there is no compromise worth having.

The question that essay left is the one a player at a gig would ask first. Between two songs a guitarist can retune — not to a new temperament, but by turning two or three pegs a few cents — and then the evening need not be served by one tuning at all. The pop songs can be played in their own tuning and the minor songs in theirs. How much is that worth, and when?

The price is the same sum, split once #

Nothing new is needed to answer it. Each song’s cost is the one the earlier essays used: the mean departure from just of the chords it is played on, in cents, plus its hand movement in frets divided by five, the exchange rate that a cent is worth five frets established. Under any fixed tuning each song is routed through its chords by the cheapest path, and the evening’s cost is the sum of its songs.

An evening with one retuning is an evening with a split point: every song before it in one tuning, every song after it in another. Try every split point and every pair of tunings, take the cheapest, and compare it with the cheapest single tuning. The difference is what the retuning is worth, in the same units as everything else, and it is worth the retuning if the retuning itself costs less — in time between songs, in the risk of a string not settling, in whatever a player puts on it.

The candidate tunings are the three the one-tuning vote used: the ground’s own, which makes the D shape exact; the C shape’s; and the E-minor shape’s. The minor songs are in A minor, the generic key, whose cost ten of the twelve minor keys share exactly. The ground’s twelve songs are in the circle-of-fifths order the earlier essays used.

Rising to the tenth song, and then flat #

Even with no minor songs at all one retuning saves 0.83, because a few keys of the ground are a little cheaper in the C shape’s tuning than in their own and an evening can be cut where they fall. With the minor songs played together after the ground, one retuning saves 2.48 for one minor song, 4.95 for two, 6.25 for three, 7.81 for five and 10.94 for nine. At the tenth it saves 11.18. At twenty it saves 11.18. It never moves again.

That ceiling is a number already in the earlier essay, though it was not read this way. Under the E-minor shape’s tuning the twelve ground songs cost 77.98; under their own they cost 66.81. The difference, 11.18, is what the pop songs pay to be played in the minor songs’ tuning. From the tenth minor song on, a guitarist who cannot retune is in the E-minor shape’s tuning all evening, and so the pop songs are paying it. A guitarist who can retune puts the pop songs back in their own tuning and the minor songs stay where they were. The saving is the refund, and the refund does not depend on how many minor songs there are.

Before the tenth song the saving has two parts, and the figure separates them. Up to two minor songs the one-tuning evening is still in the ground’s tuning, so the pop songs have nothing to get back; the whole saving is the minor songs moving to the E-minor shape’s tuning, 2.48 a song. From three to nine the one-tuning evening is in the C shape’s tuning. Retuning gives the ground back the 3.90 it was paying for that, and moves each minor song from the C shape’s tuning to the E-minor shape’s, which is worth 0.78 to each. At the tenth the one-tuning evening reaches the E-minor shape’s tuning itself; the minor songs have nothing more to gain, and the ground gets back all 11.18.

So the saving grows in three straight pieces whose slopes are 2.48, 0.78 and zero, and its jumps are the one-tuning vote’s own jumps. Read as a share of the evening, it is largest just before the tenth song: at nine minor songs one retuning takes the evening’s mean cost per song from 6.14 to 5.62, more than half a unit off every song. That is where a single tuning is serving the evening worst, and so it is where a retuning does most.

The tuning a retuning never uses #

The figures say something about the C shape’s tuning that the one-tuning vote could not. Across most of the range — three minor songs to nine — that tuning was the evening’s choice. With one retuning allowed and the minor songs together, the cheapest evening never uses it. At every count the retuned evening is the ground’s tuning followed by the E-minor shape’s.

The reason is in what the C shape’s tuning was. It makes the minor songs’ dominant exact, which is the one major chord in a minor song, and so it serves the minor songs somewhat and the pop songs somewhat: as a guitar set keeps its tuning until the third minor song priced it, it costs the ground 3.90 more than its own tuning and saves each minor song 1.69 over it. It is a compromise between the two halves of the evening, the kind of tuning that exists because one tuning has to serve two kinds of song. Once the two halves can have tunings of their own, the compromise has no job.

That is the same relation an equal temperament has to a keyboard’s keys. Where to hide the comma described every keyboard temperament as a decision about how to share an error between keys that a fixed instrument has to play in turn; a harpsichordist who could retune between pieces needed no such sharing. The C shape’s tuning is the guitar set’s temperament, and a retuning dissolves it in the same way.

The retuning itself is small, and it is not the one the earlier essay pictured. It is also a reminder that what a tuning gives up is a fingering rather than a chord: nothing about the pop songs’ chords changes at the retuning, only which of their shapes are in tune. That essay ended by imagining a guitarist moving between its two minor-song tunings, which differ only on the A and B strings, each by 11.7 cents. The retuning the cheapest evening makes is from the ground’s tuning instead, and it moves the G string and the high E string up by 11.7 cents each, the D string down by 2.0, and leaves the A and B strings exactly where the ground had them. A player who retunes the A and B strings between the pop half and the minor half has moved from a tuning nobody should be in to the right one, and left the pop half in the compromise.

The evening, song by song #

The sums hide the shape of the evening, and drawing it song by song shows where the saving comes from.

With ten minor songs the single tuning is the E-minor shape’s, and under it the pop songs cost about 6.5 each — A major, the cheapest key in the ground’s own tuning at 4.29, half as much again. The retuned evening plays the first twelve songs at between 4.3 and 5.8 and the last ten at exactly what they cost before, 5.69 each. All of the saving is in the left half of the picture.

Drag the count down and the picture changes at the vote’s own thresholds. At nine minor songs the single tuning is the C shape’s, and both halves of the evening pay something for it — the pop songs a little, the minor songs 0.78 each — so the saving is spread across the whole picture. At two, the single tuning is the ground’s own, the pop songs are already where they want to be, and all of the saving is in the minor songs on the right.

Two runs, and one retuning is worth three-fifths #

Every number so far has the minor songs played together at the end. A real evening does not have to be arranged like that, and a single retuning can only split it once.

Put the ten minor songs in two runs of five — half the ground, five minor songs, the other half of the ground, five more — and one retuning saves 6.70, sixty per cent of what it saved before. The best split now retunes once from the ground’s tuning to the E-minor shape’s after the first six pop songs, and the six pop songs after it pay the tonic’s tuning price after all. Three runs: 5.20. From four runs on the saving settles at about three, between 2.7 and 3.5, and with the minor songs alternating with the pop songs it is 2.30, a fifth of the evening’s whole refund.

The settled value is itself a change of plan. From four runs on, the cheapest single retuning does not start in the ground’s tuning at all: it starts in the C shape’s, plays the first part of the evening in that compromise, and moves to the E-minor shape’s for the rest. When the halves of an evening are not halves, the compromise tuning comes back, because no single split can separate the pop songs from the minor ones and the C shape’s tuning is the best way to play a mixture.

Drawn song by song, the two-run evening shows exactly where the lost forty per cent goes. The first six pop songs are played in their own tuning, at 4.29 to 5.78. Then come five minor songs, and the guitarist retunes for them. Then six more pop songs, which the single retuning has already been spent on: they are played in the E-minor shape’s tuning at 6.47 each, and those six songs are the whole of the difference between 11.18 and 6.70. The retuning could have been placed after the second run of pop songs instead, which would rescue them and leave the first run of minor songs in the ground’s tuning at 8.16 each; that is worse, because five minor songs lose more in the ground’s tuning than six pop songs lose in the tonic’s. Either way one half of one kind of song is played in the other kind’s tuning, and a single split cannot help it.

That is the general shape of the loss. A single retuning divides the evening into two stretches, and every song that sits in the stretch belonging to the other kind pays the full price of the wrong tuning. With the songs in one run each, no song does. With two runs of each, a quarter of the evening does, whichever way the split is made.

So the worth of a retuning is set by the running order before it is set by the mix. Ten minor songs out of twenty-two are worth 11.18 to a guitarist who plays them together and 2.30 to one who alternates them with the pop songs. The difference is a setlist decision, and it is larger than any difference in the mix of songs computed in any earlier essay of this kind.

Into each run and out again #

A guitarist who will retune more than once can recover the refund from a split evening too, and the number of retunings it takes has a simple form while the runs are long.

With the minor songs in one run, one retuning recovers the whole 11.18. In two runs it takes three: into the first run of minor songs, out of it, and into the second. In three runs it takes five. That is a retuning into each run and one out of each, less the last way out, because the evening ends in the minor songs’ tuning. Each additional retuning past that buys a few hundredths, the 0.86 that separates 11.18 from the 12.04 an evening could save if every song were played in its own cheapest tuning — a few ground songs whose keys happen to be cheaper in the C shape’s tuning than their own.

In five or ten runs the rule fails, and the reason is the length of the stretches between them. Five runs of two minor songs leave the pop songs in stretches of two and three, and a stretch of two pop songs does not save enough in its own tuning to pay for two retunings out of the tonic’s and back. The refund then comes back retuning by retuning: in five runs, 3.01 with one, 6.71 with three, 9.02 with five, 12.00 with nine. In ten runs nine retunings recover only 8.36. A guitarist alternating pop and minor songs would have to retune between almost every pair of songs to get what one retuning gives a guitarist who plays the minor songs together.

What a retuning is worth in a guitarist’s terms #

The unit throughout is the sum of the songs’ costs, each song’s mean mistuning in cents plus a fifth of its hand movement in frets. A saving of 11.18 is therefore the same as taking 11.18 cents of mistuning off one song, or half a cent off each of the twenty-two — or, at the exchange rate, sparing the hand 56 frets of movement in the evening.

A retuning between songs costs nothing in this model and something in a performance. The time is a few seconds; the risk is that a string does not hold where it was turned to. Whatever that is worth to a player, the figures give its upper bound at each arrangement: a retuning that costs more than 11.18 in these units is not worth making for any evening of this kind, and one that costs more than 2.30 is not worth making if the minor songs are spread. If a player would trade half a cent on every song of the evening against the trouble of one retuning, the minor songs had better come together.

The model the sums rest on #

A six-string guitar with each string moved by a free number of cents and the low E pinned, as in the earlier essays. Chords are fingered in five moveable major shapes and three moveable minor shapes within the fourteenth fret, open strings counting as fret zero; each song is routed by the cheapest path at five frets of movement to a cent. The ground is I–vi–IV–V in all twelve keys and the minor song i–iv–V–i in A minor, each played twice round. The three tunings are fixed — the two the search found and the E-minor family tuning — and the retuned evening chooses among them only. A search for the best pair of tunings for the two halves, rather than the best pair among three, would find a little more for each half and is not run.

The retunings are found by a dynamic programme over the songs, allowing up to a stated number of changes of tuning and choosing the cheapest sequence, which for one change is the same as trying every split point.

What the sums leave out #

The cost of the retuning itself. It is zero here and stated as a bound, not modelled. A real retuning between songs also changes the strings’ tension slightly and so their settling, which a player would know about and this model does not.

Every other reason to order a set. Keys, tempos, energy and the singer’s voice decide a running order long before tuning does. The finding is not that minor songs should be grouped; it is how much the tuning would gain if they were, and how little it gains if they are not.

What none of these figures can show #

Whether the refund is heard. No tuning on these terms puts every chord in tune — a tuned guitar has no minor chord in tune — so the comparison is between two sets of compromises, not between in tune and out. The pop songs played in their own tuning rather than the E-minor shape’s are about half a unit better on average — about half a cent of mean departure from just, at the five-frets-a-cent rate. Chord by chord, the difference is the 11.7 cents on two strings, and in a sustained chord that is beating a listener can count. Whether an audience over a whole evening would notice is not something sums of this kind can settle.

Still open: whether the halves want different tunings from the three #

The retuned evening chose among the three tunings the one-tuning vote had already found, and two of them were found for a whole evening rather than a half. With the minor songs together, the second half of the evening is minor songs only, and the tuning best for a set of minor songs alone was never searched for: the E-minor shape’s tuning makes the tonic exact, but the minor song’s subdominant and dominant pay for it, as the earlier essay found when three-quarters of the tonic’s gain was given back by those two chords. A search over the minor songs alone, at the same exchange rate, would say whether the second half wants a tuning none of the vote’s candidates is — and so whether the refund the retuning buys is all there is to gain, or only what was within reach of three tunings found for something else.