Two minor songs retune a guitar set for their dominant

One guitar tuning serves every key, and it is the D shape’s tuned a guitar once for a whole evening. The songs were the I–vi–IV–V ground of popular song, in as many keys as the evening needed, and the guitar was allowed to move each string a few cents off its fret, once, before the first song. Each song was then played by the cheapest route through it — every chord fingered in whichever moveable shape made the sum of its mistuning and its hand movement smallest, at an exchange rate of five frets of movement to a cent of error — and the tuning was the one that made the whole set’s mean cost smallest. Grown one key at a time round the circle of fifths, the set never stopped sharing a shape: from four keys to all twelve its tuning was the D shape’s, the open D chord moved up the neck made exactly just, with the same offsets to a fifth of a cent.

That result rested on every song being the same ground. Every key’s route through I–vi–IV–V contained the D shape somewhere, so the D shape’s tuning served them all. The essay ended by asking what happens to songs that break the ground — a minor-key song, whose tonic and subdominant are minor chords, and songs whose chords lie elsewhere — and how many a set could hold before its tuning stopped being the D shape’s.

A correction, made later. The count below is wrong by one. The search that produced it never tried keeping the tuning the set already had, and with two minor songs added the ground’s own D-shape tuning is cheaper than the C-shape tuning reported here (5.934 against 5.975). Priced as a vote, two minor songs save the set less than the move costs the ground songs, and three save more: the tuning moves at the third minor song. A guitar set keeps its tuning until the third minor song is the recount. Everything below about why the tuning moves — for the minor songs’ dominant, to the shape it is fingered in — stands; read the threshold of two as three.

The answer is a rule about which chord of a song decides the tuning, and it is not the chord the song is named for. Rock songs never move the tuning off the D shape. Minor-key songs do, two of them, and the chord they move it for is their dominant.

Two kinds of song outside the ground #

Two new progressions join the set. The minor-key song is i–iv–V–i, twice round: A minor, D minor, E major and A minor for the song on A — the progression of countless songs in a minor key, with the major dominant that harmonic minor supplies. The rock song is I–♭VII–IV–I: G, F, C, G for the song on G, three major chords a whole tone and a fourth apart, the progression of a great deal of rock and folk. Neither needs a chord shape the guitar model lacks; the minor chords use the moveable E-minor, A-minor and D-minor shapes, the major chords the five CAGED shapes, exactly as before. A blues, whose I7 would need a seventh-chord shape, is not among them.

Each kind of song is added to the twelve-key set one key at a time, and the set’s tuning is searched again at every count, from the same starting tunings and with the same routes, so that every difference is the songs.

Rock songs keep the shape #

The rock songs change nothing about the shape. With one, two, four, eight or twelve of them the set’s tuning is still exact for the D shape, and its mean cost falls, from 5.57 to 4.96, because a song of three major chords is cheap to route through a tuning that makes one major shape exact: every chord can be fingered in the D shape, at the price of moving the hand.

What the rock songs do move is the part of the tuning the D shape does not use.

Look at the rock songs’ tuning beside the ground’s. The four strings the D shape plays — D, G, B and high E — keep exactly the same relations among themselves: G is 1.9 cents above D in both, B level with D, high E 13.7 below D. The D shape is exactly as just as before. What has moved is the A string, from level with the D string to 11.7 cents below it — and the A string is one of the two strings the D shape leaves silent. A silent string tunes the minor chord found the same freedom used for a minor chord: a shape’s silent strings can be tuned for a different shape without costing the first anything. The rock songs spend that freedom on their ♭VII and IV chords, whichever of their fingerings sound the A string. A guitar tuned for a set is tuned for the D shape had already found the D shape’s two silent strings placed for the set rather than for any one chord; the rock songs simply place them somewhere else.

So a rock song is, to this tuning, not outside the ground at all. It is more major chords, and the tuning absorbs it in the strings it had to spare.

Two minor songs move it #

The minor-key songs are different. With one, the tuning is unmoved, still the D shape’s, and the set’s mean rises by a fifth of a cent. With two — songs on A minor and E minor — the tuning jumps: it is now exact for the C shape, with the A, G and B strings raised by fourteen to sixteen cents against the low E, which is the C shape’s tuning the earlier essays found for sets of fewer than four keys. It stays the C shape’s through eight minor songs. With all twelve, it moves again, to the E-minor shape’s tuning — and only then is a minor chord the one made exact.

The route of one minor song shows why. Under the D shape’s tuning the A-minor song is fingered around the fifth fret — A minor in the E-minor shape, D minor in the A-minor shape, E major in the C shape at the fourth fret — and none of the three chords is exact. Its mean mistuning is 7.36 cents, which is worse than playing the same song in open position on plain equal-tempered frets, 6.76. The ground’s tuning serves a minor song worse than no tuning at all.

Under the C shape’s tuning the same route has one chord made exact: the E major, the song’s dominant, fingered in the C shape. The two minor chords are no better than before. That is the whole of the change, and it is enough: the song’s mean falls to 5.67, and a second minor song in another key, whose dominant also routes through the C shape, pays the tuning’s price a second time. Two songs’ dominants outweigh what the C-shape tuning costs the ground’s twelve songs.

Only when the set holds a minor song in every key does a minor chord itself take the tuning. Under the E-minor shape’s tuning the A-minor song’s tonic is exact, its mean falls to 4.89, and its dominant goes back to seven cents out. A minor chord decides the tuning only when minor chords are most of what the set plays.

The chord that decides is the major one #

The rule underneath is the one a tuned guitar has no minor chord in tune found for a single chord shape. A tuning that makes a major shape just leaves the minor shapes well out of tune, and one that makes a minor shape just does the same to the majors; on six strings the two cannot both be had. A minor-key song’s chords are two minor and one major. For a set of mostly major chords with a few minor songs added, the cheapest thing a tuning can do for those songs is to make their one major chord exact, since the major chords are the ones it is already built to serve, and the question is only which major shape.

For the ground’s own songs the answer was the D shape, because every key’s route passes through it. A minor song’s dominant is routed through the C shape, and the C shape’s tuning also makes the ground’s C and G songs exact, as the earlier essays found for small sets. So two minor songs are enough to tip the balance to the C shape: they add two dominants in the C shape to the two ground songs that already favoured it. A minor-key song pulls a guitar’s tuning towards the shape its dominant is played in. For the ground’s I–vi–IV–V, whose own tuning is the D shape’s, that is a different shape, and the set notices at two songs.

Why one is absorbed and two are not #

The threshold at two is not an accident of these keys, and it has a simple account. A tuning for a set is a weighted vote among the songs, each song casting its weight for the tuning that serves it. The ground’s twelve songs vote for the D shape, but not unanimously: the essay before this found that the songs in C and G, taken alone, prefer the C shape, and those two songs are outvoted by the other ten. A minor song’s dominant, fingered in the C shape, is a third voice for the C shape. One is not enough to outweigh the ten. Two are.

That is also why the tuning does not move gradually. A guitar tuned for a still hand is tuned for nothing found the best tuning for a route exact for one shape or another at every valuation of a cent, never a compromise between them, because a route can move the hand to reach whichever shape is exact. So the set’s tuning is always one shape’s or another’s, and adding songs moves it only when a different shape’s tuning becomes cheaper outright — a jump, not a drift. Between two and eight minor songs nothing jumps, because the C shape’s tuning remains the cheapest; at twelve the minor chords themselves are numerous enough that making one minor shape exact beats making the dominant exact.

The chord a tuning gives up is a fingering made the point that what a tuning abandons is never a chord but a way of fingering it. The same is true of what a minor song asks for. It does not ask for its A minor to be in tune; it asks for the fingering its route happens to use for its E major, and that request is granted because the fingering is a shape other songs use too. Tune a guitar for the shape of its key’s open chord found that a single song’s best tuning is the shape its tonic is played in near the nut; for a minor song in a set, the shape that matters is its dominant’s, because its tonic’s shape is the one kind of shape the set cannot afford to favour.

What the move costs the others #

The C-shape tuning is not a good tuning for anyone in particular. The songs in C and G give up nothing — their own best tuning is the C shape’s. The song in A gives up 1.86, more than anything else in the set, and the other ground songs between a tenth and six tenths. The two minor songs that moved the tuning still give up 1.38 and 1.56 each, because the C shape’s tuning makes only their dominant exact. The tuning moves because the set as a whole is better off, not because the songs that moved it are well served; they are merely served less badly than under the D shape.

That shows up in the cost of not retuning. The ground alone, in twelve keys, loses a ninth of a cent-plus-frets to having one tuning; the earlier essay found that cost falling as keys were added, because the keys shared the D shape. One minor song trebles it. Two make it five times as large, and from three on it sits near seven tenths — six times the ground’s. Rock songs raise it too, to about a third, but slowly.

So the earlier essay’s advice — that a player with a set in many keys loses nothing worth retuning for — survives rock songs and does not survive minor ones. A player whose set mixes a few minor-key songs into a pop evening has a real choice to make between the D shape’s tuning, which serves the ground, and a retune for the minor songs, and tuned for one shape, a cent is worth five frets is the exchange rate that choice is priced in.

There is a cheaper middle course the arithmetic points to, though it is not drawn. Since the minor songs moved the tuning for their dominants, a player could keep the D shape’s tuning and re-route the minor songs so that their dominant falls in the D shape — playing the E major of the A-minor song as a D shape at the second fret rather than a C shape at the fourth. That costs hand movement, which the exchange rate prices, and it leaves the ground’s tuning untouched. Whether it is cheaper than retuning depends on the rate, and at five frets a cent the routes did not choose it; a player who values a still hand less would.

The model the numbers rest on #

A six-string guitar with its strings each moved by a free number of cents from equal temperament, the low E pinned; every chord of every song fingered in one of the five moveable major shapes or three moveable minor shapes, within the fourteenth fret; each fingering’s mistuning as the mean departure from just of its sounding pairs, as a silent string tunes the minor chord counts it; each song routed by the cheapest path through its chords at five frets of hand movement to a cent; and the set’s tuning found by alternating routing and tuning from the family tunings, equal temperament and two seeded random starts until neither changes. Songs are weighted equally. Each progression is played twice round, four beats a chord.

What the model leaves out #

Other progressions. Two progressions stand for everything outside the ground. A minor-key song with a minor v, a modal rock song with a minor chord, a blues with seventh chords — each would pull a different shape, and the rule here says which: the shape the song’s odd chord is fingered in.

Open strings. Every chord is a moveable shape, and the open-position chords a player reaches for in A minor or E major are the same shapes at fret zero. The routing is free to use them, and the fourth figure shows that equal-tempered frets in open position beat the ground’s tuning for a minor song — but the open strings’ own pitches are the tuning’s, not a separate choice.

The player’s ear. A player hears a chord’s mistuning in beats, not cents, and counting beats moves the price of a chord, not the tuning found that changes which chord is dearest without changing which tuning wins. The rule here should survive the same change, but it has not been run.

Whose evenings #

The songs are the harmonic skeletons of popular song, and the claim is about a guitarist who tunes once for an evening that mixes them: that rock songs on major chords cost the ground’s tuning nothing but the strings it does not use, that one minor-key song is absorbed and two are not, and that when minor songs move the tuning they move it for their major dominant rather than for their minor chords. The arithmetic is about just intonation on a fretted instrument — a player tuning by ear towards pure chords, or an instrument with adjustable frets — and the numbers are a few cents, which is to say audible as beating in sustained chords and not as wrong notes.

Still open: the dominant’s own shape #

The rule found here makes a prediction that can be tested by moving one chord. A minor-key song’s dominant was routed through the C shape because of where its key put it on the neck. A song whose dominant is routed through the D shape — a minor key whose V falls where the D shape is cheapest — should leave the ground’s tuning alone however many such songs are added, while a key whose V falls in the C shape should tip it at two. The computation adds minor songs key by key rather than in order round the circle and asks whether the number it takes to move the tuning is set by how many of their dominants are fingered outside the D shape — which would say that, for a guitarist’s tuning, a minor-key song is a major-key song with its tonic missing.