Looking ahead rescues the relative minor, not the bridge

The bridge is not a labelling problem took the eight bars that a key-finding reading got most wrong across a listing of six forms — the bridge of the thirty-two-bar song, two bars each of E7, A7, D7 and G7 — and relabelled them as the keys their dominants point at: A minor, D minor, G major, C major. The relabelling bought one bar. Four of the eight were read as the chord’s own root taken for a tonic, and one was unwinnable in principle, because a dominant seventh is the same four notes in the major and the minor of the key it points at.

That essay ended on the absence behind every miss. The reading assigns a key to a bar from the bar’s own notes and the cost of having changed key, and an applied dominant is a chord whose meaning is in the bar after it. The apparatus already reads in both directions, so lookahead was possible; what was missing was a cue about a pair of bars rather than about one. The question was whether the bridge’s misses are a consequence of the cue set being per-bar — so that the ceiling is an artefact of the machine’s shape — or whether even a reading that knows where each dominant goes still prefers to sit on its root.

The cue works on the bridge. It also works where nobody wanted it, and the net effect depends on something the question did not ask about: how reluctant the reading is to change key.

A cue about a pair of bars #

The cue is one term. A state that reads a bar as the dominant — the fifth degree — of some key earns a stated credit when the next bar’s chord is rooted on that key’s tonic. It is a statement about resolution, and it is deliberately about roots rather than qualities: E7 resolving to A7, as it does in the bridge, earns the credit for A minor or A major, because the next chord’s root is A. At a weight of nought every reading is exactly as it was, which the figures check.

At the setting the earlier essays found best — a cost of 3 for changing key, and the bass weighted at 3 — the cue does nothing until its weight reaches the cost of changing key, and then the bridge is read differently. Both E7 bars become the dominant of A. The first A7 bar is read as A’s tonic and the second as the dominant of D. The first D7 is D’s tonic, the second the dominant of G; the first G7 is G’s tonic, the second the dominant of C. Five of the eight bars are now read as the key their dominant points at, where two were.

The pattern in the three later pairs is the interesting part. Each chord of the chain is read twice, once as the arrival of the dominant before it and once as the dominant of the chord after it. That is what a chain of applied dominants is: A7 is the chord E7 resolves to and the chord that resolves to D, and a reading that must name one key per bar gives the first bar of each pair to the first role and the second bar to the second. The theorist’s label, which names every bridge chord by the key it points at, is half of that description. The reading’s own earlier habit, which named each chord as a tonic, was the other half.

What the cue does to the blues #

A cue that hears dominants resolving will hear them wherever a dominant-seventh chord moves up a fourth, and the listing has a form built on exactly that motion.

The blues’s tonic chord is a dominant seventh, C7, and its fifth bar is the IV chord, F7. So the opening four bars are, note for note, a dominant followed by the tonic it resolves to. Without the cue the reading hears them correctly as C major’s first degree, because nothing in their notes says F. With it, the resolution into bar five is worth the key change, and the reading hears the opening as a prolonged dominant of F arriving at F. The blues’s opening is ambiguous in exactly the way the bridge is, and a cue that resolves the bridge’s ambiguity toward the dominant resolves the blues’s the wrong way.

The first chorus loses five bars; the later choruses, which arrive at C7 from the turnaround’s V7 and so already have C as a key, keep theirs.

At the setting that read the listing best #

The cost of the cue can be set against its benefit across the whole listing.

At the earlier essays’ best setting the trade is a loss. The weight that reads three more bridge bars costs five blues bars, and from a weight of 2 onward the listing is worse than without the cue: 175 bars right with no cue, 170 at weights of 2 and 3, 155 at 4, 145 at 6. Heavier cues begin to break the other forms too, as the reading starts treating every chord a fourth below the next as a dominant.

The order in which things happen as the weight rises is itself telling. At a weight of 2, below the cost of changing key, the bridge is untouched — two bars read in the keys their dominants point at, as with no cue — and the blues has already lost its five bars. The blues breaks first because its evidence for the wrong reading is stronger: four bars of C7 followed by a bar rooted on F give the resolution credit a whole chorus’s opening to act on, while each bridge chord holds for only two bars before the next one moves the goalposts. The cue meant for the bridge reaches the blues before it reaches the bridge.

This is the same shape of result the earlier essays kept finding when a cue was sharpened. A sharper cue is worth nothing to a reading that moves found that a better bass rule only helps where the key cost holds the reading still, and two cues meet in a corner found that the profile and the bass cue do their work in different parts of the plane of settings. The resolution cue is a third axis on that plane, and like the others it cannot be judged at one setting of the rest.

So the answer to the earlier essay’s question, at that setting, is the pessimistic one in a new form. A reading that knows where each dominant goes does hear the bridge as dominants — the ceiling was partly the machine’s shape — but a reading built that way is worse overall, because the cue that hears an applied dominant cannot tell it from a tonic seventh that moves to IV.

The setting at which the cue earns its place #

The key cost is the other half of the trade, and the earlier essays chose it without the cue.

Without the cue, the listing is read best at a key cost between 3 and 5, at 175 bars — the plateau the two parameters turn out to have a ceiling between them found; at 6 the reading has become too reluctant to leave C and falls to 172. With the cue, the best cell is a key cost of 6 and a cue weight of 3, at 177 bars — two more than the reading has ever managed. At that setting the cue is worth half the cost of changing key, so it cannot pay for a key change on its own; it can only tip a change that the bars’ own notes already favour. The blues is untouched, since its opening gives no key other than C any support from its notes. And the bridge gains nothing over the no-cue reading at the same key cost: four bars either way.

The cue earns its place in a reading that is reluctant to change key, and there it helps somewhere other than the bridge.

Where the two bars come from #

The two bars come from the rondo.

One of the rondo’s episodes is in A minor, the relative minor of the refrain’s C, and a relative minor is the hardest key change for a profile to hear: A minor and C major share every note of the natural minor scale, and a reading reluctant to change key has every reason to hear the episode as C’s sixth, second and third degrees. It does, for all eight bars. What distinguishes A minor from C major is the dominant seventh E7, with its G♯, resolving to the A minor chord — and that is exactly the pair of bars the cue is about. With the cue, the episode’s middle five bars, from the first i through two V7–i cadences, are heard in A minor. The first two bars, before any E7 has appeared, and the last, which leads back to the refrain, stay in C.

The two bars at the episode’s start that stay in C are a small lesson of their own. They are the A minor chord and the D minor chord, i and iv, and nothing in them is outside C major; the reading cannot hear A minor until the first E7 supplies the G♯ and the resolution. A listener hearing the passage for the first time is in the same position — the minor key is only confirmed when its dominant arrives, and how much evidence a modulation needs measured how many bars of the new key a reading waits for before naming it — and a reading that looks ahead can place the key change back at the first A minor chord only if it credits that chord with what happens two bars later, which the cue, looking one bar ahead, does not.

The ceiling is thirteen bars with names listed the reading’s thirteen misses at its best setting and found four of them relative-key confusions. The resolution cue, in a reading set reluctant enough, is the first of the cues these essays have tried that fixes them — because what distinguishes a minor key from its relative major is precisely where its dominant goes.

One mechanism, three places #

The three results are one mechanism met three times. A dominant-seventh chord moving up a fourth is evidence for the key whose tonic it lands on. In the bridge that evidence is right, and the theorist agrees with it. In the blues it is wrong, because the style has made the dominant seventh a tonic colour, and nothing in the notes says so. In the rondo’s episode it is right and decisive, because it is the only evidence that separates A minor from C. The reading cannot tell the three apart from the chords, since the chords are the same kind of object in all three; it can only weigh the cue against its reluctance to change key, and the weighing that suits the rondo leaves the blues alone and the bridge half-read.

The surprising part is the direction. The cue was designed for the bridge — it was the bridge’s own misses that asked for it — and it is the rondo that it helps. The bridge’s applied dominants each last two bars and point at a key the music never settles in, so a reading reluctant to change key gains nothing by following them; the rondo’s dominant points at a key the music stays in for eight bars, and a reluctant reading gains a great deal by following it there.

The reading, and what was added to it #

The reading is the earlier essays’ dynamic-programming key-finder: states of key, mode and scale degree, each bar scored by how well its notes fit the degree’s triad and by whether its bass matches the triad under the register rule, and each change between bars scored by the root motion it implies and by the cost of changing key. It reads forwards and backwards and names, for each bar, the state on the best path through it. The resolution cue adds one term to the change between bars: when the earlier bar’s state is the fifth degree of a key and the later bar’s chord is rooted on that key’s tonic, the change earns the cue’s weight. The weight is swept and the key cost with it; everything else, including the bass weight of 3 and the smooth realisation of each scheme’s bass that a bass line is not a list of roots introduced, is the earlier essays’.

What one term cannot say #

The quality of the chord resolved to. The cue credits a resolution by root, so E7 to A7 counts as resolving to A. A cue that also required the arrival chord to be the key’s tonic triad would not credit the bridge’s chain at all, since no chord in it is one; it would credit the rondo’s E7 to A minor exactly as before.

Deceptive resolutions. A dominant that moves to the sixth degree instead of the tonic earns nothing here, and a reading that knew about deceptive cadences would treat a V–vi as weaker evidence for the key rather than none.

Why a blues is a blues. The reading has no notion of style. A listener who knows a twelve-bar blues knows that its opening seventh is a tonic; the arithmetic can only say that a listener without that knowledge, following resolutions, would hear the opening in the key of IV.

One weight for every resolution. A dominant resolving at a cadence and one resolving in passing earn the same credit here, where a listener plainly weighs the first more; a cue scaled by metrical position would treat the rondo’s cadences and the blues’s opening differently.

The listing is six forms. The two extra bars are two bars of 188, and the difference between the best settings with and without the cue is within what a different choice of forms could move.

Which harmonies #

The six forms are schematic versions of the thirty-two-bar popular song, a Classical rondo, the twelve-bar blues, a verse-and-chorus song, a sixteen-bar period and an ostinato ground, each with its harmony stated bar by bar. The applied dominants of the bridge are the circle-of-fifths chain of jazz standards and popular song; the blues’s tonic seventh is the defining sound of the blues; the rondo’s episode in the relative minor is the Classical rondo’s commonest contrast. A tonic bought with the function found that a reading can buy a tonic by what a chord does, and the resolution cue is the most direct version of that: a chord is heard as a dominant because of what the next chord is.

Still open: a cue that knows the chain #

The bridge is read half-right by the cue because a reading that must name one key per bar can give each chord of the chain only one of its two roles. A reading that could name a bar’s key and the key it is about to enter — a state with a second slot for the tonicization in progress — would not have to choose, and it would be the first reading in these essays whose answer for the bridge could match the theorist’s label everywhere without giving the chords’ own roots away.

The computation is the same reading with its states doubled: each state carries the key it is in and, optionally, a key it is tonicizing, with a small cost for holding a tonicization and the resolution cue paying for entering one. It would say whether the bridge’s remaining misses are the price of a one-slot state, and whether a reading with two slots still misreads the blues — which would be the test of whether the blues’s opening is ambiguous in the music or only in a machine that cannot hold two keys at once.