A second slot hears every modulation as a tonicization

Looking ahead rescues the relative minor, not the bridge gave the key-finder a cue about a pair of bars: a chord read as the fifth degree of a key earns credit when the next bar is rooted on that key’s tonic. It read the thirty-two-bar song’s bridge — E7, A7, D7, G7, two bars each, the chain of applied dominants that returns to C — as the chain it is, and still got it half wrong, because a reading that must name one key per bar can give each chord only one of its two roles. E7 is the arrival of nothing and the departure to A; A7 is the arrival from E and the departure to D. Named as the key it arrives in, a chord loses its departure; named as the key it departs to, it loses its arrival.

The shape of that failure is worth fixing before the repair, because it is the shape the repair has to escape. A one-slot reading that follows the chain has two ways to be wrong about each chord and no way to be right: name E7 as the dominant of A and it has left C, which the scheme says it has not; name it as a chord of C and it has no degree of C that holds a G♯. The pair cue made the first of those cheaper, and a tonic bought with the function had already shown the reading will buy a key with a chord’s function whenever that is the cheaper account. What was missing was an account in which a chord could keep its key and still point elsewhere.

Its closing question was a state with room for both — the key a bar is in, and, optionally, a key it is tonicizing — and a prediction that such a reading could match the theorist’s label for the bridge everywhere. It can. And at every setting where it does, it also hears two eight-bar modulations as nothing of the kind.

The state that was added #

The reading is the key-finder these essays have built up bar by bar: states of key, mode and scale degree; each bar scored by how well its notes fit the degree’s triad and whether its bass fits the triad under the register rule; each change between bars scored by the root motion it implies and by the cost of changing key; the whole passage read forwards and backwards by dynamic programming, as the ceiling is thirteen bars with names described.

The second slot adds, beside each key’s seven diatonic degrees, five more states: the dominant of its ii, iii, IV, V and vi. An applied state scores a bar against the major triad a fifth above its target — V of vi in C is E major — and it belongs to the home key: moving between it and C’s own degrees costs no change of key, because the key has not changed. What it costs is its own price for being entered, the tonicization cost, charged every time the reading moves into an applied state. The resolution cue extends to it: an applied state earns the cue when the next bar is rooted on its target, as a diatonic fifth degree earns it on the tonic.

With the tonicization cost set to infinity no applied state is ever entered, and the reading is the one-slot reading bar for bar; the figures check that before they draw anything. So every difference below is the second slot’s.

The bridge, read as it is labelled #

Set the tonicization cost low enough and the bridge comes out exactly.

At a key cost of 6, one slot hears the first six bars of the bridge in A major: E7 as its dominant, A7 as its tonic, D7 as its subdominant. That is the resolution cue doing what it was built to do — each chord pointing at the next — and it gets two bars of eight. With a second slot costing 2, the same bars are C: V of vi, V of vi, V of ii, V of ii, V of V, V of V, V, V, and every one of them is right under the scheme’s own label, III7, VI7, II7, V7 in C. Each chord keeps both roles: it is in C, so it arrives in C; and it points at its target, so it departs.

This is the result the earlier essays’ relabelling could not produce. The bridge is not a labelling problem tried scoring the reading against the keys the dominants point at and found the reading wrong under that label too. The second slot needs no relabelling. It is the first reading in these essays whose answer for the bridge is the theorist’s, bar for bar.

The blues is the test the earlier essay set, because the pair cue had heard its opening C7 as the dominant of F — a blues in the key of its own subdominant. With a second slot the opening C7 is read in C at every setting that reads the bridge right, all thirty-six bars of the blues; the chord can be the tonic and point at IV, and the reading no longer has to choose. So both of the earlier essay’s open cases are answered at once.

The rondo, heard as one key #

The cost appears in a scheme nobody was watching.

The Classical rondo in the listing has two episodes, eight bars in G major and eight in A minor, between returns of its refrain in C. With one slot, at a key cost of 6, the reading follows both: G I, G V, G I, and so on, and most of the A minor episode in A minor. With a second slot at the setting that reads the bridge, it follows neither. The G episode is C: V, V of V, V, V of V, V, I, V of V; the A minor episode is C: VI, II, VI, V of vi, VI, II, V of vi. Twelve bars that the one-slot reading got right are lost, and the rondo falls from 35 of 40 to 24.

That is not a malfunction. It is the second slot doing, to an eight-bar passage, exactly what it did to the bridge: a chord is heard in the home key and, where it points elsewhere, as an applied dominant. The difference between the bridge and the rondo’s episodes is not in any bar. It is that the bridge points at a new key for two bars at a time, and the episode stays there for eight.

The second episode goes the same way, and the figure makes it plain that the loss is structural rather than a matter of one unlucky chord.

A minor’s tonic, subdominant and dominant are C’s vi, ii and V of vi. The reading with a second slot never has to leave C to name any of them, and at a tonicization cost of 2 against a key cost of 6 it does not. The one-slot reading had been reading these bars right only because the relative minor’s dominant seventh, E7, has no degree in C and forced the key to move; given an applied state that holds it, nothing forces anything. The cadence as evidence found that a cadence is the strongest single declaration of a key a passage makes; here the episode cadences in A minor twice, E7 to A minor, and the second slot hears both cadences as a tonicization of vi.

Why a closely related key is free #

The reason the episode can be absorbed so completely is a fact about closely related keys that the arithmetic makes vivid.

G major’s chords are G, Am, Bm, C, D, Em and F♯°; C major’s are C, Dm, Em, F, G, Am and B°. They share four. The G episode uses G, C and D7: two of those are C major chords in their own right, and the one that is not — D7, with its F♯ — is exactly V of V, an applied dominant of C. Every chord of a modulation to the dominant is either a chord of the home key or an applied dominant of one of its degrees. The same is true of the relative minor: A minor’s i and iv are C’s vi and ii, and its V7 is C’s V of vi.

So a reading with applied states never needs to change key to follow a modulation to a closely related key. It can hear the whole episode inside C, paying the tonicization cost at each applied chord and never the key cost. Whether it does depends only on which is cheaper over the passage: the modulating reading pays the key cost twice, going and returning; the tonicizing reading pays the tonicization cost once for every entry into an applied state, three times in the G episode, twice in the A minor one.

The window that is not there #

The grid in the hero figure is the whole of that comparison, and its shape is the finding.

The bridge needs a cheap second slot: its chain enters an applied state three times in eight bars, so the tonicization cost has to be low against the key cost before the chain is heard in C rather than as a string of modulations. At a key cost of 6, it is read whole up to a tonicization cost of 3 and loses most of itself from 4. The rondo’s episodes need an expensive one: they enter applied states two or three times in eight bars, and the reading modulates only once the tonicizations cost more, over the episode, than two key changes. At a key cost of 6 the episodes return at a tonicization cost of 6 and are whole from 8.

Those two ranges do not meet. Across the fifty-four cells of the grid there is no setting that reads the whole bridge and the rondo’s episodes — every cell with the bridge right has the rondo at 24, and every cell with the rondo’s modulations heard has at most two bridge bars. The listing’s best score, 177 of 188, is at the one-slot end of the grid, which is where the earlier essay left it; the second slot moves the ceiling’s misses from one scheme to another rather than removing them.

And the reason is that the two costs are of the wrong kind to tell the passages apart. A price for entering a state is paid per chord that points elsewhere, whatever happens next; a key change is paid per change, however long the new key lasts. Neither sees duration. The bridge and the episode are distinguished by exactly one thing — two bars of pointing against eight bars of staying — and no pair of per-event costs can price that.

Schoenberg’s regions, arrived at by accident #

A reading that hears every modulation to a closely related key as a tonicization inside the home key has a name in the theory of harmony. Schoenberg called it monotonality: in Structural Functions of Harmony every apparent modulation within a piece is a region of its one tonic — the dominant region, the submediant region — and a movement never really leaves home. The second slot, made cheap, is a monotonal listener. It hears the rondo’s G episode as the dominant region of C and its A minor episode as the submediant region, and it hears them that way for the same reason Schoenberg’s charts do: every chord in them can be named from C.

The earlier readings in these essays were the opposite kind of listener, one who modulates whenever the chords make a new key cheaper than the old one — the listener three decisions that constrain each other built, whose key never moved when metre and segmentation were decided with it, because the key was already the most settled of the three. Neither is wrong about the harmony. They disagree about what a key is — a place a passage goes, or a region a passage colours — and the grid says that no single listener of either kind reads both the bridge and the rondo the way the schemes label them, because the schemes are themselves written by a theorist who treats the bridge as regions and the rondo’s episodes as places.

How the reading and its costs are set #

States are the twelve keys in major and minor, each with seven diatonic degrees and five applied dominants — of ii, iii, IV, V and vi. A bar’s score is the log of its notes’ overlap with the state’s triad plus the bass cue at weight 3 under the register rule, the bass realised smoothly as a bass line is not a list of roots introduced. A change scores the root motion between the two states’ roots as scale degrees of their home keys, minus the key cost if the home key or mode changes, minus the tonicization cost if the new state is applied, plus the resolution cue if the earlier state points at the later bar’s root. An applied state is reported in its home key with its root’s degree there. Bars are right when the key and degree match the scheme’s labels, as throughout.

What two slots still cannot represent #

Duration. The tonicization cost is charged on entry. A state that remembered how long it had been tonicizing — a cost that grew with every bar spent pointing away from home — would charge an eight-bar episode far more than a two-bar chain, which is the distinction the grid shows is missing. That is one more number in the state, and it is the obvious next thing.

Minor-key applied chords. The five applied states are major dominants. The rondo’s A minor episode is absorbed through V of vi; an episode in a minor key whose dominant is not an applied chord of the home key — the mediant minor, say — would not be, and a listing with such an episode would show the second slot differently.

Pivot chords. A theorist marks a modulation at a pivot, a chord that belongs to both keys. The reading has no notion of a pivot, only of costs, and so it can place the boundary of a modulation anywhere the costs allow — which is why the edge bars of the rondo’s episodes move with every setting.

Which harmonies #

The six schemes 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, as in the two parameters turn out to have a ceiling between them and every key-finding essay since. The bridge’s chain of dominants is the standard’s; the rondo’s episodes in the dominant and the relative minor are the Classical rondo’s commonest contrasts; and the difference between them — a passing chromatic chain against a sustained change of key — is the one every harmony textbook draws with the words tonicization and modulation.

Still open: a tonicization that has to be paid for by the bar #

The distinction the grid is missing is length. A reading whose applied states cost a little for every bar held, rather than once on entry, would charge the bridge’s three two-bar tonicizations six units of that cost and the G episode’s continual pointing at G far more — while the key cost, paid only on change, would stay what it is.

The computation adds one number to the state: how many bars the reading has spent in applied states since it was last on a diatonic degree of home. The prediction is specific. If the bar-by-bar cost can be set so that the bridge’s two-bar tonicizations are cheaper than two key changes and the episode’s eight bars are dearer, then there is a window in which one reading reads both — and it would be the first key-finder in these essays whose notion of a key agreed with the textbook’s, not because it was told the labels, but because it charged for time.