A modulation is a dominant that keeps coming back

A second slot hears every modulation as a tonicization gave the key-finder a state for an applied dominant — the key a bar is in, and the key its dominant points at — and it read the thirty-two-bar song’s bridge exactly as the scheme labels it, V of vi, V of ii, V of V, V, for the first time in these essays. At every setting that did so it also heard the rondo’s two eight-bar episodes, in G major and in A minor, as tonicizations inside C. Across fifty-four settings of the key cost and the tonicization cost, no reading held both.

Its diagnosis was that the costs were of the wrong kind. The tonicization was charged on entry and the key change on change, and neither could see that the bridge points away from C two bars at a time while the episodes stay away for eight. Its closing section proposed the obvious repair: a price for every bar spent in an applied state, so that time spent tonicizing is what costs.

The repair was run, and it does the opposite of what it was meant to do. The reason is worth more than the repair would have been.

A price for staying #

The reading is the second-slot key-finder exactly: states of key, mode and degree, and beside each key’s seven degrees the dominants of its ii, iii, IV, V and vi, each belonging to the home key; each bar scored against its state’s triad and its bass; each change scored by its root motion, the key cost if the home key changes, the tonicization cost if the new state is applied, and the resolution cue when a dominant’s next bar is rooted on its target. To that, one term is added: every bar read in an applied state costs a further price by the bar, whether the reading has just entered it or has been there for some time. At a price of nothing it is the earlier reading, bar for bar.

The prediction was simple arithmetic. The bridge’s chain enters an applied state for E7, again for A7, again for D7, and spends two bars in each; an episode stays away from C for eight. Charge by the bar and the eight-bar episode should cost four times a two-bar tonicization, the reading should prefer to modulate for the episodes and to tonicize for the bridge, and somewhere in the plane of costs there should be a window where it does both.

The bridge goes first #

There is no window. At a key cost of 3 and an entry price of a half, the bridge is read whole up to a price by the bar of a half. At 0.75 it falls to four bars. The rondo, meanwhile, stays at 24 of 40 — both episodes still heard as regions of C — until the price reaches 3, when the G episode is recovered and the rondo climbs to 29, and 4, when it reaches 34; by then the bridge has one bar left. At a key cost of 6 the order is the same with the numbers shifted: the bridge holds to a price of 2, is at six bars by 2.5 and three by 3, and the rondo does not move until 6.

So charging tonicizations by their length does not separate the bridge from the episodes. It separates them the wrong way round. Every price by the bar that costs the rondo’s episodes anything costs the bridge more.

Counting the bars each passage spends applied #

The explanation is a count, and it overturns the intuition the repair was built on.

The bridge spends six of its eight bars in applied states — E7, E7, A7, A7, D7, D7, each an applied dominant of C — and only its last two, G7, are the home key’s own V. The episode in G spends three: its G major chords are the home key’s dominant, its C major chord is the home key’s tonic, and only its three D7s need the applied slot, as V of V. The episode in A minor spends two: its A minor and D minor chords are C’s vi and ii, and only its two E7s are applied, as V of vi.

That is the fact a second slot hears every modulation as a tonicization found and put the other way round. A modulation to a closely related key needs the applied slot only for its dominant, because every other chord of the new key is already a chord of the old one. A chain of applied dominants needs it for every chord, because none of its chords is diatonic to anything nearby. By the bar, the chain is the longest tonicization in the listing, and the eight-bar modulations are among the shortest.

So a price on duration is a price that falls hardest on the bridge. The earlier essay’s picture — two bars of pointing against eight bars of staying — was right about the music and wrong about what the reading was doing with it: the reading is not staying in an applied state for eight bars when it hears an episode as a region. It is sitting on the home key’s diatonic degrees, and stepping into the applied slot only when the new key’s dominant sounds.

The order the prices must come in #

The count turns into a rough prediction of where each passage flips, and the prediction explains the order.

A passage is heard as tonicizations inside C when that costs less than hearing it as a change of key and back. The change of key costs twice the key cost, going and returning. The tonicizations cost the entry price once for each applied chord the passage enters and the price by the bar once for each bar it spends applied. Setting the two equal gives, for each passage, the price by the bar at which it should flip from tonicization to modulation: twice the key cost, less the entry prices, divided by the bars spent applied.

At a key cost of 6 and an entry price of a half, that is about 1.75 for the bridge, which enters three applied states and spends six bars in them; about 3.5 for the episode in G, three entries and three bars; and about 5.5 for the episode in A minor, two and two. The reading’s actual flips come later than those — the bridge between 2 and 2.5, the episodes at 6 — because the scores of the chords themselves and of their root motion also differ between the two readings, and the simple count leaves them out. The order is exact. The passage that spends most bars applied flips first, and it is the passage the schemes call a tonicization.

So the failure is structural rather than a matter of finding the right number. For any key cost and any entry price, the bridge’s break-even price by the bar is lower than either episode’s, because six bars is more than three or two. No price on duration can make a reading modulate for the episodes while it still tonicizes for the bridge, because duration, as the reading experiences it, points the other way.

The blues is untouched throughout. Its opening seventh is read in C at every price, whether as a tonic or as the dominant of its own IV, because both readings keep the home key; a price on the applied slot can move it only between two readings that the scheme’s label accepts. The bridge is not a labelling problem found the bridge unwinnable under any label for a one-slot reading; the blues, for a two-slot reading, is unlosable under any price.

What the episodes do that the bridge does not #

The same figure has a second row per passage, and it is the one that separates them.

The bridge’s three applied dominants resolve to three different places: E7 to A, A7 to D, D7 to G, and then G7 to C. Each target is reached once and left at once; no dominant ever goes back to a chord a previous dominant went to. The episode in G resolves its D7 to G three times — G, D7, G, D7, G, C, D7, G — so after the first resolution, every later one returns to a target already reached. The episode in A minor resolves its E7 to A minor twice. One target each, reached again and again.

That is close to the textbook definition of the difference, stated as a count, and it is a definition a tonic bought with the function was already circling when it found a reading buying a key with what a chord does rather than what it contains. A tonicization makes a chord momentarily a tonic by preceding it with its dominant; a modulation establishes a new key, and what establishes it is confirmation — the new tonic approached by its dominant more than once, usually at a cadence. The cadence as evidence found an ordered dominant–tonic pair to be the strongest single evidence for a key that a passage offers. The count here says that the difference between a key and a moment is not how long the passage lasts but whether the evidence repeats: a key is a place a dominant keeps coming back to.

Neither the price on entry nor the price by the bar can see that, because both are costs on the state the reading is in. A return is a relation between the state now and a state some bars ago — the same target resolved to twice — and the reading has no memory of any target but the present one.

Where the ceiling is now #

The listing’s best score is where it was: 177 of 188, at the one-slot end of the plane, with the bridge read wrong and the episodes right. The two-slot reading with a price by the bar reaches 175 at best, never more than the one-slot reading and never holding both passages.

That is informative in its own right, and it echoes the finding of three decisions that constrain each other that the key is the most settled of the reading’s decisions: it takes a passage-wide pattern to move it, and every cost tried so far is local. Three quantities have now been tried as the thing that tells a modulation from a tonicization — the cost of changing key, the cost of entering a tonicization, and the cost of staying in one — and all three fail the same test, because all three are properties of one bar or one change. The distinction the schemes draw is a property of a whole passage’s shape.

The bars themselves #

The two passages that decide everything can be set side by side, as the reading hears them at a setting that reads the bridge.

Read together they make the count concrete. The bridge’s applied states are a chain, each pointing past the last; the episode’s are a pendulum, D7 swinging back to G three times. A listener who hears the bridge as a chain and the episode as a new key is responding to exactly that difference in shape, and it is one no state that knows only its present bar can represent.

How the reading and the count are made #

The reading is the earlier essays’ dynamic-programming key-finder with the applied states and one added term, the price by the bar, subtracted from the score of every bar read in an applied state. The six schemes are read with a bass weight of 3, the register rule and the smooth bass realisation, a resolution cue of 3 and an entry price of a half, over price and key cost as the figures state. A bar is right when its key and degree match the scheme’s labels. The count of returns reads the second-slot reading’s applied bars and asks, for each, whether the next bar’s root is a fifth below its own — a dominant going where it points — and whether that root has already been reached by an earlier resolution in the same passage.

What a count of returns leaves out #

It has not been built into a reading. The count separates the three passages cleanly, but it is a description, not a cue the reading uses. A reading with a state that remembers the last target resolved to, and a bonus for returning to it, is the natural next thing, and whether it opens a window in the plane is a question the count predicts and does not answer.

Three passages are three passages. The listing has one chain and two modulations. A modulation that confirms its key only once — a brief modulation to the dominant closed by a single cadence — would count like a tonicization here, and a chain of dominants that circles back on itself, as some do, would count like a modulation.

The flip points are approximate. The rough accounting of where each passage should flip leaves out the chords’ own fit and their root motion, and the reading’s actual flips come a price or two later than it predicts. Only the order is claimed, and the order is what the sweep shows.

Cadences are not only dominants. A key is also confirmed by its subdominant, by a melodic cadence, by a change of texture at a section boundary. The count sees only dominant resolutions, which is what the applied slot represents.

Which harmonies #

The six schemes are 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, as in looking ahead rescues the relative minor, not the bridge and every key-finding essay since the ceiling is thirteen bars with names. The circle-of-fifths chain of the bridge and the tonic–dominant pendulum of a Classical episode are the two textbook shapes of the distinction, and the result here is that the reading’s trouble with them is not about their length.

The bridge in the listing is the most familiar chain there is: III7, VI7, II7, V7, two bars each, the bridge of the “rhythm changes” that jazz musicians took from Gershwin’s I Got Rhythm and wrote hundreds of tunes over. Musicians describe it as a chain of dominants, and nobody hears it as four keys; the scheme’s label says the same. The Classical episode is equally conventional in the other direction: eight bars in the dominant with its own V–I twice or three times, which every analysis calls a modulation. The count of returns is a formal statement of why those two descriptions come so easily, and the two parameters turn out to have a ceiling between them is where the listing’s first account of what cannot be read was given.

Still open: a reading that remembers where its dominants went #

The count says what the missing state is: the last target a dominant resolved to. A reading that carried it — a key, a degree, and the tonic most recently confirmed — could reward a resolution to that same tonic as confirmation, and treat a resolution to a new one as a further step in a chain.

The computation adds that slot and one weight, a bonus for returning, and sweeps it with the others. The prediction is that a modest bonus reads the episodes’ pendulums as keys, because each of their dominants returns, and leaves the bridge’s chain as tonicizations, because none of its dominants does — and that for the first time in these essays one setting holds both. If it does not, then the distinction the schemes draw needs more than one bar of memory, and the textbook’s word for what it needs — a key established — is naming something longer than any of these readings can hold.