A return is cheaper to avoid than to pay for

A modulation is a dominant that keeps coming back set out to separate two passages that every reading in these essays had confused. The thirty-two-bar song’s bridge is a chain of applied dominants — E7, A7, D7, G7 — that the schemes label as tonicizations inside C. The rondo’s two episodes, eight bars in G major and eight in A minor, are labelled as changes of key. A key-finder with a second slot for an applied dominant could read the bridge correctly, but at every setting that did so it also heard both episodes as regions of C. Charging each bar spent tonicizing made things worse, because the bridge spends more bars in applied states than either episode does.

The count that essay ended on pointed somewhere else. The bridge’s three applied dominants resolve to three different targets and none is ever revisited. The G episode’s D7 resolves to G three times, and the A minor episode’s E7 resolves to A minor twice. A key, on this account, is a place a dominant keeps coming back to, and no reading could see a return because every cost it had belonged to one bar or one change. The proposed repair was to give the reading a memory of the last target a dominant resolved to, and a single weight that rewards returning. The prediction was that a modest weight would read the episodes as keys, because their dominants return, and leave the bridge as a chain, because its dominants do not — and that for the first time in these essays one setting would hold both.

No setting does. Three ways of remembering were tried and each was swept, and in none of them does a single reading hold the bridge and recover the episodes. The reason is not that the memory fails to see the returns. It sees them, and the reading avoids every one.

A reading with a memory #

The reading is the second-slot key-finder of the two essays before. Its states are a home key, a mode and a degree, and beside each key’s seven diatonic degrees the dominants of its ii, iii, IV, V and vi. Each bar is scored by how well its notes fit its state’s triad and bass. Each change is scored by its root motion, by the key cost if the home key changes, by the tonicization cost if the new state is applied, and by a resolution cue when a dominant’s next bar is rooted on its target. The settings are the earlier essay’s: a key cost of 6, a resolution cue of 3, and a tonicization cost of half a nat.

To that the state adds one thing: the applied target the last applied dominant resolved to, inside the current home key, forgotten when the home key changes. When an applied dominant resolves again to that same target, the reading pays a return charge. The essay before proposed this as a bonus for returning, and a charge on the reading that treats a return as a tonicization is the same thing entered on the other side of the ledger: it makes the reading that treats the return as a key comparatively cheaper. With nothing charged, the reading with a memory reproduces the second-slot reading bar for bar across all 188 bars, which the figures check.

The memory is kept exactly. Since the remembered target is fixed by the previous state and the bar, the reading pushes each previous state forward to its one consistent memory rather than searching over memories.

Where the charge lands #

Charged for returns remembered through the whole key, the reading does not recover the episodes at any price. The rondo stays at 24 of 40 at charges of 2 and 4 and falls to 16 at 8. What changes instead is the blues.

The blues’s fourth bar is its I7, heard, rightly, as the dominant of IV, and it resolves to IV7 in the fifth bar. Every chorus does it. To a memory that lasts the whole key, every chorus after the first is a return to the target the previous chorus reached. The charge falls on the blues, not on the rondo, and the reading, rather than pay it once a chorus, hears most of the choruses some other way. At a charge of 2 the blues is read correctly in none of its 36 bars.

A form that repeats is a dominant that keeps coming back, too. The textbook’s word for what a modulation does, established, is about a key confirmed within a passage, and a memory that lasts the whole key cannot tell a return within an episode from a return across a form. The rondo’s refrain does the same: its third bar is heard as V of IV, every refrain.

So the memory has to forget. With the memory lasting four bars, a return counts only if it comes within four bars of the last resolution, which is true of both episodes’ dominants and of none of the blues’s or the refrain’s. Remembered that way, the blues is untouched at every charge. So are the episodes.

How long a memory the episodes need #

The length of the memory is not a free choice, and the listing fixes it. The G episode’s D7s resolve to G at its third, fifth and eighth bars, so its returns come two and three bars after the resolution before. The A minor episode’s E7s resolve three bars apart. The blues’s returns come twelve bars apart, one chorus, and the rondo’s refrain repeats its V-of-IV sixteen bars later. Any memory between four and eleven bars therefore sees every return inside an episode and none across the form, and four is the shortest that does.

So the four-bar memory is the right one, not an arbitrary one. It is exactly the memory the essay before was asking for: long enough to hold a key’s confirmation, short enough not to mistake a form’s repetition for one. If the episodes were going to be recovered by any memory, they would be recovered by this one. That they are not is the first sign that the memory is not what is missing.

It also says something about the word the textbooks use. The cadence as evidence found an ordered dominant–tonic pair to be the strongest single evidence for a key, and a tonic bought with the function found a reading paying for a key with what a chord does rather than what it contains. An established key, in these passages, is a dominant–tonic pair repeated within a few bars. The memory can count that exactly. It is the reading’s response to the count that goes wrong, and how much of the reading arrives late is a reminder that a reading which decides at the end of a passage can revise every bar before it to make the count come out cheaply.

The escapes #

To see why a four-bar memory changes nothing, it is enough to look at what the reading does with the episodes when a return is charged.

With a return charged at 8 and remembered four bars, the G episode is heard exactly as before except for one chord. Its second D7, which would have been the return, is heard not as the dominant of the dominant but as the minor ii of C. D minor is not the chord written. But the reading’s measure of a chord’s fit is the overlap of its notes with a triad’s, as a share of the notes in either, and a D7 — D, F♯, A, C — shares two of its four notes with D minor. Hearing it as D minor loses the F♯ and costs 0.63 nats of fit. The return charge was 8.

The A minor episode escapes the same way from the other side. Its two E7s resolve to A minor twice, three bars apart. The reading hears the A minor chord between them as the dominant of ii in C, an A major chord, which loses the C♮ and costs 0.69 nats of fit. The second E7 then resolves to a target the reading has not just reached, and nothing is charged.

The arithmetic of the escape is the whole result. For the reading to hear an episode as a change of key, it must pay the key cost twice, going and returning: twelve nats. To keep the episode in C and pay for its returns, it pays the return charge once for each return: 8 or 16 for the G episode at a charge of 8. To keep the episode in C and avoid the returns, it mishears one chord for about two-thirds of a nat. At any return charge above about 0.7 of a nat, avoiding the return is cheaper than paying for it, and paying for it was always cheaper than changing key. The charge never reaches the reading it was aimed at, because a cheaper escape is always in the way.

Making the escape dearer opens another #

The obvious repair is to make the escapes cost more. They are cheap because the fit is tolerant: a D7 is two-fifths of a D minor chord to this reading. Counting each bar’s fit six times over makes the D7-as-ii escape cost nearly four nats.

It then finds another. At a return charge of 8 with the fit weighted six times, the reading moves the end of the rondo’s first refrain and the first half of the G episode into F major. There the G major chords are the dominant of the dominant and the D7s are the dominant of ii, so the dominants alternate between two targets and none of them returns within four bars. It is a strange hearing of a passage in G, and it costs less than either paying for the returns or changing key twice. The rondo falls to 16 of 40.

This is not a flaw particular to the charge. The reading is an optimiser: it finds the cheapest path through all the ways each bar could be heard, and a charge on one interpretation is a reason to prefer every other interpretation that avoids it. With twelve keys, two modes and twelve states in each, there are always other interpretations, and some are nearly free.

Paying for confirmation instead #

The lead phrased its repair as a reward rather than a charge, and there is a version of reward that no escape avoids: a bonus whenever a key’s own dominant resolves to that key’s own tonic. A reading that hears the G episode in G earns it three times, once for each D7–G. A reading that keeps the episode in C earns it only for its G–C cadences.

It does move the episodes. It also moves everything else first. At a bonus of 2 the listing falls from 172 to 149, the ostinato to half its bars and the blues to 30 of 36. At 4 the verse–chorus song is read correctly in none of its 32 bars: every scheme with a dominant–tonic pair in it is re-heard in whatever key makes the most of those pairs its own V–I. At 6 the rondo reaches 29 of 40 — the G episode recovered — and the bridge has fallen to 2 of 8, read as a string of changes of key. The confirmation bonus does exactly what was feared of the one-slot reading in the bridge is not a labelling problem: it rewards hearing every dominant as a new key’s.

What the three attempts have in common #

The three repairs fail in three different ways, and the ways have one thing in common. A memory that lasts the key is spent on the form’s repeats. A memory that forgets is avoided by a mishearing that costs less than a nat. A bonus for confirmation is claimed by every scheme at once. In each case the quantity added to separate two passages is captured first by something else in the listing, because the reading will take any interpretation that is cheaper.

Three decisions that constrain each other found the key to be the reading’s most settled decision, moved only by passage-wide patterns. This essay finds the reverse side of that. The key is settled because every cost that might move it can be paid more cheaply by changing something less settled — a chord’s degree, a chord’s quality, a tonicization’s target. The key reading does not lack memory. It lacks a chord matcher that takes a chromatic note seriously.

That is the specific thing the escapes exploit. The F♯ in a D7 is, in C major, the one note that says the chord is not diatonic, and a musician hears it as decisive: it is the whole of the evidence that the D chord points at G. To this reading it is one note in four, and losing it costs two-thirds of a nat. A second slot hears every modulation as a tonicization found that every chord of a closely related key is diatonic to the home key or an applied dominant of it. The complement is that the only notes that tell the two readings apart are the chromatic ones, and the reading barely weighs them.

What the model leaves out #

One fit for every note. A chord’s fit is the overlap of its notes with a triad’s, counting every note equally. A fit that weighted a chromatic note by how unlikely it is in the home key would make every escape above expensive, and would be the next reading to try.

Returns counted by target. The memory records the last applied target, and a return is a resolution to the same one. An episode could establish its key through a cadence without an applied dominant at all, as a pivot-chord modulation often does, and nothing here would count that.

The same six schemes. Every number is from the listing the earlier essays used: six schemes, 188 bars, labelled by the schemes’ own analysis. A larger listing, with modulations of other kinds, might find a setting the six do not; it would have to find one the escapes do not reach.

Whose modulations #

The passages at stake are the ones the essay before described. A classical rondo’s episodes are a new key established by its dominant, returned to and confirmed, before the refrain comes back. A popular song’s bridge runs a chain of secondary dominants back to the home dominant. The schemes label them differently, and a trained listener hears them differently, largely, on the evidence here, by the chromatic notes. How much evidence a modulation needs found that a key change needs a certain duration of evidence to be heard. What these readings suggest is that the evidence is qualitative as well: an F♯ that a listener in C cannot hear as anything but a leading note to G.

Still open: a reading that hears a chromatic note as a leading note #

The escapes all work by giving up a chromatic note: the F♯ of D7, the G♯ implied by E7, the C♯ the reading invents when it calls A minor a dominant. A reading whose fit charged a note outside the home collection according to how strongly it implies another key — nearly nothing for a flattened seventh, a great deal for a raised fourth that is a leading note to V — would make the D7-as-ii escape cost what a musician thinks it costs. The prediction is that under such a fit the memory of returns would stop being avoidable, and the four-bar memory would do what this essay hoped. If it does, then what separates a key from a moment in these passages is two things at once, a chromatic note taken seriously and a dominant that comes back; if not, the textbook’s established is naming something no sequence of chords can carry without a listener who already expects the form.