The leading note is already inside the chord that hides the key

Three essays have tried to make a key-finder hear two passages differently, and each attempt was captured by something else in the listing before it reached its target.

The two passages are the ones every reading in this sequence has confused. A thirty-two-bar song’s bridge runs a chain of applied dominants — E7, A7, D7, G7 — that its scheme labels as tonicizations inside C. A rondo’s two episodes, eight bars in G major and eight in A minor, are labelled as changes of key. A second slot hears every modulation as a tonicization gave the reading a slot for an applied dominant and found that it read the bridge exactly and the episodes as regions of C. A modulation is a dominant that keeps coming back found what separates them — an episode’s dominant resolves to the same target again and again — and a return is cheaper to avoid than to pay for gave the reading a memory of those returns and charged it for them. The reading avoided every charge rather than pay it, most cheaply by hearing a D7 as the minor ii of C and throwing away its F♯, for two thirds of a nat.

That essay ended with a diagnosis. The F♯ of a D7 is, in C major, the one note that says the chord points at G, and a musician hears it as decisive; the reading weighs it as one note in four. A reading that charged a chromatic note by how strongly it implies another key — a great deal for a raised fourth, a leading note to V; nearly nothing for a flattened seventh — would make the escape cost what a musician thinks it costs, and the memory of returns should then do what it was built to do.

The charge was built. It closes the escape, as predicted. It changes nothing else, and the reason is in the first figure.

Where the chromatic notes are #

Before charging anything, it is worth counting what there is to charge. Across the listing’s six schemes and 188 bars, the reading of the earlier essays — no charges, a key cost of 6 — hears 41 chromatic notes: notes outside the key it hears a bar in, counted once a bar. Eleven of them are in the thirty-two-bar song and the rondo. Every one of those eleven is a note of the chord the reading hears in its bar. The F♯ of each D7 is the third of the V of V the reading hears it as; the G♯ of the A minor episode’s E7 is the third of V of vi; the bridge’s C♯, F♯ and B are the thirds of its applied dominants.

The other thirty are in the twelve-bar blues, and not one of them is inside the chord the reading hears. They are the sevenths — the B♭ of the I7, the E♭ of the IV7 — which no triad of C contains and no applied dominant supplies.

That count decides everything that follows. A charge on a chromatic note falls only on a reading that leaves the note out of the chord it hears, and in the passages the charge was aimed at, the reading already hears every chromatic note in a chord.

The charge #

Each bar is scored, as before, by how well its notes fit the triad of the state the reading hears it in, and by its bass. The new term charges each chromatic note that lies outside that triad. A note is charged at the full weight if it is a leading note — a semitone below the tonic of one of the key’s applied targets, which is where the F♯ of a D7 in C sits, below G — and at a lower weight otherwise, which is where a flattened seventh falls. With the weight at zero the reading is the earlier essay’s to the bar, which the figures check.

The price of each hearing of a D7 shows what the charge can and cannot do. Heard as the minor ii of C, the D7 loses its F♯ and cost 0.63 nats before; now it costs 8.63. That escape is closed. Heard as V of V in C it costs the half-nat of entering a tonicization, and nothing more, because its F♯ is the third of the chord heard. Heard in G, as the dominant, it costs the twelve nats of changing key there and back; heard in F, as V of ii, the same. None of those changes, because none of them leaves the F♯ out.

So the charge makes the tonicization reading no dearer at all. It was meant to make the reading prefer a change of key to a tonicization, and it can only do that by making the tonicization expensive — but the tonicization is the one reading that takes the leading note as seriously as a change of key does.

What the reading then does #

Across the whole grid, the rondo is read as its scheme labels it in 24 of its 40 bars at best — exactly where it started. With no return charged, the leading-note weight can be half a nat or eight and the rondo does not move by a bar. With a return charged 8, it falls to 16, as it did without the chromatic charge. The bridge stays at 8 of 8 throughout, which was never in doubt.

The bar-by-bar reading shows why. With the leading notes charged 8 and no return charged, not one bar of the rondo is heard differently: the G episode’s D7s are V of V in C, the A minor episode’s E7s are V of vi, and their chromatic notes are inside those chords. With a return charged as well, the reading needs a way to stop the D7s from resolving to the same target twice within four bars without dropping their F♯. It finds one that the essay before found by a different road: it moves the end of the refrain and the first half of the G episode into F major, where the G chords are V of V and the D7s are V of ii — a chord that still contains its F♯. The escape through D minor is closed, and the escape through F was never touched by the charge.

The F-major hearing is worth describing, because it looks perverse and is the cheapest thing available. In F, the G major chords of the episode are V of V and its D7s are V of ii, so the passage’s dominants now point alternately at two targets, C and G, and no dominant resolves to the same target as the one before it within four bars. The memory records no return, the return charge is never paid, and every chromatic note — the F♯ of each D7, the B of each G chord — is inside the chord heard. What the reading pays is the key cost of going into F and coming back, once, which is cheaper than paying for three returns. A passage in G heard in F is not something a listener does; it is what a reading does when every cost it has is attached to something other than where the passage actually lives. A tonic bought with the function found a reading paying for a key with what its chords do, and here the chords’ function is satisfied twice over, by two keys, neither of them the right one.

That is the structural point. A chromatic note distinguishes a tonicization from a diatonic chord; it does not distinguish a tonicization from a key. The F♯ says “this D chord points at G”, and a tonicization of G and a modulation to G both agree. What the earlier essays were missing is something that tells a passage pointing at G for a bar from a passage living in G for eight, and the note that points at G is present in both.

The blues, and what a pitch class cannot say #

The charge costs something the earlier readings had right. At no weight does it help the rondo; at a weight of a fifth of a nat it loses the whole blues. The blues’s I7 and IV7 carry B♭ and E♭, the thirty chromatic notes the reading leaves out of the chords it hears, and at a small enough price per bar it is cheaper to pay the key cost once and hear the blues in F, where the I7 is a dominant seventh and the B♭ is diatonic.

The price is small because of what the E♭ is. The flattened seventh of the IV7 was meant to be charged at a tenth of a leading note, or at nothing. But the reading works in pitch classes, and as a pitch class E♭ is D♯ — a semitone below E, which is the tonic of the key’s iii, one of its applied targets. So the E♭ is charged as a leading note however the flattened notes are priced: the blue note and the leading note are the same twelfth of the octave. That is why the figure’s two curves, with flattened notes free and charged a tenth, fall at nearly the same place — between 0.15 and 0.175 nats with the flattened notes free, a step earlier with them charged: the E♭ decides it either way, and the B♭ only brings the moment forward.

This is the diagnosis the essay before gave, turned round. It said the reading needed a chord matcher that takes a chromatic note seriously. A reading of pitch classes cannot take a chromatic note seriously without also taking seriously every note that shares its pitch class — and the blues’s sevenths, which are the most characteristic chromatic notes in the listing, share their pitch classes with leading notes. Many accounts of the blues treat its flattened notes as belonging to the melody’s inflection of a scale rather than to any key the harmony implies — a colour of the tonic rather than a pointer away from it. That is a reasonable account of what a listener hears, and it is exactly the distinction a pitch-class reading cannot make, since it has no way to know which kind of E♭ it is holding.

Why no charge on a single bar can help #

The failure is not particular to this charge, and it is worth saying why in general, because it rules out a family of repairs at once.

Any term that scores a bar by its own notes — a chord’s fit, its bass, a charge on its chromatic notes — can be satisfied, bar by bar, by the reading that hears each chord as exactly what it is. In the rondo’s G episode that reading hears G, C, D7 and G as I, IV, V of V and I of C: every note of every chord is inside the chord heard, the bass is each chord’s root, and nothing is chromatic that the chord does not explain. A change of key to G hears the same four chords as I, IV, V7 and I of G, and it too explains every note. On the bar-by-bar terms the two readings are tied, and the tie is broken only by the costs of states and changes — a half-nat for each tonicization against twelve for going to G and coming back.

So any repair that works on single bars leaves the two readings tied and cannot reverse that verdict; only a cost on states or changes can. The cadence as evidence found an ordered dominant–tonic pair the strongest single evidence for a key, and in the episode that evidence is present in both readings: V of V to I of C is a dominant–tonic pair onto G, just as V7 to I of G is. The bridge is not a labelling problem had found the same symmetry from the other side, when a one-slot reading heard every applied dominant as a new key.

The leading-note charge was the most promising of the single-bar repairs because it looked as if it would break the tie — the F♯ is the note that says G. It does say G. It says G to both readings.

What the three essays have in common now #

Three essays tried three repairs. A memory of returns was captured by the form’s repeats or avoided for less than a nat. A bonus for confirmation was claimed by every scheme at once. A charge on the leading note closes one escape and finds the rest already paid for. Three decisions that constrain each other found the key the most settled of the reading’s decisions; each repair has confirmed that the key is settled because a less settled decision — a chord’s degree, its quality, a tonicization’s target — can always absorb whatever cost was aimed at it.

This one says more precisely what is missing. The episodes and the bridge differ in how long a passage points at one target, and in how often it comes back to it. The note that points is identical in the two, and the reading already hears it. How much evidence a modulation needs found that a change of key needs a duration of evidence; the repairs so far have all been costs on single bars or single changes, and a cost on a single bar cannot see a duration, which is the one quantity that separates the passages.

What the model leaves out #

Spelling. The reading hears pitch classes, and the whole trouble with the blues is that E♭ and D♯ are one pitch class. A reading that took spelled notes — from a score, or inferred from voice leading — would charge the blues’s E♭ as a flattened note and leave the blues in C. It would still not move the rondo, because the rondo’s F♯s and G♯s are spelled as leading notes and are already inside the chords heard.

The weights. A leading note is charged at a stated weight, swept from half a nat to eight, and a flattened note at nothing or a tenth. The rondo’s result does not depend on the weight, since no weight touches it. The blues’s threshold depends on the key cost, 6, against which the per-bar charges accumulate.

The same six schemes. Every count is from the listing the earlier essays used. A modulation that establishes its key without an applied dominant — through a pivot chord — would put its new key’s leading note into a bar the reading might hear diatonically in the old key, and there the charge could act.

Whose modulations #

The passages are the ones the earlier essays described: a classical rondo’s episodes in the dominant and the relative minor, a popular song’s bridge of secondary dominants, and a blues whose dominant sevenths are its tonic colour rather than its harmonic motion. The claim is about what a chromatic note can tell a listener in those passages. It can tell them a chord is pointing somewhere; it cannot, on its own, tell them the passage has arrived there. Trained listeners hear the episodes as keys and the bridge as a chain, and on this arithmetic the note they hear as decisive is not the note that decides.

Still open: a reading that pays for time spent pointing at one target #

Every repair so far has put its cost on one bar or one change. The quantity the passages differ in is a run: an episode points at G for eight bars, the bridge at a new target every two. A reading whose tonicization state paid a cost that grew with the number of consecutive bars spent tonicizing the same target — one bar cheap, eight dear — would make a long tonicization more expensive than a change of key, and a chain of short ones cheap. The earlier essay that charged tonicization by the bar found the bridge spending more bars in applied states than either episode, but those bars were spread across three targets; charged by the run on one target, the ordering reverses. The computation is the memory reading with the run length in its state, and it would say whether duration on one target is, at last, the thing no cheaper interpretation can absorb.