The chord that did not come

The deceptive cadence has one of the best names in music theory and one of the worst explanations. A dominant chord sets up an ending, and the chord that arrives is the submediant instead of the tonic. The listener, having expected one thing, is given another.

That much is right. What is normally said next is that the arriving chord is unexpected, and unexpectedness is a statistical notion — a rare event, an ill-fitting chord, something the key did not lead one to anticipate.

Both halves of that are checkable with machinery this site already has, and neither survives.

The arriving chord fits the key #

The site’s perception field is built on the Krumhansl–Kessler probe-tone profiles, which are measurements rather than theory: establish a key in a listener’s ear, play a single note, ask how well it fits, and average over listeners. The numbers that come out are the tonal hierarchy, and they are the closest thing there is to an empirical statement of what a key means to a listener.

Take each triad of a major key and score it as the mean of its three notes’ ratings.

So the deceptive cadence does not arrive at an outsider. It arrives at the runner-up.

That should already be enough to sink the “wrong chord” account, and there is a sharper version of the same point. The submediant contains the tonic note. A C major tonic chord is C, E and G; the A minor chord that replaces it is A, C and E. Two of the three notes are shared, and one of them is the tonic itself.

A listener at a deceptive cadence has been given the tonic pitch. They have not been given it as a bass.

Two of the three voices do the expected thing #

The hero figure makes the second half of the case, and it uses the site’s voice-leading solver — the same one that measured cadences during the standard pass and refuted the usual ranking.

Set the dominant triad against each of its two continuations and look at what the individual voices do.

Going to the tonic: G stays where it is, B rises a semitone to C, D rises a tone to E.

Going to the submediant: G rises a tone to A, B rises a semitone to C, D rises a tone to E.

Two of the three moves are identical. The leading note resolves upward to the tonic exactly as it would have; the supertonic rises to the mediant exactly as it would have. The only voice that behaves differently is the one that was going to stay still and instead goes up a tone.

That is a remarkably local difference for an event described as a subversion. One voice out of three, moving one tone, in a chord that fits the key at ninety per cent of the tonic’s rating.

What the surprise actually is #

Put the two results together and the mechanism is specific.

The dominant seventh contains an obligation — a tritone whose two notes must move outward by a semitone each — and at a deceptive cadence that obligation is discharged in full. B goes to C, F goes to E. The unstable interval resolves exactly as advertised.

What does not happen is the bass. The root does not fall a fifth, and the chord it lands on is not rooted on the tonic. So the harmonic gesture of arrival is withheld while the melodic gesture of resolution is completed.

A deceptive cadence is an expectation half-met, and the half that is met is the more audible one. That is why it is a device of postponement rather than of contradiction: everything is still pointing home, the pointing has been acknowledged, and the arrival has been deferred by a single voice.

It is also why the ordinary continuation from a deceptive cadence is to try the cadence again. Nothing has been refuted. The material is intact, the obligation was honoured, and the piece is where it was two bars earlier with one more attempt spent.

The minor mode, which was a real test #

The same measurement in the minor mode is a genuine test rather than a formality, because the minor profile is a separate set of measurements. It was obtained from the same listeners with minor contexts, it is not the major profile rotated, and it differs in shape as well as in position. If the submediant’s second place in the major mode were an artefact of how the mean is taken, there is no reason for it to reappear.

So the finding holds in both modes and is more pronounced in the one where the deceptive cadence is used more freely. The reason is the same in both: the chord on the sixth degree shares two of its three notes with the tonic triad, one of them being the tonic itself, and a mean of three ratings cannot help rewarding that.

Two consequences follow. The measurement is not a quirk of one profile, which is what a test is for. And the mechanism behind it is transparent enough to be checked by hand, which is worth more than a robustness argument: the submediant scores well because it is mostly the tonic chord, and that is exactly why it can stand in for it.

Why the name is misleading, and what would deserve it #

If deception means being led to expect something that does not happen, several other progressions have a better claim.

The dominant moving to the subdominant is a much stranger event. Its voice-leading cost is 6 semitones — the largest in the key — none of its voices does what the tritone demands, and the subdominant sits below the submediant on the fit ranking. It happens all the time in blues-derived music and nobody calls it deceptive, which says something about how much the label is describing a context rather than a chord pair.

A dominant moving to a chord outside the key is stranger still, and there the surprise is genuinely statistical: the notes do not fit the profile, the listener’s hierarchy has no place for them, and the effect is a change of level rather than a postponement.

The deceptive cadence gets its name because of where it happens. Put the same two chords in the middle of a phrase and nobody notices; put them at bar eight of an eight-bar phrase, on the downbeat, with the texture thinning, and the withheld bass is the loudest absence in the music. That is the fourth column of the closure vector doing the work, and it is the argument that a cadence is an event rather than a chord pair arriving from a different direction.

Expectation, and the two things it can mean #

The word doing the most work in every account of this cadence is “expectation”, and it has two meanings that the measurements above separate.

One is statistical: how often does this follow that, in the music a listener has heard. It is a property of a corpus and of a listener’s exposure, it is learnable, and it is what the tonal hierarchy is built out of — the profile ratings track the frequency and duration of the pitch classes in the repertoire remarkably closely, which is the finding that made those numbers interesting in the first place.

The other is structural: what the material itself demands. A tritone has to resolve; a leading note has to rise; a dominant seventh is not a place to stop. Those are claims about the sonority rather than about a corpus, and they are the reason a listener who had never heard the progression before would still feel something unfinished.

A deceptive cadence separates the two cleanly, and that is the most useful thing about it. The structural expectation is met — the tritone resolves, the leading note rises. The statistical one is not, in the sense that this is not the continuation the listener’s exposure most strongly predicts. Whatever the effect is, it is produced by the second while the first is being satisfied.

Which also predicts something checkable. A listener unfamiliar with the idiom should find a deceptive cadence far less striking than a familiar one does, because only the learned half of the expectation is being denied. That is the shape of result the site’s half-learned consonance essay reports for a different judgement, and it is the sort of experiment the profiles were designed for.

The device outlives the cadence #

One more thing follows from the submediant being mostly the tonic chord, and it is the reason this device is worth a rung rather than a paragraph.

The same substitution works away from cadences. A phrase that would sit on the tonic can sit on the submediant instead, keeping two of the three notes and changing the bass, and the result is a passage that is at home and is not settled. That is one of the standard ways of extending a section without stopping it, and it is the same trick as the deceptive cadence with the closure components turned down.

Which is the general shape of everything in this phase’s closure work. A pair of chords is a resource, and what it does is decided by where it is put. The deceptive cadence, the half cadence and the tonic-to-submediant move that opens half the songs on the radio are three uses of two chord relationships, and the difference between them is entirely in the last two columns of the closure vector.

The model, and where it stops #

The fit numbers above come from the Krumhansl–Kessler profiles, and it is worth being exact about what they are and are not.

They are ratings of single notes against an established key, averaged over listeners, obtained in the laboratory in the early 1980s with Western listeners and Western contexts. They are the best-known data of their kind and they are a measurement of a population, not a law.

Aggregating three of them by taking a mean is a further step, and a crude one. There is no reason a chord’s fit should be the average of its notes’ fits; a real model would weight the root more heavily, or account for the chord as a unit, or use one of the several key-finding algorithms built on top of these profiles. What can be said for the mean is that it has no free parameters, so the result is not a consequence of a weighting somebody chose. Whether it is robust to sensible alternatives is a question with an answer, and the section below sweeps them rather than asserting it.

The aggregation, swept #

“Robust to sensible alternatives” is a claim that takes one loop, and it comes out half right, which is more useful than either whole.

The tonic is first under all seven, in both modes — that part is as robust as claimed. The submediant’s second place is not. Three aggregations put the subdominant there instead, and weighting the root triple drops the submediant to fourth, below the dominant, which is exactly the comparison the section above says “any weighting would reward”. It would not.

What survives every aggregation is the claim the argument actually rests on. The submediant is never worse than fourth of seven, in either mode, under any of these; it is second or third under six of the seven; and it is never in the bottom half. A chord that no way of scoring can put in the bottom half of its own key is not an outsider, and “not an outsider” is all the refutation of the wrong-chord account needs.

Two smaller things fall out. The minor mode is the more robust of the two — the submediant is second under five of the seven aggregations there against three in the major — which agrees with the direction of the mode comparison above and was not built in. And the correlation aggregation, which scores a triad by correlating its indicator vector against the whole twelve-entry profile and is what every key-finder on this site actually does, gives exactly the mean’s ranking. The crudest aggregation and the one with a pedigree agree, which is the better reason to keep the crude one than the absence of free parameters.

The one aggregation that breaks the second place is the one that weights the root most heavily, and its disagreement is informative rather than awkward: a root-heavy score is measuring how good a bass the chord makes, and the bass is precisely what a deceptive cadence withholds.

What the profiles cannot do is give a probability. They say how well a note fits; they do not say how often a chord follows another in any repertoire. That is the measurement this argument would really want and this site does not have. A corpus of harmonic transitions would answer directly whether the submediant is a rare continuation from the dominant, and no figure here does. What the essay claims instead is narrower and is supported: rarity cannot be the explanation offered, because the chord is not ill-fitting and its voice leading is nearly identical.

Whose music #

The deceptive cadence in this form belongs to European tonal practice from the late seventeenth century onwards, where it is common enough to be a cliché — most often at the end of an extended cadence, and very often three or four times in a row before the real ending.

The device it belongs to is more general than the repertoire. Any music that establishes a strong expectation can defer it, and the deferral is more effective the more of the expectation is satisfied on the way. That is a fact about listeners rather than about the eighteenth century, and it is the same fact that makes an interrupted resolution in a raga or a dropped beat in dance music work.

What this cannot show #

The whole argument is conducted in pitch classes, which means it cannot see the two things a performer would name first: register and voicing. A deceptive cadence with the tonic in the top voice is a very different event from one with the submediant’s own root there, and the closure vector has no column for the soprano.

It also cannot see dynamics or texture, which in practice carry a great deal of the effect — the standard deceptive cadence is louder than what precedes it, and the sudden loudness is doing work the harmony gets the credit for. Loudness is not amplitude and it is not in any of the figures above.

The second place is the mean’s answer and not the profiles’. The sweep above shows that three of seven aggregations put the subdominant second instead, so a reader who wants the strongest version of the argument should take the weaker sentence: the submediant is in the top half of its own key under every way of scoring tried, rather than that it is specifically the runner-up. The essay’s title claim is unaffected — a chord in the top half is not a chord nobody expected — and the figures draw the mean because the mean is what the site’s other essays use.

And the fit ranking is a ranking of chords in isolation against a key. It has nothing to say about a chord’s fit given the chord before it, which is the quantity that would actually bear on expectation. Building that would need transitions rather than states, and the site’s chord map is the nearest thing it has — a measure of the cost of a move, which is not the same as its likelihood.

Where the ladder goes #

This rung and the closure ladder have both been about a single moment. The variable that governs how often such moments can occur at all is the harmonic rhythm, and it turns out to be independent of which chords are used and to have a ceiling imposed from outside music altogether.