An ending that exists so a bigger one can

A phrase that ended convincingly at bar four would be a piece four bars long. Since almost no music is four bars long, almost every cadence in almost every piece has to be an ending that does not end anything, and the tradition has names for several kinds.

The usual way of describing them is by subtraction. The half cadence is a cadence without a resolution; the imperfect authentic cadence is the perfect one without the tonic in the top voice; the deceptive cadence is the authentic one with the wrong chord. All of these are true and all of them make the weak cadence sound like a defective strong one.

It is the other way round. The weak cadence is the load-bearing member. It is what makes a group of phrases into one object instead of a sequence of small pieces, and the shortfall in it is designed with some precision.

The shortfall, computed #

The hero figure runs the five-component closure vector at each of the four arrivals in a sixteen-bar plan built from two eight-bar phrases.

At bars 4 and 12, a supertonic chord arrives on the dominant. That supplies the fifth-move component — the root really does move by a fifth, and the gesture is unmistakably cadential — and it supplies neither of the other two harmonic components, because the goal is not the tonic and no leading note has resolved.

At bars 8 and 16, a dominant seventh arrives on the tonic. All three.

So the pattern along the phrase is: one signal, three signals, one signal, three signals. And the two that the weak arrivals withhold are precisely the two the strong ones add.

That is not a coincidence and it is not an observation about these particular chords. It is what “half cadence” means, and putting it in a vector makes visible something the name obscures: the weak cadence is not a diluted version of the strong one. It is a different subset, and the subset is chosen to be the one that goes as far as possible in the right direction without arriving.

Arriving is the point #

The half cadence has the fifth-move and nothing else, and that combination has a specific effect which is worth naming.

The bass gesture says an arrival is happening. The absent tonic says the arrival is not home. A listener therefore has an event that is unambiguously a punctuation mark and unambiguously not a full stop — which is exactly what is needed at the midpoint of a sixteen-bar span, and it is a state that no amount of vagueness would produce. A cadence that was merely indistinct would leave a listener unsure whether anything had happened. This one leaves them certain that something has happened and that it is not the end.

The antecedent has some of the signals and not the ones that matter most. A weak ending is not a failed strong one; it is a different vector, and the difference is concentrated in the components that say where the goal is and how long it is held — which is why the first phrase can sound finished and unfinished at once.

The economy of that is worth pausing on. Six bars of identical material, then two bars of divergence, and the two halves are heard as question and answer. Whatever is doing the work is doing it in the last quarter of each phrase, and the vector says which components those two bars turn on.

The matrix agrees, and it agrees in the way that matters #

That is the sense in which the metrical and durational signals are detachable: they can be taken away from any cadence without touching its harmony, and what is left is the same ranking with less of everything. A composer weakening an ending does not have to change a chord.

That is a small result and a satisfying one. The repetition machinery has no notion of a cadence and cannot compute a closure vector, but it can see that two passages agree and then stop agreeing, and in a period the place they stop agreeing is the cadence by construction.

It is also the clearest illustration of why the two families of measurement belong in one field. The matrix says where; the vector says what kind. Neither answers the other’s question and a description of a form needs both.

Length is the other lever, and it is continuous #

The three harmonic components are all-or-nothing. Either the root moves by a fifth or it does not. That gives a coarse instrument — seven reachable combinations, as the section on the duration lever counts, of which the tradition uses about four — and it is nowhere near enough resolution to explain the very fine gradations of closure that real music produces.

The other two components are continuous, and that is where the resolution comes from.

Note what the third row cannot show. A cadence held for twice the prevailing unit and one held for eight times both come out at the ceiling, because the component is capped at 1. That cap is a modelling decision and it hides a real distinction: the final cadence of a symphony is held for a great deal longer than the final cadence of its first phrase, and the difference is one of the loudest signals in the piece.

Uncapping it would make the number unbounded and incomparable between pieces. Capping it loses the top of the range. Neither is right — and there is a third that is, which is a saturating map rather than a truncation:

Dividing the length by itself plus one is bounded below one at every length, strictly increasing at every length, and comparable between pieces because it can never leave the unit interval. It separates the final bar of a movement from the end of its first phrase, which the cap cannot, and it does not run away, which the uncapped version does.

What it costs is the meaning of the number one. On the capped scale a goal as long as its neighbours reads 1.000, which is a natural unit and is why the cap was chosen; on the saturating scale it reads 0.500, which is arbitrary. So the trade is a readable unit against an unbounded ordering, and this rung wants the ordering, because the whole argument is that the levels of the hierarchy are distinguished by more of the continuous components and the cap is the place that argument runs out.

The three harmonic components are not independent #

One more thing falls out of writing the vector down, and it corrects the count in the paragraph above. Running every ordered pair of the seventeen chords this collection carries and recording which of the eight combinations each produces:

There are seven combinations, not eight. A root moving by a fifth onto the tonic means the chord before it was rooted on the dominant, and every dominant-rooted chord in the vocabulary contains the leading note — so a cadence cannot make the bass gesture, arrive home, and withhold the resolution. The two components the strong cadences add are not two independent signals; in a diatonic vocabulary the first implies the second.

That is a small correction and it tightens the argument. The half cadence withholds both of them because there is no way to withhold one, and the choice the tradition made is between three states rather than the seven or eight a vector of independent bits would offer.

The hierarchy this builds #

Put the two levers together and a hierarchy falls out without anything hierarchical having been built.

A four-bar unit ends with one harmonic signal and a short goal. An eight-bar unit ends with three harmonic signals and a goal about as long as its neighbours. A sixteen-bar unit ends with three signals and a goal longer than its neighbours. A movement ends with three signals, a goal several bars long, and usually a repetition of the whole cadence for good measure.

Each level is distinguished from the one below by more of the continuous components rather than by a different kind of ending. That is why a listener can tell, without counting bars, roughly how large a unit has just closed — and it is why the same chord pair serves at every level.

The novelty operator on the same sixteen bars finds the eight-bar boundary at the narrower kernel and not at the wider one, which is the structural counterpart of this essay’s point: the interior cadence is a boundary at one scale and not at another, and whether it is heard as an ending depends on how much context a listener is holding.

The figure is a reminder that the hierarchy is not visible to a change detector. Endings are not changes. In a period the boundary between the two phrases is one of the least changeful places in the sixteen bars, and it is a major structural joint.

Delay is a fifth level #

There is a device that sits between the phrase and the section, and it is the reason the deceptive cadence exists at all.

Set up an authentic cadence, arrive at the submediant instead of the tonic, and then do the cadence again properly. The first attempt supplies the leading note’s resolution and withholds the tonic goal; the second supplies everything. What a listener gets is an ending that was promised, denied and then delivered, and the delivered one is stronger than it would have been on its own — because the promise has now been made twice.

Two things follow. The first is that the deceptive cadence belongs in the same family as the half cadence rather than in a category of its own: both are engineered partial arrivals, differing in which components they hold back. The second is that the phrase “the cadence is extended” turns out to describe something specific — an arrival is attempted more than once, and the vector is filled in over several attempts rather than at one moment.

The strong beat is a hypermetric one #

The metric component has been treated so far as a fact about the beats of a bar. At the levels this essay is about it is a fact about the bars of a phrase, and the machinery is identical.

The consequence for closure is direct. A cadence lands on a bar, that bar has a weight in the hypermetre exactly as a beat has a weight in the bar, and a cadence landing on a hypermetrically weak bar is weakened by it. That is why an eight-bar phrase whose cadence arrives in bar seven and is held through bar eight reads differently from one whose cadence arrives in bar eight — and why the second is the far commoner arrangement.

The caveat in that figure’s caption is not a formality, and it is worth spending a sentence on because it is the sort of thing that quietly turns an argument into a circle. The events fed to the induction there are section beginnings, and a section begins where it does partly because of the hypermetre — so the machinery has been handed something close to the answer. Feed it the harmonic changes instead, which is the only other thing a chord scheme offers, and it returns two bars for every scheme this site encodes, because nearly every bar changes chord and an accent pattern with an accent everywhere has no structure in it. That failure is the subject of the next rung, where it turns out to say something specific about what hypermetre is made of.

It also explains an oddity that the figures above quietly contain. The arrival at bar 8 of a sixteen-bar plan is, in a strict hypermetre, a weak bar: positions 1, 5, 9 and 13 carry the weight. The music does not seem to mind, and the reason is that the durational component has taken over. A goal that is held is strong wherever it lands, which is a trade between two of the five columns and is the sort of thing a vector makes visible and a ranking cannot.

The antecedent’s route ends at the dominant, which is two steps out from the tonic on the site’s map of chord distances — far enough to be somewhere and near enough to be going home. That is the whole design of an interior cadence: it has to be an ending by every measure except the one that would stop the piece.

The rule this replaces #

There is a piece of teaching that this analysis is meant to displace, and it is worth naming it because it is nearly right.

The rule is that a phrase ending on the dominant is “open” and one ending on the tonic is “closed”, and that the pair makes an antecedent and a consequent. That is correct about this repertoire and it stops where it starts: it is a statement about two chords, it has nothing to say about the metric and durational components, and it therefore cannot explain why the same two chords are a cadence in one bar and not in the next.

The vector keeps everything the rule gets right — the tonic-goal column is the rule, and it is one of five — and adds the parts the rule cannot express. It also makes the rule falsifiable in a useful way. A phrase pair whose first half ends on the tonic and second half on the dominant would be a counter-example, and such things exist; they are rare, they sound strange, and the strangeness is exactly what the vector predicts.

Whose music, and what happens outside it #

Nested closure of this kind is a property of the tonal repertoire and of the popular styles descended from it, roughly 1700 onwards. Two observations about its edges.

The eight-bar and sixteen-bar spans are the tempo-dependent phrase lengths of a particular range of tempos, so the nesting is not four levels deep in all music. At a very slow tempo the sixteen-bar level is beyond anything a listener can hold, and composers writing at such tempos do not build it.

And a great deal of music does not nest at all. A form built on a repeating cycle has no cadence to grade, weak or strong — the whole apparatus returns zero on it — and closes by an entirely different mechanism. That is not a failure of the music. It is a different design, and it is the subject of its own rung.

The same design, one field along #

The pattern in this essay — a component deliberately withheld so that supplying it later means something — is not confined to cadences, and putting the other instances beside it makes it look less like a piece of harmonic doctrine and more like a general device.

A raised seventh exists to make one chord major, and the cost of it is an augmented second elsewhere in the scale. The minor mode does not have a leading note; it borrows one at the cadence and returns it afterwards. That is the third column of the closure vector being switched on for two bars and off again, and it is a whole scale being altered to supply one component of one ending.

A tritone is the interval that inverts to itself, and its instability is what makes the dominant seventh’s resolution feel obligatory. An instability is a withholding too: the chord is built so as not to be a place one can stay.

And the same shape runs through the site’s instrument field, where a design is forced by an arithmetic that admits no better answer and the leftover error is placed rather than eliminated. A half cadence is exactly that: the error is that a phrase has to stop halfway without stopping, and the answer is to put the shortfall in the two components that can be supplied later.

What this cannot show #

The soprano voice is missing, and for this rung more than the last. The distinction between a perfect and an imperfect authentic cadence is entirely a matter of which note the top voice arrives on, and it is one of the standard ways of grading closure within a phrase — an imperfect authentic cadence is the commonest way to end an antecedent in some repertoires, alongside the half cadence. A pitch-class encoding cannot represent it.

Nor can this see a cadence being avoided. Evaded cadences, in which everything sets up an arrival and the arrival is withheld, are among the most powerful gestures in the repertoire, and they are events in which the interesting thing is that the vector was about to be filled in and was not.

Where the ladder goes #

The levels this essay has been describing — four bars, eight, sixteen — have been treated as given. They are not: they are produced by the same metrical machinery that produces beats and bars, running one level up, and the reason it eventually stops is a number the phrase rung already measured.

And the deceptive cadence, which appears here only as a row in a figure, deserves its own treatment, because the reason it surprises is not the one usually given.