What makes an ending an ending

Every method in the repetition ladder is a method for finding change. A matrix finds where a passage has occurred before; a novelty operator finds where the neighbourhood alters; a coder finds where something new arrives. None of them can find an ending, and the reason is that an ending is frequently the least eventful bar in a piece.

The ending has its own apparatus, and it is one of the oldest parts of music theory. Cadences are named, and they are ranked: the authentic cadence is the strong one, the plagal is the “amen” ending and weaker, the deceptive is a substitution that fails to close, the half cadence stops rather than finishes.

That ranking is real, in the sense that listeners agree about it. The trouble is the explanation usually given for it.

The account that was measured, and failed #

The standard explanation is about motion. The dominant resolves to the tonic by the smallest and most efficient move available; the deceptive cadence goes almost the same way and lands somewhere else; the plagal is a weaker gesture because it lacks the leading note’s pull.

The first of those is a quantitative claim, and this site has a solver for exactly that quantity: the smallest total semitone motion that takes one chord to another, minimised over every way of assigning the voices.

Worse follows. The tonic chord to its own relative minor costs 2 semitones, which is less than any cadence on the list; the dominant to the mediant costs 2 as well. Neither ends anything. The smallest moves available in the key are not cadences, and the measurement was made during this site’s standard pass of 2026-08-08, where it withdrew two sentences from an earlier essay.

So the ranking cannot be a ranking by distance. It is not that distance is a poor proxy; it is that the ordering it produces is a different ordering.

What is left when the scalar goes #

The temptation at that point is to look for a better scalar — some other single number that comes out in the right order. That is the wrong move, and the reason is visible as soon as the candidates are laid out separately.

A cadence supplies several things at once, and they are independent. The hero figure computes five of them for six chord pairs and pointedly does not add them up.

The root moves by a fifth. The bass falling a fifth, or rising a fourth, is the single most characteristic harmonic gesture in tonal music, and it is what the dominant does to the tonic. This is computed from the chord roots.

The goal is the tonic. The chord arrived at is the one the key is named after. Computed from the roots and the key.

The leading note resolves. The seventh degree is present in the departing chord, is absent from the arriving one, and the arriving one contains the tonic a semitone above it. Computed from the two pitch-class sets.

It lands on a strong beat. The metric weight of the position the goal falls on, normalised. Not a property of the chords at all.

The goal is longer than what came before. Cadential lengthening, again not a property of the chords.

Read the hero figure across the rows and the pattern is clear. The authentic cadence is the only pair that supplies all three of the harmonic signals. Every other named cadence supplies some proper subset — and each of those subsets is a different subset.

The subsets are the names #

This is where the vector earns its keep, because the traditional names turn out to be names for particular subsets rather than for points on a scale.

The plagal cadence has the tonic goal and neither the fifth-fall nor the leading note. That is precisely why it sounds settled but not concluded: it arrives home without having been sent there.

The deceptive cadence has the leading note’s resolution and neither the fifth-fall nor the tonic goal. It is the mirror image of the plagal — everything that pushes, nothing that arrives.

The half cadence has the fifth-fall and nothing else. It goes somewhere with the right gesture and the destination is not home.

And the control, a tonic chord moving to its relative minor, has none of the three. It is one of the closest moves in the key by every distance measure the site owns, and it ends nothing, which is the case the scalar account cannot explain and this one does not have to.

A dominant seventh contains a tritone, and in the resolution its two notes move outward by a semitone each into the tonic’s third and root: two voices, contrary motion, from the most unstable interval in the scale to the most stable. It is the component easiest to mistake for the whole, because it is a genuine mechanism and an audible one. It is also one signal among five, and a cadence that has it and nothing else is the one called deceptive.

Why there is no total #

A reader who wants a number will now be looking for weights. Give the leading note two points and the tonic goal three, add them up, and there is a strength.

The reason not to is that the ordering such a sum produces is decided entirely by the weights, and nothing in the music decides the weights.

Weight the tonic goal most heavily and the plagal cadence outranks the deceptive, which matches the tradition. Weight the leading note most heavily and the deceptive outranks the plagal, which matches the fact that a deceptive cadence at the end of a phrase is a far more dramatic event than a plagal one and is used at moments of much higher tension. Both weightings describe something true, and the two orderings disagree.

A quantity whose ordering is an artefact of a free parameter is not a measurement, and this site has said the same thing before in a different field: a summed roughness does not select the triad either, and what selected it there was requiring a chord to be in the best fraction on two axes at once rather than good on a weighted blend of them.

The same discipline applies here. The authentic cadence is not the highest-scoring cadence. It is the only one that scores on every harmonic axis, which is a stronger statement and does not need a weight.

How much freedom the weights actually have #

That is worth pressing, because “the ordering is an artefact of the weights” is itself a claim that can be checked, and checking it makes the refusal to total look better rather than worse. Draw two hundred thousand random non-negative weightings of the five components, rank the six pairs under each, and count the distinct orderings.

Six of the seven hundred and twenty possible.

That is the whole of the freedom a weighting has. The six orderings are the six arrangements of the plagal, the deceptive and the half cadence among themselves, each turning up about a sixth of the time; everything else in the ranking is fixed. In particular:

• The authentic cadence is first in a hundred per cent of the weightings. Not usually, not on the standard ones: in all of them. It scores on every component the others score on and on components they do not, so no non-negative weight vector can put anything above it. Its vector dominates.
• The control is last in a hundred per cent. A tonic moving to its relative minor scores on none of the three harmonic components, so it is dominated by everything.
• Plagal above deceptive in 50.0 per cent and below in 50.0. Exactly the coin toss the paragraph above describes, and the two are never tied, because the two components that separate them are exactly one apiece.

So the honest statement is sharper than “there is no total”. There is a partial order, it has a top and a bottom, and the weights decide only the middle — the three cadences that each supply one signal and no other, which is exactly the set the tradition disagrees about and describes with words rather than with a rank.

That also explains why the naming survives without a scale. The two ends of the ordering need no weighting to be established, so a tradition can agree that the authentic cadence is the strongest and that a I–vi is not a cadence at all, while having no settled view on whether a plagal ending is more final than a deceptive one. It has no settled view because there is nothing to have a view about: the answer is whatever the listener is weighting on the day, and the two are exactly balanced across all possible weightings.

Where the cadences sit on the map the site already had #

The site’s harmony field placed the seven triads of a key by how far their voices must travel from the tonic, and that map is worth returning to now that the distance account has failed, because the map is still right about something.

On that map the submediant sits closer to the tonic than the dominant does, and no amount of proximity makes it an ending. The map is a true picture of a real quantity and the quantity is not closure — what a route shows is a route, and a cadence is what happens at the end of one.

The distinction is worth having in one sentence. Distance describes the cost of getting somewhere; closure describes whether arriving there settles anything. The first is symmetric — the cost from the dominant to the tonic is the cost from the tonic to the dominant — and the second is emphatically not. A measure built out of a symmetric quantity was never going to produce an asymmetric result, and that, rather than any inaccuracy, is why it could not have worked.

Three of the five components in the hero figure are asymmetric by construction. The root moves by a fifth in one direction; the goal is the tonic; the leading note resolves upward. Reverse any of the pairs and all three go to zero. That is the property a description of an ending has to have.

Two of the five are not about the chords #

The last two components deserve separating out, because they are what make the vector a description of a musical event rather than of a pair of chords.

That figure is the argument that closure is a property of an event and not of a harmony. A dominant seventh resolving to a tonic on the second beat of a bar, held for a quaver, in the middle of a run, is not an ending. Every textbook knows this and few of them say it, because the chord pair has a name and the name travels with it.

The duration component works the same way. A cadence chord is normally longer than the chords before it — often much longer, and often with the whole texture thinning at the same time. Take that away and the closure goes with it.

Where the endings actually fall #

A vector describes a candidate. Whether a piece uses it is a separate question, and the chord schemes this field is built on answer it directly.

That is the duration component doing its work in a real plan rather than in an abstraction. Three of those four arrivals are followed immediately by the next thing; the fourth is followed by nothing. The chords at bars 8 and 30 are the same chords. What separates them is that one of them is allowed to last, which is the fifth column of the hero figure and is why it is there.

It also shows how thoroughly the components can be dissociated in ordinary music. The bar-30 cadence has all five. The bar-8 cadence has three of five and is heard as a link rather than a stop, and the listener’s reading of it is not a mistake being corrected later — it is available immediately, from the length of the chord.

Whose music, and when #

Everything above is about tonal harmony in the European tradition, roughly 1650 to 1900 and the popular styles descended from it. The five components are the components of that system’s endings.

The point is worth pressing rather than noting, because the components are not universal and the ways they fail are informative.

A modal repertoire has no leading note on most degrees — that is close to the definition of modal — so the third component is unavailable, and modal cadences close by other means: by the approach to the final, by duration, by the octave and fifth in the sonority. Renaissance polyphony manufactures a leading note where the mode does not supply one, which is the practice called musica ficta, and the fact that singers were prepared to alter the mode to get one says how much of the work that single component does.

A raga’s closure is on the sa and is a matter of arrival and of dwelling; a gamelan’s is signalled by the gong, which is an instrument rather than a harmony. Neither has anything corresponding to the first three columns of the hero figure, and both close unmistakably.

So the vector is not a theory of endings. It is a description of one repertoire’s endings, and the last two components — arrive on a strong beat, and stay there — are the only two of the five that all of these traditions share.

The listener’s side of it #

None of the five components is a claim about how a cadence feels, and it is worth saying what the missing half looks like, because there is a body of measurement on it that this site has already used.

An ending is an expectation satisfied, and expectation is not a property of a chord pair either — it is a property of a listener who has been hearing this key for a while. Counting produced the tonal hierarchy: the reason the tonic is a resting place is that it has been the most frequent and most emphasised note in everything the listener has heard, in this piece and in every piece before it. Take a listener with no exposure to the idiom and the arriving chord is a chord.

That puts closure in the same class as a metre that is inferred rather than received and a category boundary the sound does not contain. The five columns describe what the music supplies. What turns a supply into an ending is a listener who has learned what to do with it, and the learning is measurable and has been measured.

The reason the vector is still worth computing is that it is the public half. Two people can disagree about whether a passage sounds final; they cannot disagree about whether the root moved by a fifth.

What this cannot show #

The pitch-class representation is doing real work and it has one hard limit that has to be stated, because it affects the first component.

In pitch-class space, a falling fifth and a rising fourth are the same move. The computation cannot tell them apart, and the consequence is visible in the hero figure: the tonic chord moving to the subdominant scores on the fifth-move component exactly as the dominant moving to the tonic does. It takes the second component beside it to separate them. Register is what a real bass line has and a pitch-class set has not.

The vector also cannot see the soprano. Whether a cadence’s top voice arrives on the tonic or on the third is the difference between a perfect and an imperfect authentic cadence, which is one of the sharpest distinctions in the whole apparatus, and it is a fact about voicing rather than about chords.

And it says nothing about what a cadence closes. A cadence at the end of an eight-bar phrase and the same cadence at the end of a movement are the same five numbers and enormously different events. That difference is a matter of nesting, and it is the next rung.

Where the ladder goes #

The weak cadence exists so that a strong one can mean something, and the shortfall in a half cadence’s vector turns out to be a deliberate one on exactly the components its partner supplies.

The deceptive cadence has a rung of its own too, in the harmony field, because the standard explanation of it — that the arriving chord is a surprise — does not survive being measured against the site’s own tonal hierarchy.