The resolution the metric cannot find

The dominant seventh is the clearest case of directed motion the tonal system has. Sound one and something is owed; the leading note is going up, the seventh is going down, and the chord that follows is the tonic unless a composer is doing something on purpose.

It is also, in principle, a voice-leading fact. Four voices have to arrive somewhere, and this site can compute how far they must travel to arrive at any triad at all. If the resolution is a matter of how little the voices move, the tonic ought to be at or very near the minimum.

Ranking all twenty-four triads settles it, and the answer is not the expected one in any respect.

What the ranking actually measures #

The nearest triads to G7 are G major and B minor. Neither is a resolution of anything.

G major is three of the four notes of G7 with the seventh removed, so one voice moves a tone and nothing else happens at all. B minor holds two notes and moves the other two by a semitone each. Both are cheap for the same reason: they barely leave the chord.

That is what a distance measures. It is a measure of how little changes, and the chords nearest a dominant seventh are the chords most like it — which is exactly what a resolution is not. An arrival is defined by what has to change, and a metric of least motion is a metric of least change, so it ranks the chords that resolve nothing at the top.

Written out in four parts under the ordinary rules, V7 to I has one realisation the search prefers and it is the textbook’s: the leading note rises, the seventh falls, the fifth is dropped. Nothing had to be added to the metric to get that — it is the cheapest legal arrangement.

Four semitones is a perfectly ordinary distance. Eleven other triads are exactly as close, and among them are C sharp major, F sharp major, E flat minor and A flat minor — chords with no function in the key at all. On motion alone the tonic is indistinguishable from a chord a tritone away from it.

A twelve-way tie is worth pausing on, because it is half the system. Twenty-four triads sort into five distinct distances from G7 — two, three, four, five and six semitones — and the middle value takes half of them. A measurement whose entire range is five values, over a set of twenty-four objects, cannot rank those objects; it can only sort them into a few heaps. The metric that separated chord pairs so usefully on the first rung of this ladder was separating triads from triads, where the range runs from one to six and the heaps are small. Ask it about a four-note chord against a three-note one and the resolution of the answer collapses.

The chords at three semitones are a better group than the ones at four, and they are still not resolutions: B major, B flat major, E minor and G minor. E minor is the closest thing in that list to a functional destination — it is the tonic’s relative minor, and it shares two notes with G7 — but no repertoire treats a dominant seventh as owing anything to the mediant.

The deceptive cadence is the furthest chord there is #

The result that a slate would not have predicted is at the other end of the list.

A minor is the joint most distant triad from G7, at six semitones, tied with A major and beaten by every other chord in the system. The deceptive cadence — the one described as a substitution, as the tonic’s near relative arriving in its place — is by this measure the least available destination the dominant seventh has.

The extra two semitones are in the bass, and that is the whole of the difference. The essay on the deceptive cadence reached the same place from a different direction — measuring the arrival chord against the tonal hierarchy rather than measuring the motion — and found the surprise to be local to one voice and that voice to be the bass. Two independent measurements, one conclusion, and neither of them a claim about how far the voices go.

So the deceptive cadence is not cheap. It is expensive, and it works anyway, which means whatever makes a resolution a resolution is not being measured here.

Trying the other criterion of the previous rung does not rescue it either. Counting common tones instead of semitones, A minor shares one note with G7 and so do eleven other triads, while G major shares three. Both criteria put the deceptive destination near the bottom and neither puts the tonic near the top, which is the useful part of the failure: two different ways of measuring nearness agree, and the thing they agree about has nothing to do with which chord a listener is waiting for.

On the criteria plane the same two destinations sit apart on the common-tone axis and together on the motion axis, which is the disagreement the ladder’s other rung is about — and here the motion axis is the one that matches practice.

That second figure is where the deceptive cadence’s real economy shows up. The chord is far from where the music is and near to where the music was going, and a listener who has already predicted the tonic is not measuring the move from the dominant seventh at all. The cheapness is in the relation to the expectation, not in the relation to the chord actually sounding.

What does select the two chords #

Distance fails, so the constraint has to be stated the way the tradition states it: as a direction rather than a magnitude.

The dominant seventh contains a tritone — in G7, the B and the F — and that interval is the only one that inverts to itself, which is why a single tritone belongs to two different dominant sevenths. The rule about it is not that its notes move a little. It is that they move by one semitone each, in opposite directions: outwards, so that B goes up to C and F goes down to E, or inwards, so that they close to B flat and F sharp. “Outwards” is a statement about pitch classes and about spelling — an augmented fourth expands, and its inversion the diminished fifth, which is the same two notes with the lower one on top, contracts. Which of the two a listener hears depends on the register the chord is voiced in, and the figures below draw the registers a four-part texture actually uses.

Take that as a constraint and ask which triads can receive it. The outward resolution lands on the notes C and E, so the target must contain both, and exactly two of the twenty-four triads do: C major and A minor.

The two is not a coincidence of this chord and it is worth seeing why, because the reason turns the observation into a theorem. Both resolutions of a tritone land on a major third — B up to C and F down to E gives C and E; B down to B♭ and F up to F♯ gives F♯ and B♭, which is the same interval spelled the other way. And a pair of notes a major third apart lies in exactly two triads: the major triad rooted on the lower and the minor triad rooted a minor third below it. So any tritone in any chord admits exactly two destinations per direction, always, and the inward resolution of G7’s tritone selects F♯ major and D♯ minor with the same rigidity.

Running the constraint over the other four-note chords says why the dominant seventh is the one that has this property:

A chord with no tritone has no direction and a chord with two has too many, which is the whole family sorted by one count. The major and minor sevenths are not weak dominants; the constraint that makes a dominant a dominant does not apply to them at all. The diminished seventh’s famous ambiguity is not vagueness — it is eight sharply specified destinations rather than two, which is a different thing and is why it is a modulating chord rather than a cadential one.

Set that beside the metric this rung began with. The distance sorts twenty-four triads into five heaps and puts half of them in the middle one; the tritone constraint admits two, for every chord that has a tritone at all, and the two are the authentic and deceptive destinations. One quantity has a resolution of five values over twenty-four objects and the other has a resolution of two, which is the difference between a measurement and a rule.

Not two out of a handful of plausible candidates — two out of the complete set. The authentic cadence and the deceptive cadence are the entire list of triads the outward resolution of that tritone permits, and the enumeration produces them without being told anything about keys, functions, expectations or cadences.

And the tritone’s own two notes moving outward by a semitone each is the mechanism underneath: both resolutions discharge it, and what differs is the bass.

There is a second tendency tone in the chord and it is doing the same work. The seventh — the F in G7 — is the note that makes the chord a seventh rather than a triad, and it is one half of the tritone; the leading note is the other. So the two obligations the tradition states separately, the leading note rises and the seventh falls, are one obligation stated twice: the tritone expands. Nothing else in the chord is under any constraint at all, which is why the two survivors differ in a note that was never obliged to go anywhere.

The inward resolution is the other half of the same enumeration and gives E flat minor and F sharp major — the pair a tritone away, which is the substitution jazz harmony is built on. Four triads in all, from twenty-four, selected by a rule that never mentions distance.

On a keyboard the difference between the two resolutions is one key: C and E are common to both destinations, and what changes is whether the bass takes G to C or G to A.

What separates the two survivors #

C major and A minor share C and E. The difference between them is one note and, more to the point, one root and one bass.

The pitch-class metric can see neither. It reported that the two chords are two semitones apart and that they share two of three notes, which is true and is the same measurement that made vi one of the three cheap chords in the key. It has no way to say that one of them is the tonic and the other is not.

Root and bass are function, and function is asymmetric in a way none of the measurements on this ladder are. The distance from V to I equals the distance from I to V; the cadence runs one way. Closure turned out to be a five-component object with no total, and three of those components — does the root move by a fifth, is the goal the tonic, does the leading note resolve — are exactly the ones separating the two chords that survive the tritone constraint. The deceptive cadence satisfies the third and fails the first two.

So the tendency-tone account is not underdetermined by motion, as it first appeared to be. It is underdetermined by distance and determined by direction, and the residue that direction leaves — a choice between two triads — is settled by the bass, which is the one thing a pitch-class model was always going to be silent about.

The same ranking from a plain major triad has no such structure — its nearest neighbours are A minor and E minor, which are the chords sharing two notes with it, and neither is an ending. The dominant seventh’s ranking is special and a triad’s is not.

Which computation produced the numbers #

Four voices arriving on a triad is a map onto three notes, so one of them is doubled. Every such map is tried — all the ways of assigning four voices to three destinations with none of the destinations left empty — and the smallest total is taken, with each voice going the shorter way round the circle of twelve.

Requiring every note of the target to sound is the modelling decision that matters, and it is the four-part one: a realisation that dropped the third of the tonic to save a semitone would not be the chord. The doubling is not chosen in advance either; which note gets two voices is part of what is minimised.

The result survives a second, independent construction. Realising V7–I and V7–vi in a register, under all seven prohibitions of the previous rung but one, gives four semitones and six — the same two numbers, from a solver that knows about compasses, doublings and parallel fifths and nothing about pitch-class distance. The two pictures in this essay showing voices on a stave-like grid are that solver’s output, not a transcription.

Whose cadence, and when #

The dominant seventh as an object with an obligation is a specific historical formation, and its jurisdiction is narrower than its reputation.

The chord exists in the sixteenth century as a passing dissonance and becomes a functional object over the seventeenth, with Rameau’s Traité of 1722 giving it the theoretical status it still has. Its treatment as a chord that must resolve, with the seventh prepared and falling and the leading note rising, belongs to the eighteenth and nineteenth centuries and to the repertoire the counterpoint prohibitions were codified for.

Three well-populated repertoires ignore it entirely. In the blues, a dominant seventh sits on the tonic and resolves to nothing, which makes the chord’s obligation optional rather than definitional. In jazz after about 1940 the tritone substitution treats the inward resolution as freely available, so a dominant seventh may go to either of two keys and the listener follows. And in modal writing — much film music, a great deal of popular music since the 1960s — the chord is used for colour with no cadential force at all.

The enumeration in this essay is silent about all of that, because it is an enumeration of what the arithmetic permits rather than of what a style does. What it does establish is that the permission is narrow: four triads out of twenty-four, and the two that tonal practice uses are the two the outward direction allows.

Where the model stops #

No register, no bass, no doubling that a chorale would recognise. The ranking works in pitch classes, and the thing that separates the two surviving triads is precisely the register information the ranking discards.

The tritone is not exactly a tritone. On a keyboard the interval is 600 cents; in just intonation the seventh of a dominant seventh has several defensible tunings, and the resolution’s two semitones are of unequal size in every historical temperament. None of that is in a semitone count, and it is the subject of most of this site’s tuning field.

Nothing here is about expectation. The claim is about which chords the tritone’s motion can reach, not about what a listener predicts. The prediction is a matter for the measured tonal hierarchy, which is a different kind of evidence with a different kind of limit.

Ties are not ranks. Twelve triads sit at four semitones, and the figure lists them in an order that is alphabetical rather than musical. Any account that reads significance into the position of the tonic within that block is reading the sorting.

In the minor the same realisation appears with the raised seventh, and the cheapest legal arrangement is again the textbook’s — which is the same result one mode along.

And ranked against the measured tonal hierarchy the submediant sits second of seven, at 90 per cent of the tonic’s own fit — which is why a listener accepts it as an arrival at all, and is a different measurement agreeing with this one.

What the ranking establishes #

A resolution is not a short move. Ranking by motion puts the chords that change least at the top, and a cadence is defined by what changes. The tonic’s position in the middle of the ranking is not a defect of the metric; it is a demonstration of what the metric is for.

The deceptive cadence is dear, not cheap. Six semitones against the tonic’s four, and the extra two are in one voice. Every account calling it a near substitution is measuring from the wrong chord.

Direction selects where distance cannot. Two semitone moves in opposite directions cut twenty-four triads down to two, and the two are the pair tonal practice actually uses. That is the strongest result available here and it required no notion of a key.

Where the ladder goes #

Every rung of this ladder so far has assigned voices — to chords, to registers, to destinations — and has assumed that a voice is a thing a listener follows. The last rung asks whether that is true, and the answer has thresholds attached: a part is only heard as a part if it is far enough from its neighbours and does not leap too far at speed. Whether the ear takes the assignment at all turns out to be decided by numbers measured in a laboratory in 1975, and the smoothest part-writing is not the safest by that standard.