Dominant seventh — where it appears
Named by 20 essays across 5 fields — each of them below, with the objects they name alongside it.
The interval that inverts to itself
Six is the only number that divides twelve into two equal halves, so the tritone is the only interval unchanged by being turned over. That symmetry is not a curiosity — it is why one tritone belongs to two dominant chords, and why they resolve to keys a tritone apart.
What makes an ending an ending
Cadences are ranked. The authentic one is strong, the plagal weaker, the deceptive weaker still — and the solver here measured the quantity that ranking is usually explained by and found it says something else entirely. What survives is not a weaker version of the ranking but a different kind of object, with five components and no total.
The chord that did not come
A deceptive cadence is described as a surprise, and the explanation offered is that the wrong chord arrived. Measured against the tonal hierarchy already in use, the wrong chord is the second best-fitting triad in the key — and two of its three voices do exactly what they would have done in the right one. The surprise is not statistical. It is one voice, and it is the bass.
Two criteria that are taught as one
Move the least and keep the common tones arrive in one breath, as though they were one instruction. They are two different functions, and enumerating every pair of chords settles exactly where they agree: as a way of connecting two given chords they never once disagree, and as a way of saying which chord is nearer they disagree about one comparison in eighty.
The resolution the metric cannot find
Rank every triad by how far its voices are from the dominant seventh and the tonic is not at the minimum. It sits in a twelve-way tie in the middle, the chord a deceptive cadence goes to is the furthest away of all twenty-four, and the two nearest are chords no cadence ever uses. What selects the two resolutions is not a distance — it is a direction, and it picks out exactly two triads from the twenty-four.
A seventh chord cannot be spaced to last like a triad
A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.
The seventh has two places to be
Tune a dominant seventh's seventh as the seventh partial and its coincidences last 0.87 seconds instead of 0.66 — exactly the diminished seventh's figure, which is the highest any four-note chord in this census reaches. It gets there by levelling rather than by improving: the root's pair with the seventh gains a third of a second and the fifth's loses one, and all three pairs that contain the seventh arrive at the same number. The route is a valley. A seventh tuned six cents flat has lost the ninth-partial coincidence and not yet found the seventh, and lasts 0.46 seconds, which is worse than leaving it alone. And the same retuning halves the minor and half-diminished sevenths.
The bridge is not a labelling problem
Seven of the key-finder's thirteen remaining misses are the thirty-two-bar song's bridge, and the essay before this one proposed that they might be an argument about labels rather than a failure: relabel the eight bars with the keys their dominants point at, and the reading's E and D majors might turn out to be right. They do not. The reading gets one bar of eight under either label, because four of the eight are read as the chord's own root taken for a tonic — a dominant seventh heard as a chord in its own right. And one of the bars the relabelling could have bought is unwinnable in principle: an applied dominant is the same four notes in the major and the minor of the key it points at.
Looking ahead rescues the relative minor, not the bridge
A key-finding reading that scores each bar by its own notes cannot hear an applied dominant, whose meaning is in the bar after it. Given one cue about a pair of bars — a dominant earns credit when the next bar is rooted on its key's tonic — the thirty-two-bar song's bridge is read as the chain of dominants it is, each chord both the arrival of the last and the departure to the next. But the same cue hears a blues's opening tonic seventh as the dominant of its own IV, and at the setting that read the listing best it costs five bars for three. The cue earns its place only in a reading reluctant to change key, and there what it rescues is the rondo's episode in the relative minor.
Four notes cannot get below the seventh partial
Tune a dominant seventh as 4 : 5 : 6 : 7 — every note moved, not only the seventh — and its coincidences last 0.868 seconds, which is the one-note retuning's figure to the millisecond. Tuning the third and fifth buys nothing at the top and costs seven voicings a fifth of their life. Every other chord of four notes a harmonic series holds does worse, and the reason is not temperament: no four pitch classes avoid a semitone, a tone and a tritone together, and none of those three can be tuned to coincide below the seventh partial. The half-diminished seventh, which the one-note retuning ruined, reaches the same ceiling read down the series.
A tuned seventh chord does not beat; it buzzes
Three four-note chords share a ceiling of 0.87 seconds on how long their partial coincidences last, and nothing in that measure could tell them apart. Their beating can. A chord tuned exactly to one harmonic series is one periodic sound, so no two of its partials can differ by less than the series' fundamental: the tuned dominant seventh has no slow beating at all, where the tempered diminished seventh has 125 beating pairs at the strike. Its roughness is another matter. Tuning removes the beats and changes the roughness by a few per cent, in either direction, and a seventh on its way to 7 : 4 beats more, not less, until it arrives.
A tempered seventh chord already has the tuned one's low note
A dominant seventh tuned 4 : 5 : 6 : 7 is one periodic sound, and its period is a note below the chord that nobody plays. The question left open was whether tuning adds that note. Run through the temporal account of residue pitch, the tempered chord turns out to repeat at the same place, a few cents sharp, at 0.82 of the tuned chord's exactness at the strike and 0.96 a second later. Tuning does not create the low note. It makes it exact, and it gets there by a route on which every step helps, which is not true of the beating.
A tuned close seventh chord low on the keyboard flutters at one rate
A dominant seventh tuned 4 : 5 : 6 : 7 in close position repeats exactly at a frequency two octaves below its bass, and below B2 that frequency is too slow to be a pitch. The question left was whether the repeat is heard as a flutter. In every auditory filter the chord reaches, its envelope pulses at that one rate, forty times over the threshold for detecting a fluctuation in the filter where it is strongest. The tempered chord fluctuates too, but a third as deeply and at rates that change from filter to filter. Tuning adds something here rather than sharpening it, and it adds it only within two cents of 7 : 4.
A tuned chord's flutter binds it only if its notes start together
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates at one rate in every auditory filter, and one rate everywhere looked like the ear's evidence that a sound is one thing. It is only half of that evidence. The filters also have to rise and fall at the same moments, and whether they do depends on when each note starts — which tuning does not decide. Spread the notes' onsets by two milliseconds and the tuned chord's filters move together no more than a tempered chord's.
A tuned chord stays in step for half a cent-second
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates in every auditory filter at once, but only if its notes start within a millisecond of each other — and a tuned chord, it was said, keeps its phases for as long as it sounds. It keeps them only as long as it stays exactly tuned. Mistune its notes by a fraction of a cent and they slide against each other, and the chord drifts out of step in a time inversely proportional to the mistuning: about half a cent-second. A tenth of a cent holds it for four seconds, a cent for half a second. A chord that starts out of step never drifts into it.
A shared vibrato cannot hold a tuned chord together
A tuned close seventh chord binds its auditory filters together only while its notes stay within a third of a millisecond of each other, and mistuned by a fraction of a cent it drifts out of step in half a cent-second. A vibrato shared by every voice looked like a way to hold it: if all four notes swing together, they slide together. They do not stop sliding. A perfectly locked vibrato leaves the drift exactly as fast as before, because it changes every note's frequency by the same factor and leaves their differences alone; a vibrato locked ninety-eight per cent takes an exactly tuned chord out of step in six seconds, and one locked ninety per cent in one.
A chord comes back into step too briefly to bind
A vibrato that is shared imperfectly does not carry a tuned chord's notes ever further apart: each note's time is warped back and forth, and the chord passes back through step twice in every swing. Locked ninety-seven or ninety-eight per cent, it does so about ten times a second. But each return lasts on average less than one cycle of the chord's own flutter, which is the rhythm the ear's filters would have to share to show the binding — fewer than one return a second lasts that long. The chord is in step often and never for long enough.
Two octaves up, a sung chord can bind at every swing
A vibrato's swing between two notes is the same number of milliseconds in every register; the tolerance for being in step halves each octave, and so does the period of the chord's own flutter. The first two make a scaling law — an octave higher is the same chord with its vibratos disagreeing half as much — and the third breaks the tie. Matched for time in step, the higher chord's returns carry twice as many flutter cycles: over C2 fewer than one return a second is long enough to bind, over C4 with the vibratos locked ninety-nine and a half per cent nearly every return is.
A narrow vibrato needs no lock
A tuned close seventh binds and releases at the vibrato's swings only where its notes' timing swings by about the width of its onset tolerance, and the swing is the vibrato's width times how far the singers' vibratos disagree. Down to a ninety-five per cent lock the two trade one for one: a 71-cent vibrato locked ninety-nine and a half per cent is the same chord as a 7-cent one locked ninety-five. Below that the trade fails, in the singers' favour. With no lock at all the swing stops growing with the disagreement, and a vibrato of three cents over middle C — twelve over the bass — binds the chord at most of its swings with each singer's vibrato its own.
A straight tone drifts out of step and stays there
A tuned close seventh binds — its filters flutter together — only while its notes keep their timing within a fraction of a millisecond, and a narrow vibrato keeps them there because a sinusoid's timing offset swings inside a fixed range. A straight-tone singer's pitch is never quite straight: it wanders slowly, a cent or less, at random. That wander's timing offset is a random walk. Over C4 a wander of half a cent takes the chord out of step within a fifth of a second and leaves it in step four per cent of the time, where a three-cent vibrato holds it nearly half. What matters is the wander's slow part, and adding a vibrato on top never helps.
Named alongside it
The objects these essays reach for when they reach for this one.
Harmonic seriesJust intonationPeriodicityAuditory filterSeptimalVoice-leadingBeatingConsonanceTonal functionVibratoCadenceLeading note