Acoustics & Music Theory
why it sounds like that
Concept

Dominant seventh — where it appears

The four-note chord on the fifth degree, containing the key's only tritone and the obligation that tritone carries. Adding the seventh does not create the tension; it makes explicit the tension the triad already had by containing the leading note.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Every interval, and what it becomes when it is turned over. Each interval within the octave paired with its inversion — the interval left when the lower note is raised by an octave. The two always add to twelve semitones, so exactly one interval can be its own inversion, and it is the one at six.

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.

harmony · The triad Five signals, computed separately, and no total. The five components of closure for 6 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted.

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.

form · Closure How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire.

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.

harmony · Tonal-expectation Both criteria, measured from C major. The 23 chords other than C major placed by the smallest total voice motion and by how many notes are held in common. Agreement between the two criteria would be a staircase; the cells drawn in the discrepancy colour are the ones that break it, and over the whole set of 24 chords the two criteria order a pair of destinations oppositely in 72 of 6072 comparisons. A minimal assignment drops a common tone it could have held in 0 of the 552 pairs.

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.

harmony · Voice-leading Every triad, by how far it is from G7. All 24 chords ranked by the smallest total motion that takes G7 onto each, every note of the target sounded and one voice doubled where the sizes differ. The spread is narrow — 2 semitones at the nearest and 6 at the furthest — and the ranking is flat in the middle: 12 chords tie at four semitones.

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.

harmony · Voice-leading No seventh chord can be spaced to last as long as a triad. Every inversion and spacing within 2 octaves over C3, struck at 80 dB, for two triads and five seventh chords: the bar is the longest any spacing keeps every pair's partial coincidence, and the tick is the bound set by the chord's worst pitch-class distance — the longest any presentation of that distance lasts. major triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; minor triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; dominant seventh: 0.66 s over 32 voicings, bound 0.64 set by the tone; major seventh: 0.42 s over 32 voicings, bound 0.38 set by the semitone; minor seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; half-diminished seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; diminished seventh: 0.86 s over 32 voicings, bound 0.87 set by the tritone. Every seventh chord contains a distance worse than any a triad contains, except the diminished seventh, whose distances are only minor thirds and tritones.

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.

timbre · Harmonic series

Named alongside it

The objects these essays reach for when they reach for this one.

Voice-leadingCadenceLeading noteClosureExpectationInversionTonal functionTritoneCommon-toneConsonanceDecayEnumeration

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