Acoustics & Music Theory
why it sounds like that
Concept

Tritone — where it appears

The interval of six semitones, the only one that divides the octave evenly and so the only one that inverts to itself. It occurs exactly once in the major scale, which is what makes the chord containing it unambiguous about its key.

Named by 4 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 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 Which rotations have a fifth above their own tonic. Each rotation drawn by scale degree, with the degree a perfect fifth above the tonic marked. 1 of the 7 does not have one: Locrian. A tonic without a fifth above it has no triad of its own, which is a stronger disqualification than any preference.

The mode with no fifth

Six of the seven rotations have a perfect fifth above their own first degree and one does not, which is a structural disqualification rather than a preference. And the set's single tritone sits on a different pair of degrees in every rotation — on the fourth and seventh in Ionian, on the tonic itself in two others — which decides most of what each mode can do.

intervals · Modes The mode is its tritone. Every set of degrees that lies inside at least one of the seven modes — 510 of them — scored by how many modes it leaves standing, with the 15 minimal sets that decide printed in full. All 15 have two members and every one of them either is a tritone or contains the tritone above the tonic. The only degree every mode has is the tonic itself; there is no degree belonging to just one mode. Averaged over every order the degrees could arrive in, 5.67 of the seven have to have been heard before the mode is settled — most of the scale, because the deciding pair arrives when it arrives.

The one note that decides the mode

A key signature names the seven and not the rotation, so the mode has to be heard. Enumerate every set of degrees that lies inside at least one of the seven modes — 510 of them — and count how many modes each leaves standing. Fifteen minimal sets decide, every one has two members, and every one is the mode's own tritone. The only degree all seven share is the tonic; no degree belongs to a single mode; and averaged over the orders the degrees could arrive in, 5.67 of the seven have to have been heard first.

scales · Modes

Named alongside it

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

Dominant seventhLeading noteModeRotationVoice-leadingCadenceEnumerationInterval contentInterval patternInversionKey signaturePerfect fifth

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