The interval that inverts to itself

Turn an interval over — move its lower note up an octave — and it becomes a different interval. A fifth becomes a fourth. A major third becomes a minor sixth. A semitone becomes a major seventh. The two always add to twelve semitones, because that is what an octave is, and the pairing is exact.

Which means there can be at most one interval that turns over into itself, and there is exactly one, because twelve is even.

What follows immediately #

The tritone divides the octave into two equal parts, so it has three properties no other interval has, and all three are consequences of the same arithmetic.

It is its own inversion. C to F sharp is six semitones; F sharp to C is six semitones. The interval does not care which note is on the bottom.

There are only six of them. Every other interval class has twelve distinct instances — twelve major thirds, one on each root. A tritone on C and a tritone on F sharp are the same pair of notes, so there are six tritones in the system rather than twelve.

Transposing one by a tritone leaves it unchanged. Move C–F♯ up six semitones and the result is F♯–C, which is the same set. No other interval has this.

The three are one fact and it is worth saying which. An interval of k semitones is fixed by transposition by t exactly when t is a multiple of the interval’s own period, and a two-note set’s period is the smaller of k and 12 − k only when those are equal. So all three properties are the statement 2k = 12, and the tritone is the unique interval for which the octave is an exact multiple of the interval — which is also why the diminished seventh, at 3k = 12, is the only other chord with the same kind of symmetry and has four members rather than two.

Every use tonal practice makes of the tritone is one of these three facts wearing a hat.

The chord it belongs to #

A dominant seventh chord — the chord a minor key has to raise a note to obtain — contains exactly one tritone: the interval between its third and its seventh. In G7 — G, B, D, F — the tritone is B to F.

Because there are six tritones and twelve dominant sevenths, each tritone belongs to two of them. B and F are the third and seventh of G7. They are also, respelled as C flat and F, the seventh and third of D flat 7.

So G7 and D♭7 share their defining interval. Their roots are a tritone apart, which is not a coincidence either — it is the third property above, restated.

Which computation produced the number #

The figure does not draw a resolution from a rule about which note goes where. It computes each voice’s destination.

For each note of the seventh chord it takes the three notes of the target triad, transposes each into the octave nearest the starting note, and picks whichever is closest. The distance moved is then measured and printed, and the total is summed.

For G7 to C major the result is: B moves up one semitone to C, F moves down one to E, G stays where it is, and D moves down two to C. Total motion, four semitones across four voices.

For D♭7 to G♭ major: F moves up one to G flat, C flat — the same pitch class as B — moves down one to B flat, and the remaining voices settle similarly. The same two pitch classes have moved by the same distances in opposite directions and arrived somewhere else entirely.

That is the whole of tritone substitution, and it is arithmetic rather than convention. The tritone does not know which chord it is in. The other two notes decide, and they can be changed.

It is worth being precise about what the two notes do, because they are not passengers. In G7 the root G and the fifth D sit a fifth apart above the tritone; in D♭7 the root D♭ and fifth A♭ sit a fifth apart, and the two pairs are a tritone apart from each other. So the substitution is not keep the tritone and change the rest arbitrarily — the rest is a perfect fifth in both cases, and there are exactly two places a perfect fifth can be put so that the result is a dominant seventh containing that tritone. Two chords, not two of many, which is what makes the partition into six pairs exhaustive.

Six tritones, twelve chords #

The pairing can be written out completely, and it is worth doing because the result is a small closed structure rather than a list of special cases.

The six tritones are C–F♯, C♯–G, D–G♯, E♭–A, E–B♭ and F–B. Every dominant seventh contains exactly one of them, as the interval between its third and its seventh. Working backwards from each tritone to the two chords that own it:

F–B belongs to G7, whose third is B and seventh is F, and to D♭7, whose third is F and seventh is C♭. C–F♯ belongs to D7 and A♭7. C♯–G belongs to E♭7 and A7. And so on: twelve chords partitioned into six pairs, with the two members of each pair a tritone apart.

The partition is complete and it has no exceptions, because it is forced. A dominant seventh is determined by its tritone plus the decision about which of the two notes is the third — and there are exactly two ways to make that decision.

On the keyboard a tritone is six semitones up from C to F♯ and six more from F♯ back to C — the same distance in both directions, which is what “half an octave” means and what no other interval manages.

Everything usually presented as a collection of chromatic devices — tritone substitution, the German sixth, enharmonic modulation through a dominant seventh — is a consequence of that one partition. There are not several tricks. There is one structural fact and several names for exploiting it.

That claim is checkable by the same nearest-note computation the resolution figure uses, and running it produces an invariant and a signature.

Four semitones every time, and all twelve dominant sevenths give four to their own tonic and four to the substitute’s. The total is an invariant of the chord type rather than a property of any resolution, which is worth knowing before any of it is read as an explanation: the substitution is not cheaper than the ordinary resolution and never was.

What separates them is the distribution, and it is exact. An ordinary dominant resolution has a common tone that does not move at all and a voice that moves two semitones. A tritone substitution moves every voice by exactly one semitone — three down and one up — with no common tone and no larger step. That is the arithmetic behind the thing players say about the device, which is that it slides rather than resolves.

And the German sixth’s signature is identical to the substitution’s, note for note: −1, −1, −1, +1. The two are not analogous devices with a shared ancestor; they are the same four semitones moving the same four ways, under two names given by two repertoires that were not talking to each other. The German sixth is a tritone substitution resolving to the dominant rather than to the tonic, and the only difference between them is which chord the four notes are aimed at.

Where it sits in the map of intervals #

There is a further way to see the tritone’s oddity, and it uses the picture this site draws chord relationships on.

On this lattice the notes reachable in one step from any given note are its fifth, its fourth, its major and minor thirds and their inversions. The tritone is reachable by no single step at all: it takes six moves east, or a combination of thirds, and it is the furthest thing on the lattice from where it started.

The tritone is therefore simultaneously the most remote interval in the fifths-and-thirds geometry and the interval that most reliably forces a resolution. Those two facts look contradictory and are the same fact: an interval that belongs nowhere on the map is an interval whose position has to be decided by something else, and in tonal practice the something else is where it goes next.

What the chord sounds like #

It is worth separating the tritone’s structural role from its sound, because they get conflated and only one of them is doing the work.

On the lattice of fifths and thirds the tritone is reachable by no single step at all — six moves east, or a combination of thirds — so it is simultaneously the most remote interval in that geometry and the one that most reliably forces a resolution. Those two facts are the same fact: an interval that belongs nowhere on the map is one whose position has to be decided by where it goes next.

An equal-tempered tritone is 600 cents. The nearest simple ratios are 7:5 at 582.5 cents and 10:7 at 617.5, which straddle it at seventeen and a half cents either side. Neither is especially close, and both involve a seven — which is the first prime the five-limit lattice has no axis for.

So the tritone is the interval that European harmony uses most pointedly and the one its own tuning theory is least equipped to describe. It has no good place in a system of threes and fives, its equal-tempered size approximates nothing well — a failure no division of the octave fixes cheaply, and the chord it defines became the engine of an entire tonal practice anyway. That is not a paradox: its usefulness comes from the arithmetic of twelve, not from the arithmetic of ratios, and those are two different arguments that this site keeps carefully apart.

Why it resolves at all #

The symmetry explains why one tritone serves two keys. It does not explain why the tritone wants to go anywhere, and that is a separate question with a partly different answer.

Two things are going on.

The first is voice leading. Both notes of the tritone are a semitone away from a note of the target triad, and a semitone is the smallest move available. A resolution in which every voice moves as little as possible is efficient in a measurable sense, and the figure measures it: four semitones of total motion for a four-note chord going to a three-note one is about as economical as a chord change gets.

The second is roughness, and it is weaker than the textbooks imply.

A bare tritone is not a rough interval. It is much smoother than a semitone and smoother than a minor seventh. The instability that theory attributes to it is not sensory dissonance; it is the fact that the interval is ambiguous — it belongs equally to two keys, and does not by itself say which — combined with the fact that both its notes have a semitone available.

That distinction matters because it explains why the tritone stops being unstable the moment the surrounding practice stops caring which key anything is in. In whole-tone and octatonic writing the tritone is a stable interval and gets used as one, and nothing about its acoustics changed — any more than the notes of a scale change when the mode does.

What was built on top #

Once one tritone belongs to two dominants, a set of standard devices follows and they were all in use long before anybody described them this way.

Tritone substitution. Replace a dominant with the dominant a tritone away and the tritone survives, so the resolution still works. Its bass moves by a semitone instead of a fifth. This is the single most characteristic reharmonisation in jazz, and it is the symmetry cashed in directly.

The augmented sixth chords. The Italian, French and German sixths are all chords containing an interval that is a tritone in sound and is spelled as an augmented sixth. The German sixth is enharmonically identical to a dominant seventh, which is precisely the ambiguity above being used as a modulation device: the chord is heard as one thing, resolved as another.

The diminished seventh. Stack the symmetry twice and the result is four notes in equal three-semitone steps, which contains two tritones and is unchanged by transposition up a minor third. There are three of them in the system, each spellable four ways, each able to resolve to four different keys — and each is as short a move from its neighbours as a chord change gets.

And the shortest-move metric is the one that does supply it, up to a point: both notes of the tritone are a semitone from a note of the target triad, and four semitones of total motion for a four-note chord going to a three-note one is about as economical as a chord change gets.

The tritone as an axis. Bartók’s writing, and a good deal of twentieth-century harmonic theory after it, treats the tritone as a structural distance rather than an interval to be resolved. Keys a tritone apart are maximally remote on the circle of fifths, and a piece that treats that pair as its two poles is using the same symmetry to a different end.

Where the model stops #

Nothing here explains why the tritone was avoided. Medieval theory’s prohibition on the melodic tritone — mi contra fa — is about melody, not harmony, and about singability in a modal system rather than about instability in a tonal one. The famous diabolus in musica tag is a later invention and the frightening reputation is largely nineteenth-century.

The equal division is a fact about twelve equal semitones. In meantone or just intonation an augmented fourth and a diminished fifth are different sizes, and the tritone is not its own inversion. The symmetry that this whole essay rests on was created by equal temperament, and it did not exist for most of the repertoire in which the tritone acquired its reputation.

Roughness models say little here. The dissonance curve treats a chord as a sum of pairwise interactions between partials, which is a reasonable first model and demonstrably incomplete for anything that depends on context. Nothing in it can distinguish a dominant seventh from the same four pitches used as a stable sonority.

Voice-leading cost is not the same as musical necessity. The four-semitone total for G7 to C is small, and there are other small ones. The computation ranks moves; it does not say which will be made, and a great deal of what is interesting in a progression is a move that is not the cheapest.

Six tritones is a claim about pitch classes. Registrally, C–F♯ and F♯–C′ are different events with different roughness at different registers. Everything above works in pitch classes and quietly discards the octave a note is in.

Whose music, and when #

The tritone’s career divides into three periods, and its status in each is different.

Modal polyphony, before about 1600. The tritone is chiefly a melodic problem. Singers reading unaccompanied lines are expected to avoid the augmented fourth between F and B, and the standard remedy is musica ficta — flattening the B, unnotated. Harmonically the interval occurs and is treated carefully rather than as a special object.

Tonal practice, roughly 1600 to 1900. The dominant seventh becomes the standard cadential chord and the tritone inside it becomes the mechanism of tonal function. Every device listed above is developed within this period, and the augmented sixth chords in particular are an eighteenth- and nineteenth-century speciality that depends entirely on the enharmonic ambiguity.

After tonality. The symmetry that made the tritone useful for pointing at a key makes it useful for refusing to. Debussy’s whole-tone writing, in which every interval is even and the tritone is ordinary, and the octatonic writing of Rimsky-Korsakov, Stravinsky and Bartók, both use it as a stable interval. Jazz then reintroduces its functional use and pushes the substitution device much harder than nineteenth-century practice did.

On the circle of fifths the tritone is the one position diametrically opposite home, which is the same remoteness one geometry along — and the reason a tritone substitution is described as a move to the far side rather than as a step.

The pattern across all three is that the interval did not change and its function did. Six semitones is six semitones in 1400 and in 1940; what varies is whether the surrounding practice is trying to establish a key, and the tritone is only unstable relative to that project.

The ladder from here #

Later rungs on this anchor: the dominant seventh’s other three notes, and what happens when they are altered. The augmented sixth chords in detail, and what the three national names actually distinguish. The diminished seventh as a modulating pivot, and how many keys a single one reaches. Octatonic and whole-tone collections, and what their symmetry does to the idea of a tonic. And the whole space of triads, in which the tritone is a distance rather than an interval.

Twelve is even, and everything above is that fact having consequences for four hundred years.