A progression is a path, and the map can be drawn

A major key contains seven triads. They are usually presented as a numbered list — I, ii, iii, IV, V, vi, vii° — which correctly says what they are and says nothing about how they relate.

Place them instead by how far their voices have to travel from the tonic chord, and a map appears.

The radii are computed, not chosen, and printing them is worth the line: iii at 1, vi at 2, IV and V at 3, ii and vii° at 5. The first two share two notes with the tonic, IV and V share one, and ii and vii° share none — so the two furthest chords are the two that have nothing in common with home, and they are tied rather than vii° being furthest on its own.

A progression is a walk on that map, and its character is largely a matter of the route.

The three functions #

Distance from home is not the only structure in the space. The chords also sort into three groups by what they do, and that grouping is not derivable from distance.

Tonic function — I, vi, iii. Chords containing the tonic note, which sound like home or like substitutes for it.

Subdominant function — IV, ii. Chords containing the fourth degree, which sound like departure.

Dominant function — V, vii°. Chords containing the leading note and the tritone, which sound like they must resolve.

The standard tonal progression is T–S–D–T: home, away, tension, home. Nearly every common sequence is an instance of it. I–IV–V–I is the bare form. I–vi–IV–V is the same with a tonic substitute inserted. ii–V–I is the last three-quarters of it and is the fundamental unit of jazz harmony.

That last point is worth dwelling on. Function is not a property of a chord. A C major triad is a tonic in C, a dominant in F, and a subdominant in G, and it is the same three notes in all three cases. Function is a relationship between the chord and a key, and it exists only because a key has been established.

What makes the dominant work #

The V–I cadence is the strongest gesture in the system, and its strength is entirely mechanical.

The dominant seventh chord in C contains B and F — a tritone, and the only tritone in the scale. When it resolves to the tonic, the B rises a semitone to C and the F falls a semitone to E. Two voices, moving a semitone each, in contrary motion, from the most unstable interval available to the most stable.

The measurement is worth pausing on, because the usual account runs the other way — the cadence is said to be final because the motion is small. It is not the smallest motion available, and the account survives only in a narrower form: every part of the resolution is doing work. The tritone is unstable, so it needs to go somewhere. It has exactly one satisfying resolution, so where it goes is not ambiguous. The motion is by semitone, which is the smallest step available. And the resolution lands on the tonic triad, which is the most consonant chord in the key.

The result is a gesture with only one plausible continuation, and that is what finality means: not that the music has run out, but that the alternative continuations have.

The circle-of-fifths progression #

One route across the map is used more than any other, and it has a simple description.

Move the root down a fifth, repeatedly: vi–ii–V–I, or the longer iii–vi–ii–V–I. Each step feels like a dominant resolving — and the two other things usually said about it are both false, which is worth establishing before the explanation is offered.

A descending fifth does not share two notes. Every one of the seven diatonic fifth-down moves shares exactly one, and none shares two. The moves that share two are the third-related ones — I to iii, I to vi — which are precisely the moves the progression does not use.

And it is not minimal voice leading. Ranking each chord’s six available moves by total motion and asking where its fifth-down move sits:

Mean rank 3.43 of six, and never the cheapest from any chord. The descending fifth is a mid-table move by voice leading and a poor one by common tones, in every position in the key.

So the explanation that closes this section — that two independent properties coincide at the fifth — has neither property to work with. What the progression has is the third thing on the list: every step is a dominant resolving, and that is the whole of it. The move is chosen for its function and pays for the function in motion, which is the same verdict the modal cadences reach on a different ladder and the dominant seventh’s resolution on a third. Three ladders, three measurements of nearness, and in every case the move the tradition uses is not the near one.

Pachelbel’s canon, the jazz turnaround, autumn leaves, half the Baroque repertoire — all the same walk. It works for one reason rather than two, and the reason is the one that survived the arithmetic above: every step is heard as a dominant resolving to its own tonic, so a chain of them is a chain of resolutions, each of which is complete and each of which turns out to be the dominant of the next.

The path that does not come home #

There is a consequence of walking these paths in pure ratios that the map does not show, and it is one of the more startling facts in the subject.

Take I–IV–ii–V–I and tune every move to the simplest available ratio. Each individual move is beautiful. The tonic at the end sits 21.5 cents below the tonic at the beginning — the syntonic comma, arriving because the ii chord can be tuned pure against IV or pure against V but not both.

This is the comma pump, and it is not an abstraction. Choirs singing this progression in just intonation drift flat, measurably, and repeat the drift on every cycle. What they do about it is compromise, unconsciously, in a way that is functionally a temperament they never chose.

The map above, drawn in pitch classes, cannot show this at all. It assumes that returning to the tonic chord is returning to the same place, and in exact ratios it is not.

What the map does not have #

The picture places chords by distance and it is missing the one thing that makes a progression a progression.

It has no direction. The distance from V to I equals the distance from I to V, and the two are completely different musical events. One is an arrival and the other is a departure, and asymmetry is the essence of tonal function.

It has no time. A progression is an ordered sequence and the map is a static space. The same set of chords in a different order is a different progression, and often not a good one: V–IV–I is the reverse of the standard cadence and sounds, to a listener trained on the tradition, wrong.

It has no rhythm. Which chord falls on the downbeat, how long each is held, and where the phrase ends do at least as much work as the choice of chords. A cadence on a weak beat is not a cadence.

It has no bass. Inversions are invisible in pitch-class space, and the bass line is very nearly a second melody with its own logic.

Cadences, graded #

Not every arrival is equally final, and the gradations are the punctuation of the system.

Perfect authentic — V to I, both in root position, melody landing on the tonic. Maximum closure, and the only ending that sounds like a full stop.

Imperfect authentic — the same chords with an inversion or with the melody landing elsewhere. Closure with a qualification, like a semicolon.

Half — ending on V rather than moving to it. An open question, and the standard way to end a phrase that expects an answer.

Deceptive — V moving to vi instead of I. The dominant resolves, and it resolves somewhere unexpected: vi shares two of its three notes with I, so the destination is nearly the expected chord and is not it. The received description goes one step further and says the voice leading is almost identical too, and the figure below is where that stops being true — V–vi costs five semitones against V–I’s three. What survives is the leading note: B still rises to C in both, and it is that one voice, not the total, that makes the gesture read as a resolution to the wrong place.

Plagal — IV to I, without a dominant. Weaker, because there is no tritone to resolve, and it works as a coda after closure has already happened rather than as closure itself.

Prolongation, and the claim that a piece is one chord #

The strongest structural claim in tonal theory is worth stating even though it is contested, because it is what large-scale harmonic analysis rests on.

Schenker’s argument is that a tonal piece is an elaboration of a single tonic harmony. The surface chords are not a sequence of equal events but a hierarchy: some are structural and others decorate them, and stripping away the decoration repeatedly leaves a skeleton that is, ultimately, one chord with a descending melodic line over it.

The appeal is that it explains why long pieces hold together, and why a piece ending in a different key than it started sounds unfinished. The objections are substantial: the reductions are not unique, the method is difficult to falsify, and the perceptual evidence that listeners track tonal structure over ten-minute spans is weak.

What survives regardless of the theoretical dispute is the observation that harmony operates at more than one time scale. A chord can be local decoration or structural arrival, the difference is real, and no map that treats every chord as a point can express it.

Whose music, and when #

Everything above describes functional tonal harmony, which is a European practice of roughly 1650 to 1900 and its descendants in popular music.

Before it, harmony was modal and driven by voice leading rather than by root progressions, and the machinery of function does not apply. Rameau’s Traité de l’harmonie of 1722 is where chord roots and functional relationships get their first systematic statement, and it postdates most of the music people think of as tonal.

After it, composers dismantled it deliberately. Wagner’s chromaticism strains function past the point where a key can be located; Debussy uses chords for colour rather than for function; jazz retains ii–V–I while allowing substitutions that a functional analysis handles awkwardly.

Outside Europe, harmony in this sense is usually not the organising principle at all. A great deal of the world’s music is melodic over a drone, or built from interlocking rhythmic parts, or organised by melodic mode. The map on this page is a map of one repertoire, and its usefulness elsewhere is a question to be asked rather than assumed.

Harmonic rhythm, which is half the effect #

The same chords in the same order can produce completely different music depending on when they change, and no map of chord space contains that.

Rate. One chord a bar is a different style from four. Renaissance polyphony changes harmony slowly; classical style changes it fast and regularly; a modal jazz piece may hold one for sixteen bars. The rate is as characteristic of a period as the chord vocabulary is.

Placement. A chord change on a downbeat is an arrival. The same change on the last quaver of a bar is an anticipation, and it sounds like a push rather than a landing. Syncopated harmony is one of the defining features of a great deal of popular music, and it uses exactly the same chords as the unsyncopated version.

Acceleration. Harmonic rhythm speeding up toward a cadence is one of the most reliable signals of an approaching arrival in classical style, and it is done with the chords already in use rather than with new ones.

That means a complete account of a progression needs a horizontal axis the figures here do not have. The map says which chords are near; the music is made of when.

Substitution, and why it works #

Jazz practice replaces chords with others that a functional analysis would call different, and the replacements are predictable once distance is available.

Tritone substitution. Replace a dominant seventh with the dominant seventh whose root is a tritone away. The two chords share their third and seventh — the tritone itself, whose two notes swap roles — so the tension-carrying interval is identical and only the root has moved. That is why the substitution preserves function while changing the bass line completely.

Relative substitution. Replace a chord with one sharing two of its notes: iii for I, vi for I, ii for IV. The voice leading barely changes and the harmonic colour does.

Both are the same move: keep what carries the function and change what carries the colour. The functional account explains why they work and the voice-leading account explains why they are smooth, which is a good illustration of the two theories being complementary rather than rival.

Where the model stops #

Triads only. Real tonal harmony uses sevenths constantly, and the dominant seventh’s tritone is the mechanism the whole cadence rests on — yet the map places triads.

One key. The figures show the seven chords of one key, so modulation leaves the diagram entirely. Modulation, secondary dominants and borrowed chords all leave the diagram, which is most of what makes a piece longer than sixteen bars interesting.

Distance is voice leading only. The radius measures semitone motion. Function is a separate structure that distance does not predict — V and vii° are at different radii and do the same job.

Equal temperament, except where it isn’t. The map is drawn in twelve pitch classes; the comma-pump figure is not. The two figures on this page are drawn in incompatible spaces, which is honest about the subject and awkward in a single essay.

The map is one repertoire’s map #

A closing caution, because the diagram is seductive and it is a diagram of something specific.

It places seven triads of one major key by voice-leading distance from the tonic, which presupposes: that the triad is the unit of harmony, that a key has a tonic, that the seven diatonic chords are the vocabulary, and that distance from home is the relevant coordinate.

Every one of those is a property of functional tonal practice. Take any away and the picture stops applying. Modal jazz has a tonic and no functional progression. Blues has a repertoire of three chords used in an order that a functional analysis calls wrong and that has been standard for a century. Chromatic nineteenth-century harmony uses chords that are not in the key at all.

The map is genuinely useful and it maps a territory with borders. Stating them is not a hedge — it is the difference between a description and a rule.

The ladder from here #

Later rungs: the dominant seventh in full. Secondary dominants. Cadence types and their strengths. Harmonic rhythm. The bass line as structure. Substitution in jazz. Sequences and the descending fifths pattern. Modulation as a change of map. Prolongation, and Schenker’s claim that a whole piece is one chord. The comma pump measured in real choirs. And chromatic harmony, where function stops predicting and voice-leading distance starts.

Rameau derived the whole of functional harmony from the harmonic series, in an argument that does not work — the minor triad defeated him, and he spent decades on increasingly strained fixes. The theory outlived its justification by three hundred years and is still what everybody is taught.