How often the chord changes

A progression is a list of chords. The site has a map of them, placed by how far the voices must travel, and it explains a great deal about which sequences are common.

It explains nothing at all about the difference between a chorale and a piece of dance music, and the two can be built on the same list. Take the tonic, the submediant, the subdominant and the dominant. Put one in each bar of a slow four and it is a hymn; put four to a bar and it is a Bach chorale phrase; put one every two bars at 120 and it is most of the popular music of the last sixty years. Same chords, same order, same map.

What differs is the rate, and the rate has a name — harmonic rhythm — and rather less attention than it deserves.

The arithmetic, and the range #

A style’s harmonic rhythm is normally described as a rate against the bar: one chord a bar, two a bar, one every four bars. Multiply by the tempo and divide by the metre and it becomes a rate against the clock, which is the form in which it can be compared with things that are not music.

The hero figure does that for seven styles, on a logarithmic axis because the range demands one.

At the fast end, a chorale at 66 crotchets a minute with a chord under every crotchet is 1.10 chords a second. At the slow end, a modal jazz head holding one mode for sixteen bars at 132 is 0.034. That is a factor of thirty-two, which is wider than the range of tempos those styles use and considerably wider than any other harmonic variable this site measures.

Harmonic rhythm is therefore the largest single axis of variation in tonal music, and it is orthogonal to the one that gets the attention. The chords a modal jazz head uses and the chords a chorale uses are drawn from overlapping vocabularies; what separates the two is a factor of thirty in how often they change.

Where the floor is #

At the slow end something has to stop the rate falling, and it is not a harmonic constraint.

A phrase is a few seconds long, because a series of events is heard as one thing only inside a window of about two to eight seconds. Divide one by that window and a rate falls out: below about 0.3 chords a second, one chord spans an entire perceptual present, and a listener has no change of harmony inside the unit they are holding.

Three of the seven rates are below that line. A four-chord song at one chord every two bars is 0.25; a modal head at 0.034 is thirty seconds a chord; organum is slower still.

That is not a failure. It is a different design, and the music that sits there is systematically different in a way the rate predicts. When the harmony does not change inside a phrase, something else has to carry the phrase’s shape — and in each of those three cases something does. In a modal jazz head it is the melodic line and the improvisation over it. In a four-chord song it is the vocal. In organum it is the chant, moving above a held note that is not functioning as a chord at all.

A slow harmonic rhythm demotes harmony from the thing that moves to the thing that is stood on. That is why the term “modal” attaches to so much of this music: with the harmony static, what remains to be organised is the scale, and a scale that is not going anywhere is a mode.

Where the ceiling is #

At the fast end the constraint is even further from harmony, and it gets an essay of its own because it is a piece of physics.

A chord does not stop when it stops being played. In a reverberant room it decays at a rate fixed by the room, and while it decays the next chord arrives on top of it. Beyond a certain rate a listener is not hearing a sequence of harmonies at all; they are hearing several at once.

The vertical dashed lines in the hero figure are that limit, computed for six rooms. Left of a line, in that room, a chord has fallen twenty decibels before its successor arrives; right of it, two or more chords are audible together.

The numbers are stark. A cathedral with an eight-second reverberation time allows 0.375 chords a second. A shoebox concert hall allows 1.5. A recording studio allows 8.6.

Put the two sets of numbers side by side and something falls out that neither alone would give. The chorale, at 1.10, is above what a cathedral allows and below what a concert hall allows. The repertoire written for the great stone rooms sits comfortably to the left of the cathedral line; the repertoire written after those rooms stopped being the venue does not.

Why the middle of the range is where the theory is #

There is a band on the hero figure — roughly 0.5 to 1.1 chords a second — inside which sit the chorale, the baroque allegro, the classical allegro and the blues. That band is where almost every claim in the whole of tonal theory was made, and its width is not an accident.

Above it, harmony smears. Below it, harmony stops moving. In between, a chord lasts long enough to be identified and not so long that a phrase contains only one — which is the condition under which a progression is a perceptible object at all.

That is a strong claim and it is checkable against the arithmetic already on the page. Take the fastest of them: a chorale at 1.10 chords a second is 0.91 seconds a chord, and 0.91 seconds is comfortably inside the range where the ear will accept a pulse. Take the slowest of the four: a blues at 0.5 is two seconds a chord, which is at the slow edge of that range — and its true rate is slower still, for a reason the section on rate against size sets out, which puts it just outside. That does not weaken the claim about the band; it moves one of the four styles from the inside edge to the outside one, and the band was drawn from the other three. So across the whole band the chord change is itself a metrical event — it happens at a rate the listener can entrain to.

Below the band it stops being one. At 0.25 a chord change happens every four seconds, which is outside the pulse range and inside the phrase window, so it becomes a grouping event rather than a beat. Above the band it happens faster than the room allows and the changes stop being individually resolvable.

So the middle of the range is not a stylistic preference. It is where a chord change is fast enough to be felt and slow enough to be heard, and the width of that window is set by two measured properties of listeners and one of buildings.

The two limits are not symmetric #

There is an asymmetry between the floor and the ceiling that is worth drawing out, because it says which of them is a constraint and which is a choice.

The ceiling is physical. Exceed it and the music is smeared: the harmonies overlap, the counterpoint becomes a wash, and no amount of skill on anyone’s part recovers the clarity. A composer writing for a specific building is working under it whether or not they can state it.

The floor is perceptual and much softer. Go under it and nothing breaks; the music simply stops using harmony as its agent of change and uses something else. Whole traditions live below the floor and are none the worse for it.

So the sentence to carry away is that harmonic rhythm has a hard ceiling and a soft floor, and the interesting music of any period tends to sit just under whichever ceiling it has.

In a similarity matrix the rate is visible as a scale rather than as a pattern: the blues changes chord about once a bar, which is what makes its blocks four cells across rather than sixteen, and a piece at a quarter of the rate on the same encoding gives a matrix with blocks four times as large and nothing else different.

It is a rhythm, so it can be syncopated #

The word rhythm in “harmonic rhythm” is doing real work, and one consequence is worth stating because it is where the variable stops being a single number.

A chord change is an accent. It marks the bar it lands on, and where it lands relative to the metre is a choice — which is why a chord that changes on the fourth beat rather than the first is one of the standard devices for pushing a phrase forward, and why the anticipated chord change is the defining rhythmic gesture of several popular styles.

So harmonic rhythm has a rate and a phase, and both are compositional variables. The site’s metric weight profile is what a phase is measured against, and a chord change on a weak position is doing exactly what a syncopated drum hit does.

It also means the rate need not be constant, and usually is not. The commonest single use of harmonic rhythm as a device is to double it approaching a cadence, which is the acceleration a classical sentence is built on and which the unit-length ratio measures from the other side. Fragmentation and harmonic acceleration are two descriptions of one event.

The same variable, one field along #

Rate against pattern is a distinction the site has drawn before, in a field with no harmony in it at all.

A Euclidean rhythm is a pattern of onsets, and the same pattern at 90 and at 180 beats a minute is two entirely different musical objects. Swing is a ratio that changes with tempo, because the short note holds a roughly constant absolute length. A polyrhythm’s least common multiple is a count of steps and its period is a number of seconds.

In each case the structure is one thing and the rate at which it is delivered is another, and both are compositional. What is unusual about harmony is how thoroughly the theory has concentrated on the first and ignored the second: there are whole textbooks of chord progressions with no systematic treatment of how fast the chords go.

The rate is a period, and a period is a cycle #

One more reading of the same axis connects this essay to the rhythm field, and it is worth a figure because the conversion is not obvious.

Invert a rate and it becomes a period. A four-chord song at 0.25 chords a second is one chord every four seconds and four chords every sixteen — which is a cycle sixteen seconds long, repeated. Written that way it is the same kind of object as a drum pattern, differing only in the units.

That is not a metaphor. A harmonic loop with no cadence in it genuinely has no first chord, and the commonest four-chord loop in popular music is played starting from at least three of its four positions in songs that are otherwise indistinguishable. A cycle has no beginning applies to harmony the moment the harmony stops going anywhere.

What the rates are, and are not #

The numbers on these figures come from stated conventions multiplied by stated tempos. That is arithmetic, and it is not a measurement of any repertoire.

The distinction matters and this site cannot fudge it. To say that a chorale changes chord on every crotchet is a claim about a style, made from general knowledge of it, and somebody could reasonably say that a great many chorale phrases hold a chord for two beats. To say that a chord every crotchet at 66 is 1.10 a second is arithmetic that cannot be wrong.

What would settle the first half is a corpus: several hundred pieces, encoded, with chord changes counted and tempo taken from the marking or from a recording. Such corpora exist and this site does not have one. Every claim above of the form “this style changes chord at this rate” should be read as a convention rather than as a finding, and every claim of the form “at this rate and this tempo, that is this many a second” is exact.

The room numbers are on the other side of that line. A reverberation time is a measurement of a building, the decay is linear in decibels by definition, and the rate at which a chord has fallen twenty decibels is arithmetic.

Rate times size, which is the quantity the caveat is about #

The obvious objection to everything above is that a chord change is not a unit. A move sharing two notes with its predecessor is a smaller event than one sharing none, so a slow rate of large changes and a fast rate of small ones might come to the same thing — and if they did, the factor of thirty would be an artefact of counting the wrong thing.

Both halves are computable. The voice-leading distance between two triads is a number of semitones, and multiplying it by the rate gives semitones per second, which is an amount of harmonic motion rather than a count of events.

The objection does not survive, and the reason is that the sizes are too close together to cancel anything. Between the seven diatonic triads the voice-leading cost runs from 1 to 6 semitones, so the widest possible size ratio is six; across these five progressions the mean cost runs from 2.00 to 4.00, a ratio of two. A rate difference of thirty-two cannot be undone by a size difference of two, and could not be undone by the extreme of six either. Rate dominates because rate has more room to vary.

It does reorder one pair, and the reordering is instructive. The blues changes chord half as often as its stated rate says. Written out, five of its twelve bar-to-bar transitions are the same chord twice — the four opening tonic bars, the repeated subdominant, the second pair of tonic bars — so “one a bar by construction” counts twelve changes where there are seven. At 0.29 actual changes a second it sits below the four-chord song’s rate rather than above it, and only its larger average step brings the two back level in semitones per second.

That is a fault in the convention rather than in the arithmetic, and it is the kind the whole “conventions, not corpora” caution above was written for. The other four progressions have no repeated chord in them, so their stated rates and their change rates are the same number.

What this cannot show #

There is no such thing as the harmonic rhythm of a piece. It varies within a phrase, within a movement and between the hands of a keyboard player, and a single number for a style is an average over something that is not distributed evenly.

The five progressions in the section above are representative rather than measured. Two of them are definitional — a twelve-bar blues is its own chord sequence, and the four-chord loop is named for its four chords — and three are chosen as typical of their style, which is the same kind of claim as the rates themselves and carries the same warning.

Where the ladder goes #

The ceiling has been asserted here and is computed properly in the room’s own rung, where the decay curves are drawn and the count of simultaneously audible chords comes out of them.

And the floor leads somewhere else entirely. Music that sits far below it — a repeating cycle with no chord changes at all — has no harmonic events to organise a form with, no cadence to compute, and uses a different variable altogether.