A phrase is a number of seconds

Ask what a phrase is and the answer comes back in bars. Four of them, usually, or eight; the four-bar phrase is so standard a unit that whole styles are described by their departures from it.

Ask what fixes the number four and the answers get vague, and they get vague in a particular direction. They appeal to convention, to the practice of a repertoire, sometimes to the length of a line of verse. All of those are real and none of them is a constraint — a convention explains why everybody does it and not why it works.

There is a constraint, it is not a musical one, and it is measured in seconds.

The window that does not move #

A series of events becomes a single perceived thing only if it is short enough. Beyond a certain span the events stop being a group and become a sequence — still ordered, still comprehensible, no longer one object. The span has been measured many times under many names: the perceptual present, the psychological present, the integration interval.

The measurements do not agree on a number, and the disagreement is instructive rather than embarrassing. Paul Fraisse’s reviews put the upper bound of a perceived group at around five seconds. Ernst Pöppel’s integration interval is nearer three. Depending on the task, on the material and on the listener, published values run from about two seconds to about eight.

That range is what the shaded band in the hero figure is. It is wide, it is soft at both edges, and it is the same for a listener whatever music is being played to them — which is exactly what makes it useful here, because everything else in this essay moves and it does not.

The arithmetic, which is the whole argument #

A phrase of $b$ bars in a metre of $m$ beats at a tempo of $t$ beats a minute lasts

$$\frac{60\,b\,m}{t} \ \text{seconds}.$$

That is all. The hero figure evaluates it for seven ordinary tempos and five bar counts, and plots the results against the band.

Read it along a row and the dots march rightwards as the phrase gets longer. Read it down a column and something more interesting happens: the bar count that lands inside the band changes.

The window is fixed and the bar count runs from one to eight. A four-bar phrase in an adagio lasts eighteen and a half seconds, which is not a perceptual group by any measurement anybody has published; a four-bar phrase in a scherzo lasts four seconds, which is comfortably one.

This is what “felt in one” means #

There is a piece of ordinary musical language that this arithmetic explains, and the explanation is exact rather than suggestive.

A scherzo is written in three-four and is said to be felt in one, which is a claim the time signature is making and not a fact the notation records: the player counts one beat to the bar rather than three. At 180 beats a minute a beat lasts a third of a second, which is well inside the range where the ear will accept a pulse but at the fast end of it, and a bar lasts a second, which is nearer the middle. So the unit a listener locks on to is the bar.

Once the bar is the felt beat, the four-bar group is the felt bar — and that is the level at which a scherzo’s structure is actually organised. Its phrases are eight bars and sixteen bars because those are, in the units a listener is working in, two and four.

The window is a property of the listener and does not move; which bar count lands inside it does. At a scherzo’s tempo the eight-bar phrase is comfortably inside; at an adagio’s it is far outside — so the notated phrase length is not what is being perceived, and the number of seconds is.

The same figure drawn for an adagio comes out the other way up. The notated beat is already slow enough to be near the edge of the pulse range, the bar is well outside it, and the four-bar group is not a rhythmic object at all. The metrical ladder has a top, and where it stops is set by seconds.

Both land their eight-bar unit inside the window, which is the useful negative result: the tempo conventions of the repertoire are not arbitrary with respect to this bound. A minuet and an allegro differ in almost everything and agree in putting their phrase where a listener can hold it.

The same sixteen bars, twice #

The cleanest demonstration is to take one written structure and put two tempos on it.

So the unit a listener holds is not the notated phrase at all at slow tempos — it is whatever subdivision of it happens to last a few seconds. The phrase moves down the hierarchy as the tempo falls, which is why an adagio’s sixteen bars are heard as a succession of two-bar units and its written phrasing is a fact about the page.

This is a case where the notation is actively misleading, and it belongs beside the site’s other examples of that. Staff notation records what is written and not what is heard in the matter of timing; here it records a count of bars and hides the fact that a bar is not a fixed amount of anything.

Two passages written as eight-bar phrases can be, perceptually, two entirely different kinds of object — and where a phrase ends is decided by the same seconds — one a single gesture, the other a paragraph. Nothing on the page distinguishes them and the tempo marking, four inches away at the top of the movement, is what decides.

Why four, then #

The window explains why phrase lengths vary. It does not by itself explain why four is the number that gets taught, and there is an answer to that which is not a convention.

Look at where the ordinary tempos of the common-practice repertoire sit. Allegro is somewhere around 120 to 150, andante around 70 to 100, allegretto and moderato in between; the great bulk of eighteenth- and nineteenth-century movements are in four-four or three-four somewhere in that range. Put 138 in four-four into the arithmetic and four bars come to 7.0 seconds. Put 120 in three-four in and four bars come to 6.0. Put 100 in four-four in and it is 9.6, which is where a composer starts writing in two-bar units instead.

So a four-bar phrase is six to eight seconds at the fast end of that range. Three worked tempos are not an argument, though, and running the same arithmetic across the whole range says something the three examples hide.

Two bars, not four, is the count that spends most of the tempo range inside the window. In four-four, sweeping every tempo from 40 to 200, the two-bar group lands between two and eight seconds at 88 per cent of them and the four-bar phrase at 50; narrowing to the 100-to-150 band where the bulk of the common-practice repertoire sits makes it worse rather than better — two bars is inside at every tempo in that band and four bars at 60 per cent of them. On the plain criterion the norm ought to be two.

What rescues the four is the second half of the criterion, which is where a phrase sits within the window rather than whether it is in it. Asking for the top half — five to eight seconds, long enough to say something — turns each count into a tempo band:

The four-bar phrase is the allegro answer and the two-bar phrase is the andante answer, and neither is the answer everywhere. The two four-bar bands together cover 90 to 192, which is most of what an eighteenth- or nineteenth-century movement is marked — allegretto through presto in triple time, allegro through presto in duple — and that is a specific and checkable claim rather than “most of the range”.

It also puts the essay’s own aside in its proper place. “Put 100 in four-four in and it is 9.6, which is where a composer starts writing in two-bar units instead” was offered as an exception; the sweep says it is the rule below 120, and that the four-bar norm is what the arithmetic returns for fast music and nothing else. A slow movement written in two-bar units is not departing from the norm — it is at a tempo where two bars is what four bars is at allegro.

And the last row is the scherzo again, arriving from the arithmetic rather than from the argument: eight bars of three-four sits in the top half of the window from 180 upward, which is exactly the tempo at which a scherzo’s phrases are eight and sixteen bars. Three counts, three tempo bands, one window.

One thing the band table should not be read as saying is that the boundaries are sharp. Every number in it inherits the five-second edge of a window whose published estimates run from two to eight, so “120 to 192” means the middle of the band is where four bars is comfortably right and the ends are arguments rather than findings. What the sweep establishes is the ordering and the separation — that the two-bar and four-bar bands do not overlap, that they are about an octave of tempo apart, and that the gap between them falls near 100 beats a minute, which is where the repertoire’s own vocabulary changes word.

The convention is real. It is a convention that encodes a constraint, at the tempos the repertoire happens to use, and it stops being a good rule the moment the tempo leaves that range. Which is exactly what is observed: chant has no bar lines at all, an adagio’s phrase marks cover one or two bars, and a fast dance movement groups its bars in eights and sixteens without anybody calling those phrases long.

Long phrases are made of short ones #

The obvious objection is that music is full of phrases far longer than eight seconds. A Wagner paragraph, a Bach ritornello, a raga’s exposition: none of these is over in five seconds and all of them are single spans of musical argument.

They are, and the window does not deny it. What the window says is that such a span cannot be one perceptual object, and the way that shows up in the music is that long spans are audibly built out of shorter ones.

A metre drawn as a tree is recursive by construction — a bar divides into beats and beats into subdivisions, all the way down — and nothing in that structure says how long any of it lasts. That is the gap this essay is about: the notation supplies the divisions and the tempo supplies the seconds, and only the second of them is what a listener’s memory is measured in.

A tree is the right picture for this, and the same tree serves at every level. Four bars group into an eight-bar sentence; two sentences make a sixteen-bar period; periods make a section. Each level is a group of two, three or four of the level below, and each level below the top is short enough to be held. The long paragraph is real and it is constructed — assembled by a listener from units, each one inside the window, in the same way a metre is assembled from onsets that do not contain it.

That is why the objection strengthens the case rather than weakening it. A span of forty seconds that is not divisible into perceptible units is not a long phrase; it is a stretch of music with no phrase structure, which is a real and recognisable thing — and it is what a listener means when a passage is said to wander.

The second constraint, which arrives at the same answer #

There is an independent reason for phrases to be a few seconds long, and it is worth setting beside the first because it is physical rather than perceptual.

A great deal of music is sung, and most of the rest was written by people who sang. A singer’s phrase ends when the breath does. Vital capacity for an adult is around four to five litres, of which perhaps three are usable in a phrase; a trained singer at a moderate dynamic uses on the order of 0.1 to 0.2 litres a second. That gives something between fifteen and thirty seconds at the extreme, and a great deal less at any volume worth hearing — comfortable singing phrases run to five or ten seconds, and that is the number every choral director works with.

The wind instruments impose the same constraint with different constants, and one of them inverts it: an oboe uses so little air that the player’s difficulty is getting rid of the old breath rather than finding new. But the flute at full volume empties a player in a few seconds, and the brass are in between.

So there are two independent constraints — one in the listener and one in the lungs — and they land on the same order of magnitude. That is a coincidence in the sense that neither causes the other, and not a coincidence in the sense that both are properties of the same animal.

What makes it worth stating is that the constraint outlived its cause. A string quartet has no lungs and a piano has fewer still, and both write in phrases of the same length as a song. The keyboard repertoire inherited a unit from vocal practice and then found it was also the unit the listener wanted, which is the more interesting half.

The bar is a variable, and the notation hides it #

One consequence deserves separating out, because it undermines a habit of description that this site has itself relied on.

Every measurement in the repetition ladder took the bar as its unit. The matrix compares bars, the novelty operator counts bars, the coder codes bars. That is convenient, it is what the encodings carry, and the figures above show it is not a perceptually meaningful unit at all: the same bar is a beat in one movement and a phrase in another.

A matrix built on bars is therefore measuring at a different perceptual scale depending on the tempo of the piece it is handed, and comparing two pieces at different tempos on such a measure is comparing two different questions. The fix is easy to state and awkward to apply — sample at a fixed number of frames a second rather than a fixed number a bar, which is what the method does on real audio and one of the reasons it does — and it is not applied here, because the encodings are bar-level and inventing a tempo for a scheme would be inventing data.

So the limitation is recorded rather than repaired. Everything in that ladder is a statement about a bar-level description, and it inherits the wobble that this essay has just put under the bar.

What the window cannot bear #

The band is two to eight seconds, and a factor of four is a wide band. Any argument that leans on a precise value of the perceptual present is leaning on a number that does not exist.

That is why the claims above are all of the form “this count is inside and that one is a long way outside”. An eighteen-second phrase is outside the window under every published estimate, and a four-second one is inside under every published estimate, and it is only those unambiguous cases that the figure is being asked to decide. Where a value sits near an edge — the scherzo’s eight-bar phrase at exactly eight seconds — the figure marks it and the argument does not use it.

The spread between listeners is real too, and it varies with training, with attention and with the material. Nothing here is a claim about an individual.

And the tempo sweep asks for one thing the window alone does not supply: a preference for the top of it. Nothing measured says a six-second group is better than a three-second one, and the whole four-versus-two result rests on that preference. It is a reasonable one — a phrase has to carry an argument as well as be held — but it is an assumption imported from musical practice into a perceptual measurement, and the bands would be different octaves of tempo if the target were the middle of the window instead.

And the window bounds a phrase; it does not determine one. Which four bars group together is decided by harmony, by melodic shape, by where the cadence falls and by the text if there is one. The window says only that whatever the grouping is, it has a few seconds to happen in. That is a boundary condition, and this site’s whole instrument case is an argument that boundary conditions decide more than they look as though they should.

Where the ladder goes #

If a phrase is a span of seconds, the next question is what happens inside it. Two eight-bar phrases can occupy identical spans and be built from completely different arithmetic, and the difference between them turns out to be measurable in one number — one of them accelerates and the other does not.

Above the phrase there is more metre, not less. The same induction machinery that produces a four-beat bar produces a four-bar hypermeasure when it is fed bars instead of beats, and the level at which that ladder stops is the number this essay has been about.