The bar above the bar

Musicians count bars in fours. Conductors beat four-bar groups. A drummer’s fill lands at the end of a four-bar unit and a phrase begins at the start of one, and none of this is written down: the score has bar lines and no lines above them.

The word for it is hypermetre, and the usual way of introducing it is by analogy — a four-bar group behaves like a bar of four beats.

It is not an analogy. The site already has machinery for finding metre in a pattern of onsets, and for giving each position in a metre a weight. Neither of them contains any statement about what the positions are. Hand them bars and they produce hypermetre, by the identical computation, with nothing modified.

The same weight profile #

Start with the simpler of the two. A metric weight profile assigns a strength to each position in a repeating unit, and it is built by counting: a position’s weight is the number of levels at which it begins a group, with the whole unit counting as one level.

For a bar of four beats, divided in two and then in two again, the profile is 3, 1, 2, 1. The downbeat begins the bar, begins the first half and begins the first beat, so it scores three; the third beat begins the second half and itself, so it scores two; the second and fourth begin nothing but themselves.

That is the profile every account of metre gives, and it is what makes the difference between a cadence on the downbeat and the same chords on the second beat.

Now ask the same function for eight positions, divided in two three times: 4, 1, 2, 1, 3, 1, 2, 1. Bar one of an eight-bar group is the strongest, bar five is next, bars three and seven next, and the even-numbered bars are weak.

Every musician who has counted an eight-bar phrase recognises that shape, and the function that produced it does not know what a bar is.

The same induction #

The second piece of machinery is harder to fool, and it is where this essay found something.

A metre is inferred rather than received: given a pattern of onsets, the site scores candidate metres by asking how many strong positions have onsets on them, how many are empty, and how many onsets fall off the grid. The tresillo supports several readings and the scoring says which fit best.

Run the identical function over bars. What is needed is a definition of a bar-level onset — some event that marks a bar as accented — and a chord scheme offers exactly one candidate: the bar changes chord.

That is not what a musician counting this music would say, and the reason is visible in the top row of the figure. Nearly every bar is marked. In this plan the chord changes in twenty-seven of thirty-two bars; in the rondo it changes in all forty.

An accent pattern in which almost everything is accented contains no metre. The scoring rewards a candidate whose strong positions have onsets and penalises it for strong positions that do not, and when every position has an onset the shortest period wins automatically — it has the most strong positions and every one of them is filled.

What that failure is actually saying #

The temptation is to call this a defect in the scoring and reach for a fix. It is not a defect. It is a correct answer to a question that was badly posed, and the bad posing is the finding.

Harmonic change is not the evidence for hypermetre. A listener counting a rock song in fours is not counting chord changes; the chords may change every bar, every two bars or every four, and the four-bar count is stable across all of them. What marks a hypermetric downbeat is where the melody restarts, where the drum pattern resets, where the bass returns to its root, where the texture thickens — a bundle of accents that a chord scheme does not carry any of.

So the honest position for this site is that it has the machinery and does not have the input. That is worth stating plainly rather than papering over, because the alternative was available: feed the induction the section beginnings instead, and it returns eight bars, which is the answer everyone expects.

The distinction matters. The hero figure shows that the machinery is scale-free, which is this essay’s claim. It does not show that hypermetre can be derived from a chord scheme, which would be a different and much stronger claim, and which the chord figure refutes.

The same figure one level down #

Putting the beat-level version beside the bar-level one makes the identity concrete rather than asserted.

Everything in that figure has a bar-level twin. The onsets become bars that carry an accent; the candidate periods become candidate hypermetres; the strong positions become hypermetric downbeats; the score is computed by the same three terms with the same three weights. The function in the site’s library is called once by each figure and does not take an argument saying which it is being used for.

What differs is the input’s density, and that is the whole of this essay’s negative result. At the beat level a pattern of onsets is sparse — three onsets in eight steps is typical, and the empty steps are what make one candidate fit better than another. At the bar level, using harmony as the accent, the pattern is nearly solid, and a nearly solid pattern is uninformative at every period.

So the machinery transfers and the evidence does not. That is a more useful thing to have found than a confirmation would have been, because it says exactly what a bar-level metre-finder would need: a sparse accent, of the kind a drum kit and a melody supply and a chord chart does not.

Where the ladder stops, and why #

If the arithmetic is the same at every level, something outside the arithmetic must decide where the levels end. Nothing about a metric grid prevents a sixty-four-bar hypermeasure. Nothing about the induction prevents a candidate period of thirty-two. Yet metre plainly does stop somewhere, and everybody agrees roughly where.

The answer is in seconds, and the site has both of the relevant numbers already.

The pulse range runs from about 100 ms to about 2 seconds, with a preference near half a second. That is the beat’s preferred rate, and it is a measured property of listeners.

The perceptual present runs from about 2 to about 8 seconds. That is what bounds a phrase, and it is a different measured property of the same listeners.

Put them side by side and hypermetre has a precise home. It lives in the gap between the two windows — too slow to be felt as a pulse, fast enough to be held as one thing. A four-bar group at an ordinary allegro is seven seconds, which is not a beat by any measurement and is a group by most of them.

And the ladder stops where the second window does. A sixteen-bar hypermeasure at that tempo is twenty-eight seconds, which is outside every published estimate of the present, and it is not something a listener can count. Musicians do count sixteen-bar and thirty-two-bar structures, and the way they do it is by counting four-bar groups — that is, by introducing yet another level whose unit is inside the window again.

Which makes it a different kind of thing above the bar #

There is a real discontinuity in the ladder, and it is worth being clear that this essay’s scale-free claim is about the arithmetic and not about the experience.

Below about two seconds, a level is a pulse. It can be tapped, it entrains, and it has the properties the rhythm field measured — a preferred rate, a resistance to change, an ability to continue when the sound stops.

Above two seconds, a level is a group. It is counted rather than felt, it does not entrain, and it is held in memory rather than in the body. Somebody can tap a beat for a minute without effort and cannot tap a four-bar group at all except by counting the beats inside it.

So the metrical hierarchy is one construction with two regimes inside it, and the boundary between the regimes is a number about human beings. That is the same shape as several other results on this site: the point where a rhythm becomes a pitch is a rate rather than a category, and the point at which two sounds become one event is a delay.

Where the count starts #

One more thing follows from the weight profile, and it is the commonest hypermetric argument in the analytical literature.

A profile of 4, 1, 2, 1, 3, 1, 2, 1 says which position is strongest; it does not say which bar of the music is position one. That is a phase question, and the same music admits more than one answer.

A phrase that begins with an upbeat bar puts its first notated bar on hyperbeat four rather than hyperbeat one. A repeat that begins one bar early elides two hypermeasures into seven bars. An introduction of an odd number of bars shifts everything after it. In each case the notation is unchanged and the count is different, and performers disagree about such passages in exactly the way the ambiguity predicts.

This is the same ambiguity the beat-level essay is about, arriving one level up and with the same resolution: the listener commits to a reading, the reading is not uniquely determined by the sound, and a composer who wants a particular one has to supply evidence for it.

The evidence a listener actually uses #

If not harmony, then what? The literature on hypermetric perception points at a short list, and each item is measurable in principle.

Parallelism is the strongest of them. A hypermetric downbeat is where something starts that started before — the same melodic figure, the same bass pattern, the same drum bar. That is a repetition at a fixed offset, which is exactly what an off-diagonal stripe in a similarity matrix is, and it is available from any encoding rich enough to have a melody in it.

Duration comes next: a longer note or a longer chord marks the position it begins. Density and texture follow — more onsets, more instruments, a fuller sound at the group’s start. And cadence works backwards: a cadence marks the end of a group and therefore places the next downbeat.

None of these is harmonic change, and three of the five are unavailable in a chord scheme. That is the input problem stated as a list.

What this cannot show #

The whole essay rests on a bar being a fixed unit, and the previous rung established that it is not. The seconds in the tempo-window figure are for one tempo, and at a slow tempo the bar itself is already outside the pulse range, which means the hypermetric ladder has one fewer rung — the four-bar group in an adagio is not a metrical level in any sense.

The figures also assume a hypermetre that is regular, and much music is not. Extra bars, elided bars and phrases of five or seven bars are ordinary in some repertoires and universal in others. A regular hypermetre is a common-practice habit and a strong one, not a law, and the induction scoring above would report the irregularity as a poor fit rather than as a different structure.

And the scoring has a defect which the chord-event figure exposes: it has no penalty for a candidate period being implausibly short. The section below connects the measured constraint to it and reports what that does, which is half of what was wanted.

The missing constraint, connected #

The scoring above has nothing in it about how long a candidate period lasts, which is why a saturated accent pattern lets a period of one bar win. The constraint that is missing is measured and is already in this collection: a hypermetric level is a group rather than a pulse, so its period should sit inside the perceptual present. Weighting each candidate by how near its period is to the middle of that window — linear in log period between two seconds and eight, peaking at three and a half, exactly the shape the rate preference uses one level down — takes no new number and no new parameter.

It removes the defect. On the rondo and on the verse-and-chorus plan the unweighted winner is a period of one bar, which is not a hypermetre at all; weighted, both return two. So the repair does the job it was named for, on every scheme, and the degenerate answer is gone.

It does not deliver a four-bar group. Two bars wins on all four schemes, and the reason is in the fourth row: at 108 beats a minute a four-bar group lasts 8.9 seconds, which is outside the published present entirely and therefore scores zero. Sweeping the tempo does not rescue it — a four-bar group first gets any weight at all above about 130 beats a minute, and even at 180 its weight is 0.49 against a two-bar group’s 0.51.

So on these numbers the natural hypermetric unit is two bars and not four, which is not what any musician counting this music would say. Three readings are available and the essay can only rule out one of them. The accent evidence may still be wrong, which is this rung’s own diagnosis and is untouched by the repair. The present window may be the wrong constraint for a counted group, in which case the number to use is not one this collection has. Or musicians counting in fours are counting two two-bar groups, which is precisely the move the section on where the ladder stops attributes to them one level up, where sixteen-bar structures are counted as four four-bar groups.

The third is the tidiest and it is not evidence for itself. What the repair establishes is narrower and worth having: the scoring’s degenerate answer was a missing constraint rather than a fact about the music, the constraint costs no free parameter, and closing it leaves a gap of exactly one binary level between what the arithmetic prefers and what musicians report.

Why this level has no notation #

Every other level of the metrical hierarchy has a symbol. A beat has a note value, a bar has a bar line, a subdivision has a beam. The four-bar group has nothing at all, and the absence is worth a paragraph because it is not an oversight.

Bar lines were introduced to keep parts together in ensemble music, and they mark a level at which players need to agree. Above that level nobody needs to agree in the same way: the four-bar group is a shared expectation rather than a coordination device, and a piece where it goes wrong produces confusion rather than a collision.

So the notation stops where the coordination problem stops, and the level above is left to be inferred. Which means hypermetre is a place where notation hides something — not by distorting it, as it does with timing, but by declining to represent it at all.

Rehearsal marks and letters do the job informally, and so do the conductor’s hands. Both are outside the score’s own system, and both mark the level this essay is about.

Where the ladder goes #

Hypermetre is the last of the levels that a listener holds directly. Above it, structure is a matter of memory and of key rather than of counting — and the key plan is where the large shape of an eighteenth-century movement actually lives.

Below it, everything in this essay depends on a listener supplying a regular grid to music that only partly specifies one, which is the site’s oldest claim about metre applied one level up.