The beat has a preferred rate, and it is not the notation's

Metre is inferred rather than received, and the previous rung was about which metre a listener settles on. This one is about a prior question that the preference rules do not answer: metre is a hierarchy with several levels, and only one of them gets tapped. Which one?

Not the one the time signature names. That is a notational fact, and the tapping is not — a bar of 4/4 at 60 and a bar of 4/4 at 200 have identical notation and are felt at different levels of the same hierarchy.

Three numbers, and where they come from #

The preferred rate: about 500 to 600 milliseconds. Asked to tap at a comfortable rate with no music present, people cluster near two taps a second. The distribution is broad — there is real individual variation, and it drifts slowly with age towards slower rates — but the mode is remarkably stable across studies and populations.

The fast limit: about 100 milliseconds. Below roughly a tenth of a second between onsets, a series stops supporting a beat. The events are still individually audible for a while past that, but they cannot be tapped along with and they are not felt as a pulse; the sequence becomes a texture with a rate rather than a beat with a tempo.

The slow limit: about 2 seconds. Above roughly two seconds between events, the sense of a connecting pulse fails. A listener hears a series of separate events rather than a beat, and asked to continue tapping after the sound stops, their timing degrades sharply.

All three are properties of the listener. None of them moves when the music does, and it is worth pausing on how unusual that is in this subject: almost every other quantity on this site is a property of a sound, a string or a system of ratios. These three are not measured on any instrument.

What follows immediately #

Put a notated tempo against that window and the consequences are mechanical.

At 120 beats a minute the notated beat is 500 ms — right at the preferred rate. The listener taps the notated beat, everybody agrees on where the pulse is, and the notation and the feeling coincide. It is not an accident that a very large fraction of dance music sits between about 100 and 140 beats a minute.

At 200 beats a minute the notated beat is 300 ms, which is inside the window but well faster than preferred; two beats is 600 ms, which is bang on it. So listeners tap every other notated beat. A fast piece is felt in half, and the notated bar effectively becomes the beat.

At 40 beats a minute the notated beat is 1,500 ms, near the slow edge. The subdivision at 750 ms is much closer to preferred, so listeners subdivide: they tap the quavers, or feel them, and the notated beat becomes a level above the pulse. This is why very slow music is harder rather than easier to keep together, and why performers of slow movements almost always report counting subdivisions.

Where the felt level changes, exactly #

Three worked tempi are three points, and the rule that produced them — take the metrical level whose duration is nearest 550 milliseconds — has boundaries that can be solved for. A level loses to the one above or below it when it is a factor of √2 away from the preference, so the switches are at 550·√2 and 550/√2 milliseconds and every octave from there:

The notation and the feeling coincide over exactly one octave of tempo, from 77 to 154 beats a minute, and each band above and below is another octave. The three cases this essay works — 40, 120 and 200 — land in three different bands, which is why they behave differently, and the boundaries say where the behaviour changes rather than that it does.

Set that beside the dance figure and the two agree more closely than either was measured to. The tempo surveys put popular dance music between about 90 and 140 beats a minute; the band in which the notated beat is the felt beat is 77 to 154. The dance repertoire sits inside the notated-beat band with a little room at each end, which is what a repertoire written to be counted, taught and danced to should do, and it is a correspondence between a psychophysical constant and a publishing convention that nothing connects.

One thing that does not vary with tempo is worth recording as a negative. The window runs from 100 to 2,000 milliseconds, a factor of twenty, which is between four and five doublings — so at every tempo from 30 to 400 beats a minute there are four or five metrical levels inside it, never three and never six. The amount of metrical ambiguity available to a listener is a property of the window’s width and not of the tempo; what tempo decides is which of those four or five is nearest the preference, not how many there are to choose from.

What a conductor is doing #

The clearest working demonstration of all this is the practice of changing what a beat pattern subdivides.

A conductor beating a fast movement “in one” is giving a single gesture per bar rather than three. Read against the window, that is not a stylistic choice: at 180 beats a minute a notated crotchet is 333 ms and a bar of three is 1,000 ms, and neither is at the preference — but the bar is closer to a rate a body can move at, and the gesture has to be physically producible. Beating three at 333 ms is possible and exhausting; beating one at 1,000 ms is comfortable.

Conversely a conductor subdividing a slow movement is supplying, with a visible gesture, the level the players’ own clocks want and the notation does not name. It is the same act as a performer counting quavers, done for a group.

That is the situation the preferred rate exists to resolve. A piece offers several candidate pulses and the window admits more than one of them, so what picks the beat is not the music’s structure but where in the window each candidate falls — which is a fact about the listener and not about the piece.

That figure is the argument in its most compact form. Everything about the pattern — its proportions, its symmetry, the shape it makes on the circle — is unchanged. What changes is which of its levels lands near half a second.

Why the notation cannot fix it #

A composer can write any tempo and any time signature, and neither is a claim about which level will be felt. That is not a defect of notation; it is a consequence of notation recording proportions while the window is a fact about absolute durations.

The point generalises to something this site has said before from another direction. A groove is unwritable in principle because a fixed physical delay is a different note value at every tempo. The same asymmetry appears here: the tempo window is fixed in milliseconds and the metrical hierarchy is fixed in proportions, so the mapping between them changes with every tempo marking, and nothing in the notation records the result.

Where dance music sits, and why #

The claim that dance tempi cluster in the preferred region is checkable, and it is the strongest external evidence for the window.

Surveys of tempo across popular and dance repertoires find a strong concentration between about 90 and 140 beats a minute — that is, 430 to 670 ms a beat, which brackets the preferred rate closely. The correspondence holds across genres that have nothing else in common, and across periods.

There is a mechanical reason as well as a perceptual one. Dancing is walking with structure, and a comfortable walking pace is about two steps a second, which is the same 500 ms. Whether the tapping preference reflects the walking rate, or both reflect something about the mechanics of a body of that size, is not settled — but the correspondence is close enough that any account of one has to say something about the other.

Whose music. The tempo surveys behind that claim are overwhelmingly of recorded Western popular music of the last seventy years, and generalising from them requires care. What can be said more widely is that traditions with dance repertoires put their dance repertoires at rates a body can move at, which is unsurprising, and that traditions with non-metrical or very slow repertoires — an ālāp, a Japanese shakuhachi honkyoku, a plainchant — exist and are not failures. Music that sits outside the window is music that has declined to have a beat, which is a legitimate thing to do and is done deliberately.

Where it comes from, and what happens to it #

Two facts about the preferred rate’s origin are worth stating because they narrow the possibilities.

It is present very early. Infants show sensitivity to beat at rates in the same broad region as adults, and the preference for moderate rates is present well before any musical training. Whatever sets it, it is not learned from a repertoire in the way that the tonal hierarchy is.

It slows with age. Spontaneous tapping rate drifts slower across the lifespan, by a substantial fraction. So the constant is not fixed for an individual either, and a piece of music sits at a slightly different place in the window for a listener of seventy than for the same listener at twenty.

The combination — early, unlearned, and slowly drifting — points towards something about the mechanics and metabolism of a body rather than towards a specialised musical faculty. Which is consistent with the walking correspondence and with the observation that larger animals move more slowly at every gait.

Which computation produced the numbers #

The window’s three boundaries are quoted from the literature and carried as a small table. The metrical levels laid on it are computed from the tempo: at 120 beats a minute the beat is 60,000/120 = 500 ms and each level is a doubling or a halving of that, so every mark’s position in the figure follows from one number.

The band boundaries come from one further assumption, stated here because it is the only place the preference enters as a rule rather than as a mark on a figure: the felt level is taken to be the one nearest the preference in ratio rather than in milliseconds. That is the right comparison for a quantity whose levels are related by doublings, and it is what puts the switches at √2 either side. Comparing in milliseconds instead would put them at the arithmetic midpoints — 825 and 275 milliseconds rather than 778 and 389 — which moves the notated-beat band from 77–154 to 73–218. The dance repertoire sits inside either, so nothing above depends on the choice; the boundaries themselves do, to about ten per cent at the bottom and forty at the top.

That division is the honest one. Nothing here derives why the preferred rate is 500 ms — that is a fact about people and this site cannot compute it — and everything about where a given piece’s levels fall relative to it is arithmetic.

The site’s habit is that every claim gets a test it could fail, and this figure’s is built in: it reports which levels fall inside the window and which do not, computed rather than asserted. A tempo whose levels all fell outside would be drawn as such, and the assertion that some tempi have no level near the preference is testable by handing the generator one.

What the picture cannot show #

The distribution. Spontaneous tempo is drawn here as a line and it is a broad distribution with substantial individual variation. Two listeners in a room may genuinely tap different levels of the same piece, and it is not a disagreement about the music.

Hysteresis. A listener already entrained to one level tends to stay there when the tempo changes, well past the point at which they would have chosen differently from cold. An accelerando is followed at the same level for longer than the window alone predicts, and then switches abruptly.

Everything about the content. The figure has only tempo on it. In practice which level is felt also depends on where the events are — a level with no onsets on it is much harder to feel than one with onsets — and on harmonic rhythm, which frequently marks a level of its own. A piece can be pushed towards a level the tempo alone would not favour, simply by putting events there.

And the window is not a hard edge. The three numbers are the middles of transitions. Beat perception degrades gradually towards both limits rather than switching off, and the degradation is measured differently by every task — tapping accuracy, tempo discrimination and reported presence of a pulse all fail at somewhat different rates, so “the limit” is three limits reported as one.

The polyrhythm case, which the window sorts out #

A polyrhythm is two rates at once, and the window decides which of them a listener can be in.

A three-against-two at this tempo puts its two layers at 417 and 625 milliseconds — both inside the window and on opposite sides of the preferred rate. Neither layer wins on rate, which is exactly why the choice between them is available to a listener at will and why it can be switched without the music changing.

Now change the tempo and the reversibility disappears. Speed the same figure up until the three-layer is at 150 ms and it drops far from the preference while the two-layer at 225 ms is still awkward but nearer; the ambiguity collapses and one reading dominates. Slow it down until the two-layer is at 2 seconds and that layer stops being a beat at all.

So a polyrhythm’s characteristic effect — two clocks running at once with the listener able to choose — is available only in a band of tempi where both layers are inside the window. That band can be computed for any ratio, and it gets narrower as the ratio gets more extreme: seven against five has its two layers a factor of 1.4 apart and works over a wide range, while seven against two has them 3.5 apart and only works in a narrow one.

The generalisation #

The result is a specific case of the theme running through this whole field: the listener supplies a constant, and the music has to be written around it.

The categories that make temperament possible are a hundred cents wide because of a listener. The window in which two events are one event is a few tens of milliseconds because of a listener. The stability hierarchy that makes a tonic a destination is a statistic accumulated by a listener. And the beat is half a second because of a listener.

None of those numbers is in the music, all of them are constraints the music is written against, and in every case the traditions that flourished are the ones that landed on the right side of them. That is not a claim that perception determines music — a great deal of music sits deliberately at the edges of every one of these windows, and the edges are where the interesting things happen. It is a claim that the constraints are real, measurable, and worth knowing about before making a general statement about what music is like.

Where the ladder goes next #

The metre-induction anchor now has a rung about which metre is chosen and one about which level is felt. What it does not have is the mechanism that keeps time once the choice is made — entrainment as an oscillator locking to an input, which explains the anticipation of taps, the survival of a beat through silence, and the smooth tracking of a tempo change, none of which a preference-rule model can do.

Sideways, the fast edge of this window is the slow edge of another. Speed a rhythm through about twenty events a second and it stops being a rhythm and becomes a pitch; the beat fails at about ten a second, so there is a band between the two where a series of events is neither a rhythm nor a tone. That band is where a good deal of twentieth-century experimental music went looking, and it is a gap in the listener rather than in the sound.