One player is not two clocks

The first rung of this ladder ends on an observation and does not follow it up. Fluent players of five against four report that they are not counting anything; what they do instead is entrain to one part and place the other against it by feel. The rung then says, correctly, that the least-common-multiple analysis “describes the sound correctly and describes the production not at all”.

The production has two obvious accounts and they are not equally likely. One is that each hand runs its own timekeeper. The other is that there is one timekeeper, running at the composite step, and both hands are scheduled off it.

Those make different predictions, and the difference is enormous.

The two accounts, stated precisely #

Both use the standard decomposition of a timed movement, due to Wing and Kristofferson in 1973: an interval is produced by a timekeeper, which decides when the next stroke is due, and the stroke itself arrives after a motor delay which varies independently from stroke to stroke.

The two-clock account gives each hand a timekeeper of its own. The left hand’s onsets are the running sum of intervals drawn about one third of a bar; the right hand’s are the running sum of intervals drawn about one half of a bar. Nothing connects them.

The one-clock account has a single timekeeper running at the composite step — the sixth of a bar that the composite of three against two lives on — and both hands take their onsets from it. Neither hand has a clock; the pair has one.

The simulations below give the two accounts the same total timing noise per bar, so that they differ in structure and not in how sloppy they are.

A random walk has nothing to come back to #

The two-clock account’s problem is not that it is noisy. It is that its noise accumulates.

Each hand’s onset time is the sum of every interval it has produced so far, so an interval that came out ten milliseconds long is not corrected on the next stroke — it is carried, permanently, into every onset after it. The difference between two such sums is a random walk, and the spread of a random walk grows with the square root of the number of steps.

After twenty-four bars at a hundred to the minute — about fifty seconds of music — the two-clock spread is a hundred and six milliseconds. A hundred milliseconds at that tempo is a sixth of a bar, which is one whole step of the composite grid. The hands are not slightly out; they are a different rhythm.

And it is still climbing. There is no bar number at which it settles, because nothing in the model pulls the hands together.

The one-clock account cannot drift at all #

Under the other account, the only thing that separates the hands is the motor delay on each individual stroke, and a motor delay does not accumulate: it is added to one onset and gone.

So the spread is flat, and its size is set entirely by the motor variance. Set the motor delay to zero and the two hands become mathematically identical at every shared step.

That control is worth running because it identifies which term does what. The drift belongs entirely to the timekeeper structure; the residual spread belongs entirely to the motor delay. They are separable and the figure separates them.

What players actually do #

Measurements of pianists playing three against two, and of drummers playing four against three, put the asynchrony between the hands in the region of ten to thirty milliseconds, and — this is the part that matters — stable. It does not grow over a passage. Players hold these figures for minutes.

The shaded band in the figures is that range. The one-clock account sits inside it at any reasonable motor variance. The two-clock account leaves it before the fourth bar and never returns.

That is about as decisive as a comparison between two models gets, and it is decisive for a reason that has nothing to do with fitting: the two accounts differ in whether their error accumulates, and accumulation is visible over a long enough passage without any parameter being tuned.

What it is not is a comparison between the accounts available, because there is a third and this collection already has it. Two players and no clock is a pair of independent timekeepers that nudge toward each other by a gain, and nothing about that model requires the two timekeepers to be in two people. Two hands with their own clocks and a correction at each bar line is a perfectly ordinary hypothesis, and it is bounded rather than accumulating for the same reason the ensemble model is.

A coupling of a tenth is enough to stop the drift, and it lands the spread at 20 to 32 milliseconds — inside the measured band across the whole passage. At a quarter it sits at 17 and at a half at 11, and by three quarters it is indistinguishable from the one-clock account at 9.

So what the measurement separates is not one clock from two. It is accumulating from bounded, and every account with any correction in it at all is on the bounded side. The two-clock account fails because it has no correction, not because it has two clocks, and adding the smallest correction anybody has measured in an ensemble rescues it completely.

That leaves the rung’s title stronger than its evidence, and the honest version is worth having because it says what a further measurement would need. A one-clock account predicts a specific residual — the motor variance and nothing else, which the zero-motor control above pins at exactly zero — while a coupled account predicts a residual that grows as the coupling weakens. Telling them apart needs the residual’s size rather than its trend, and the size depends on a motor variance nobody here has measured. The trend is decisive and free; the size is what is owed.

How long the two accounts stay indistinguishable #

The interesting number is not the drift after twenty-four bars. It is the bar at which the drift leaves the range measured players stay in, because that is the first moment at which listening could tell the two accounts apart.

For three against two it is the second bar. The two-clock spread is thirty-one milliseconds by then, and thirty is the outside of what has been measured in real playing.

For seven against five it is the first. The drift grows faster at wider ratios — a hundred and eight milliseconds at bar twenty-four for 3:2 against a hundred and sixty-four for 7:5 — because a hand playing seven strokes to the bar takes seven independent timekeeper draws per bar rather than three, so its sum accumulates variance faster.

So the two accounts are not hard to distinguish and they do not need a long passage. They need about four seconds. That is a strong result and it is also a warning about the simulation: if uncorrected two-clock playing were this bad, nobody would ever have proposed it, and the version worth arguing with is always the corrected one.

What a listener would hear #

A hundred milliseconds is not a subtle quantity. It is roughly the shortest gap that can be a beat at all, it is three times the asynchrony at which two attacks stop fusing into one event, and at a bar of 1.2 seconds it is a full step of the composite grid.

The composite is the reason the drift is audible rather than merely measurable. Its gaps are one and two and three steps, and a drift of one step turns a gap of one into a gap of zero. The sequence of durations that the figure is — the thing a player has learned and a listener recognises — stops being that sequence.

Which is also why the drift cannot be dismissed as a laboratory artefact. It is not a claim that two-clock playing would sound imprecise. It is a claim that it would sound like a different rhythm, within a few seconds, every time.

The measurement that cannot decide #

There is a second signature in the Wing–Kristofferson decomposition and it is the one that gets cited: the lag-one autocorrelation of a hand’s own inter-onset intervals is negative.

The reason is easy. A motor delay that comes out long pushes one onset late, which makes the interval before it long and the interval after it short. Consecutive intervals are therefore anticorrelated, and the size of the effect is the motor variance over the timekeeper variance plus twice the motor variance — a quantity that cannot be positive and cannot go below minus a half.

The slate for this essay assumed that this would separate the two accounts — that one clock would give a negative value and two would give zero. It does not, and the reason it does not is obvious once written down: both accounts have a motor delay after a clock. The negative autocorrelation is a signature of that arrangement, not of how many clocks are in it.

So the number that the literature reaches for as evidence of an internal timekeeper is evidence of an internal timekeeper, and it is silent on the question this rung is about. The two accounts give values a little apart — the one-clock case runs more negative, because the timekeeper variance seen by one hand is smaller under it — but the difference is a function of how the variance was divided up, which is a modelling choice rather than a measurement.

The drift decides and the autocorrelation does not, and a measurement that decides nothing is worth identifying as such. What it does measure is worth having — it is the only route to the motor and timekeeper variances separately, from timing data alone, which is why it was invented — and it is a route to a different quantity than the one this rung needs.

Why the composite has to be represented #

If the one-clock account is right, something has to be holding the composite, because that is what the clock is running on.

That fits what the previous rungs found uncomfortably well. The mnemonics taught for these figures are mnemonics of the composite. What learners are drilled on is a durational sequence. Performers trained on one ratio are documented as not generalising to a neighbouring one — which is what one would expect if what was learned is a specific sequence of durations rather than a general capacity to divide.

Three against two is notated as two layers and described as two layers, and if the one-clock account holds then that drawing is a picture of the notation rather than of the performer — for whom there is one line of four unequal gaps and no second layer anywhere.

The awkward part is that this contradicts what players say. Asked what they are doing, fluent players describe holding one part and feeling the other against it — two things, not one. That report is evidence and it should not be waved away, and the honest reading is that the introspective report and the timing data are about different levels: what is attended may well be one hand while what is scheduled is the composite.

Where a third account lives #

Two hands belonging to one person is the easy case, because there is a plausible common clock. Two people is not, and the models above have nothing to say about it.

Measured deviations against a grid run to tens of milliseconds and are systematic rather than noise — which is a different claim from anything here, and is where the microtiming ladder starts rather than where this one does.

Two players cannot share a timekeeper and they do not drift apart either, which means something is correcting. That third account — mutual error correction, where each player adjusts toward what they heard the other do — is a real model with its own signature, and it belongs to a different anchor on this site. It is worth naming here because it shows that the dichotomy this essay set up is a dichotomy about one performer, and generalising it to an ensemble would be wrong.

Where the model stops #

Nothing here is a measurement. Every number in this essay comes out of a simulation with parameters chosen by hand — a timekeeper deviation of ten milliseconds, a motor deviation of six. Those are the right order of magnitude for the published literature and they are not fitted to any dataset. What the simulation establishes is a qualitative difference in behaviour that no choice of parameters removes; it establishes nothing quantitative.

The two-clock account is a straw man in one respect. Nobody claims that a performer runs two clocks with no feedback whatever. The realistic version has each hand’s clock corrected toward the other, which is the ensemble model above, and which behaves like the one-clock account over long spans while differing from it over short ones. What has been ruled out is uncorrected independence, which is worth ruling out because it is the picture the notation invites.

One shared onset per bar is doing a lot of work. The error is measured at the point where both hands are supposed to strike together, which happens once a cycle. A model in which the hands are corrected only at that point would also stay bounded, and would not be a single clock. Telling those apart needs the within-bar onsets measured against a fitted grid rather than against each other, and that is a harder experiment than the one this figure describes.

The measurement point is once a bar. Everything above measures where each hand’s own bar-start falls, which is one sample per cycle. A drift that is slow compared with a bar is invisible to a measure taken once a bar, and one that is fast is aliased by it. The figures are honest about the quantity they draw and it is a coarse quantity.

And the composite is an inference, not an observation. No figure here shows a representation. What the drift argument establishes is that the hands’ errors do not accumulate independently; that they are scheduled off a common list is the most natural way for that to be true and it is not the only way.

Whose playing this is about #

The measurements cited are of conservatory-trained pianists and percussionists playing notated polyrhythms in a laboratory, mostly in Europe and North America, mostly since 1980. The pedagogy the mnemonics come from is anglophone conservatoire teaching of roughly the same period.

That is a narrow base, and it is narrow in a way that matters here. In traditions where the layers are played by different people, the one-clock account is unavailable by construction — there is no shared timekeeper to be had — and the coordination has to be done by correction. The bell-and-ensemble arrangement is the clearest case: nobody there is playing three against two with two hands, and the whole texture is people listening to each other.

So the finding is that one performer producing a polyrhythm is probably running one clock. It is not a finding about polyrhythm.

The ladder from here #

This ladder began with two cycles that coincide every so many steps, and every rung since has assumed that they do coincide. The last one asks what happens when they never do — when the ratio of the two cycles is not a ratio of whole numbers at all, so there is no common grid, no composite and no lowest common multiple.

The answer is that the leftover has a name, it is 23.46 cents, and it has had an essay on this site since the very first phase.

Two things this rung has left open are worth naming. Whether the beat a performer is holding is the one the rules would choose is not established by anything here — the clock in the one-clock account runs at the composite step, which is nobody’s beat. And a performance in which the whole point is that the parts do drift, as in a phase piece, is a case where two clocks is not a wrong model of the performer but a correct description of the score.