The deviations are not noise

The second rung of this ladder established that a fixed physical delay is a different note value at every tempo, so a consistent feel is unwritable in principle. It took its numbers from two published profiles and treated them as one kind of thing.

They are not one kind of thing. Split each profile into the quantity that is constant across the bar and the quantity that varies by position, and the two profiles come apart completely.

Two quantities that share a word #

The arithmetic is the ordinary one for splitting a series into a mean and a variation about it.

The offset is the average deviation across the bar. It says the whole part is early or late, uniformly. A soloist who sits thirty milliseconds behind the ride cymbal on every beat has an offset of thirty and no pattern at all: the relationship to the timekeeper is identical on beat one and on beat four.

The pattern is the spread about that average. It says which positions in the bar are displaced and by how much. A Viennese waltz’s second beat arrives 55 milliseconds early and its third 25 late, which is a large pattern about an offset of almost nothing.

Reported as “expressive timing of up to fifty milliseconds”, those are indistinguishable. They are not the same phenomenon and they almost certainly do not have the same mechanism.

What each one is doing #

An offset is a relationship between two parts. It only exists if there is something to be behind: a soloist thirty milliseconds behind the ride is thirty milliseconds behind the ride, and playing alone the same soloist has an offset of zero by definition, because there is nothing for the offset to be measured from.

A pattern is a property of the bar. It exists in a solo performance, it is reproducible without any reference part, and it is what a listener would call the shape of the metre rather than the placement of a voice.

That distinction predicts something checkable: an offset should disappear when the reference does, and a pattern should not. A soloist recorded without a rhythm section has no offset to have; a waltz played by one pianist still has its early second beat, because the second beat is early relative to the bar rather than to another player.

Both have to be read against the motor floor #

A deviation is only systematic if it is larger than the deviation the same performer produces when trying to be exact, and that floor has a published size.

Skilled performers asked to play a metronomic sequence produce timing deviations with a standard deviation of roughly 8 to 20 milliseconds, depending on tempo, instrument and how the measurement is made. That is the motor floor, and it is the number every claim about expressive timing has to clear.

The Viennese pattern clears it comfortably. A pattern deviation of 33.4 milliseconds against a motor deviation of 8 to 20 leaves between 74 and 95 per cent of the variation systematic — the range is the range of the floor, not of the finding, and at either end the conclusion is the same.

The jazz soloist’s pattern does not clear it at all. A pattern of 1.9 milliseconds against a floor of 8 to 20 is 1 to 5 per cent systematic, which is to say: the beat-to-beat variation in that profile is motor noise, and the profile’s content is entirely in its offset.

The two quantities do not clear the floor at the same rate #

Comparing each to the motor deviation of one note is the right first check and it is not the whole one, because both quantities are averages and an average has a standard error. The two average down differently, and the difference is structural.

An offset is a mean over every beat, so n beats of music give it a standard error of the motor deviation over the square root of n. A pattern is one mean per position in the bar, so eight bars of four-four give the offset thirty-two samples and each position of the pattern only eight.

From the same eight-bar phrase an offset is establishable at half the size a pattern is, and the factor is exactly the square root of the beats in a bar. So a literature reporting patterns is working with less statistical power per note than one reporting offsets, and a small pattern is a harder claim than a small offset of the same size.

Turned round, the same arithmetic says how much music each of the measured effects needs at three standard errors with a fourteen-millisecond floor: the jazz offset needs three beats, the Viennese second beat one bar, the Viennese third beat three bars — all of them comfortable. The jazz pattern of 1.9 milliseconds would need 489 bars, which is more of one performance than anybody has recorded.

That last figure is the strongest form of the conclusion above. The jazz pattern is not merely below the floor; establishing it as anything other than noise would take a continuous performance about twenty minutes long at the tempo in question, and the measurement it comes from is nothing like that.

That is a real finding about a real dataset and it is only visible once the two quantities are separated.

The same offset is a different quantity at every tempo #

A duration expressed in milliseconds has a fixed physical size and a proportion of the beat that moves.

Thirty milliseconds behind the ride is 4 per cent of the beat at 80 to the minute, 6 per cent at 120, and 10 per cent at 200. Whether a listener notices a lag presumably depends on the proportion rather than the duration, so the same physical habit is a different musical effect across the tempo range — and there is no evidence that players adjust it.

There is a second and larger constraint hiding in that. The beat itself has a preferred rate, and at the fast end of the usable range a beat is 300 milliseconds and a thirty-millisecond offset is a tenth of it, which is comparable to the whole swing displacement. At the slow end a beat is 1,500 and the same offset is 2 per cent, which is near the threshold of anything.

So the offset a player produces is, without any change in behaviour, a strong effect in fast music and a marginal one in slow — which is the sort of prediction that could be tested by asking listeners to detect it at several tempos, and which as far as this site can establish has not been.

What a sequencer cannot do about it #

The clearest evidence that these are two different quantities is what happens when a machine tries to add them.

Quantisation removes both: every onset goes to the nearest grid position, offsets and patterns alike, which is why a quantised performance is described as stiff. The standard remedy — a humanise control — adds random deviation, drawn from a distribution, independently at each onset.

That adds neither. Random deviation is by construction not an offset, since its mean is zero, and it is not a pattern, since it does not depend on position. It is precisely the motor noise that the two real quantities have to be distinguished from, added deliberately.

The correct machine version of either quantity is trivial to write and rarely offered: add a constant to one track, or add a fixed per-position table to every bar. Both are deterministic. Neither is what the word humanise means in any product, which is a small piece of evidence that the two quantities are not widely distinguished outside the measurement literature.

The test the literature actually uses #

Neither of the numbers above is how a study establishes that a deviation is systematic. The real test is replication: record several takes, and ask whether the same positions are displaced in the same directions each time.

That is the right test and it is stronger than a comparison against a floor, because it does not require the floor to be known. If a performer’s deviations replicate across takes far above chance, they are being produced rather than escaping, whatever their size.

It also decomposes cleanly. Across takes, the variance at each metrical position splits into the between-position variance — which is the pattern — and the within-position variance, which is the motor floor measured on the same performer at the same session rather than quoted from a paper.

Which of the three is which #

Putting the three kinds together is worth doing explicitly, because the word microtiming covers all of them and they have different units, different mechanisms and different jurisdictions.

An offset is one part’s constant displacement relative to another. Units: milliseconds. It requires two parts. It is the thing described as laid back, on top, in the pocket.

A pattern is a bar-position-dependent displacement. Units: milliseconds, one per position. It requires only a bar. It is what makes a Viennese waltz a Viennese waltz and is the clearest case of a feel that cannot be notated.

A subdivision ratio is the proportion in which a beat is divided. Units: a ratio, dimensionless. It requires only a beat. It is swing, and it is the one of the three that has a notated approximation.

The third is the odd one out in a way worth stating: because it is a ratio rather than a duration, it is the only one of the three that is not automatically a different note value at every tempo. Which makes it the only one notation has any purchase on, and even then the purchase is poor, because the measured ratio does not stay put as the tempo changes.

What this does to the earlier rung #

The claim rung two made — that a fixed physical delay is a different note value at every tempo, so a consistent feel is unwritable in principle — survives the split intact, and it applies to two of the three kinds rather than to microtiming as a whole.

An offset in milliseconds is unwritable, because it is a duration. A bar-position pattern in milliseconds is unwritable, for the same reason. A subdivision ratio is writable, in principle, and the reason it is not written is different: notation has only a few ratios available, none of which is 2.75:1.

So the earlier rung’s argument was right and slightly over-general. Two of the three kinds are unwritable because they are durations; the third is unwritable because the alphabet is too small.

The third kind’s own picture makes the distinction visible without any arithmetic. A swung bar drawn against the grid displaces its offbeats by a proportion of the beat, so the same drawing at a different tempo is a different picture in milliseconds and the identical picture in proportion — which is exactly the opposite of what the first two kinds do. An offset of thirty milliseconds is a fixed distance that becomes a different fraction of every beat it is measured against; a ratio of 2.75 to one is a fixed fraction that becomes a different distance. Only one of those two can be written down, and the reason the writable one is not written is that notation carries three ratios and none of them is 2.75.

Where the deviations carry information #

There is a use for these deviations that the split makes visible, and it points at the two rungs of the metre ladder that ended in a shortfall.

A bar-position pattern is, by construction, a signal about which position of the bar this is. If the second beat of a waltz is reliably 55 milliseconds early, then a note 55 milliseconds early is evidence that this is a second beat — which is metrical information, carried by timing rather than by presence.

That is exactly the evidence the metre-induction machinery throws away: it takes a list of steps, and a note played early is on the step it was on. The phase ambiguities that ladder found so intractable might be much less ambiguous in a real performance, and the figures that found them were fed quantised patterns.

Whether listeners use it is a separate question with a partial answer: they do, at least for the largest effects, and the experiments that show it are the ones where a passage is quantised and listeners find the metre harder to locate. That is not the same as showing they use a specific table of deviations, and nobody has that.

Whose playing, and how many people #

Both profiles come from a handful of studies of European art music and of American jazz, recorded in the last fifty years, mostly from small numbers of highly trained performers.

The Viennese waltz result in particular rests on a specific tradition of playing a specific repertoire; it is a claim about how Viennese orchestras play Viennese waltzes and not about waltzes. Groups outside that tradition play the same notation with a much smaller pattern, and there is no suggestion anywhere that they are playing it wrong.

The jazz offset is on firmer ground in one respect — it has been found repeatedly, across performers and recordings — and on weaker ground in another, which is that the magnitude varies enormously between players and is one of the things a player is identified by.

A cycle has no first beat to be offset from #

One more thing the split exposes, and it is a limit on where any of this applies.

An offset needs a reference part. A bar-position pattern needs a bar with positions in it — a first beat, a second, a third — which is to say it needs a metre with a phase. Drawn as a cycle, a rhythm has neither: the positions are all equivalent until somebody decides where one is.

So a pattern is only measurable in a repertoire that agrees where the bar starts, and the measured profiles above are all from repertoires that do. For a cyclic timeline the equivalent quantity exists — measurements of West African and Cuban performance find systematic departures from the even grid, and they are consistent within a tradition — but it is a pattern relative to the cycle, and which rotation of the cycle it is stated in is a decision made by the transcriber.

Drawn as a cycle rather than a bar, the twelve-step timeline has no step that is step one — every position is equivalent until somebody names one — so an offset has nothing to be an offset from and a bar-position pattern has no positions. The equivalent quantity does exist: measurements of West African and Cuban performance find systematic departures from the even grid, and they are consistent within a tradition. But it is a pattern relative to the cycle, and which rotation of the cycle it is stated in is a decision made by the transcriber rather than a fact about the playing.

What the picture cannot show #

It cannot show the reference. An offset is measured against something, and the figures here draw the reference as a grid. In a real ensemble the reference is another player who is also deviating, and an offset against a moving reference is a different quantity from an offset against a grid.

It cannot show tempo dependence. Each profile is drawn at one tempo. The same performer at half the speed does not double the deviations and does not keep them constant either; what happens is somewhere between and is measured differently in every study.

It cannot show note lengths. Everything here is onset timing. Expressive performance also varies duration and articulation, which interact with onset timing in ways no onset list records.

It cannot show that the deviation is intended. Systematic is not the same as deliberate. A pattern that replicates across takes is being produced reliably; whether the performer could suppress it, and whether they know it is there, are separate questions and the answers are usually no and no.

It cannot show what an unequal bar does to any of this. A bar-position pattern assumes the positions are evenly spaced, and a bar of unequal beats has positions that are not — so its measured deviations are deviations from a grid that is itself an idealisation, and separating the two is a harder problem than the one solved here.

And the motor floor is quoted rather than measured. Eight to twenty milliseconds is a range from the literature, applied to profiles measured by other people in other rooms. The proper comparison uses the same performer’s own control condition, which is what a study does and what a figure built from published summary numbers cannot.

The ladder from here #

The offset needs two parts, and two parts raise a question this rung has stepped around: how two people who cannot share a timekeeper manage not to come apart. The answer is that each corrects toward the other, and the correction leaves a signature in the asynchrony.

That signature turns out to measure exactly one thing, and it is not the thing everybody wants from it.

And after that, a boundary. The ear sorts pitch into categories with boundaries that can be measured, and time has categories too — the simple ratios. A deviation lives inside one of them, which is why it reads as expression rather than as a wrong note, and the swing ratio turns out to walk straight across two boundaries as the tempo rises.