Field

Rhythm and metre

Time divided, subdivided and deliberately misaligned — drawn as a circle rather than a line.

E(3, 8) as a cycle. A rhythm drawn round a circle, one equal arc per step, with the struck steps filled. On a circle the pattern has no beginning, which is why the same necklace of onsets is several different named rhythms depending on where a listener decides bar one is.

Rhythm is a circle, and the bar line is a choice

Draw a rhythm as a line and it looks like a sequence of decisions. Draw it as a cycle and the same pattern turns out to be shared across continents, differing only in where somebody decided to start counting.

Six rhythms, all from the same construction. Euclidean rhythms generated by Bjorklund's algorithm, which spreads a number of onsets as evenly as a number of steps allows. The patterns were not transcribed from recordings; they are what the algorithm produces, and the names beside them are where each one is already in use.

As evenly as possible, which turns out to be a famous rhythm

Ask an algorithm to space five strikes over eight beats as evenly as it can. It produces the cinquillo. Ask for three over eight and it produces the tresillo. Nobody told it about Cuba.

3 against 2. Two evenly spaced pulses over the same span, one in 3 and one in 2. They agree only where both grids land together, and the gap between agreements is the least common multiple — which is what makes one polyrhythm sound like a shape and another like two unrelated things.

Two clocks at once, and where they agree

Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number.

How much swing there is, against how fast the music goes. The ratio of the long note to the short note of a swung pair, as a function of tempo, derived from the finding that the short note holds a roughly constant hundred milliseconds. The notated readings are horizontal lines, and each is correct at exactly one tempo. Above about three hundred beats a minute the ratio reaches one and the swing has gone.

Swing is a ratio, and it is not two to one

Swung quavers are notated as a triplet figure, which is a ratio of two to one. Measurements of what drummers actually play give anything from three and a half to one down to one to one, and which of those it is depends almost entirely on the tempo.

Three metres as trees. Three metres — 4/4, 2 + 2 + 2 + 2; 6/8, 3 + 3; 5/8, 3 + 2 — each drawn as the bar dividing into beats and the beats dividing again. Simple and compound metres have the same number of beats and differ in the second division; an additive metre has beats of unequal length, which no single division produces.

Beats of unequal length

A bar of nine in Balkan practice is not nine of anything. It is four beats, three short and one long, and the inequality is at the beat level rather than inside it — which is a thing no single division of a bar can produce.

$Where the beats actually fall. Measured timing deviations from a strict grid, in milliseconds, for three published profiles. The right-hand column converts each deviation into the note value it would have to be written as, at three tempi — and because a fixed number of milliseconds is a different fraction of the beat at every tempo, no single notated rhythm describes any of these.$

The milliseconds that are the groove

A Viennese orchestra plays the second beat of a waltz about fifty milliseconds early, every bar. A jazz soloist sits thirty behind the ride cymbal. Neither deviation can be notated, and not because notation is coarse — because it measures the wrong quantity.

3 against 2, from a rhythm to an interval. The same 3:2 pattern at 6 rates, an event rate from 1.5 to 300 per second, on a logarithmic axis. Below about 20 events a second the two streams are counted; above about 40 they are heard as two pitches a 3:2 apart, which for 3:2 is 702 cents. Nothing in the pattern changes across that boundary. The slowest rate drawn is a tempo of 90 events a minute; the fastest is a pitch of 300 Hz against 450 Hz.

A rhythm fast enough to be a chord

Three against two is a polyrhythm. Speed the same pattern up until the events arrive faster than about twenty a second and it is a perfect fifth. The ratio never changed, the figure never changed, and the only thing that moved is the rate — which makes rhythm and pitch one continuum with a perceptual boundary across the middle of it.

One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -4, 3/4 12, 6/8 12, 12/8 as 3+3+3+3 12, so 3/4 and 6/8 and 12/8 as 3+3+3+3 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

The beat is inferred, and sometimes wrongly

A bar line is not in the sound. One sequence of onsets supports a reading in four, in three and in six, and which one a listener settles on comes from a small set of preference rules rather than from anything in the signal — which is why the same recording can be heard two ways and why nobody can be talked out of either.

The metrical levels of 120 bpm, against the window. The range of inter-onset intervals that can be heard as a beat at all, from about 100 to 2000 milliseconds, with the preferred rate near 550. Each mark is one metrical level of a piece at 120 beats a minute. Which of them a listener taps is decided by which falls nearest the preferred rate, not by which one the notation calls the beat.

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

Metre is a hierarchy of levels and only one of them is tapped. Which one is decided by the listener's own clock rather than by the time signature: the window in which a series of events can be a beat at all runs from about a tenth of a second to two seconds, with a marked preference near half a second — so at the extremes of tempo the notated beat and the felt one part company, predictably.

Twelve stages, and the composer chose the rule. Every rotation of a 12-step pattern with 8 onsets against the unrotated original. Each row is one stage of a phase piece: the filled cells are what is heard when the two parts sound together, and the count beside it is how many of the two parts' onsets coincide. The number of stages is the length of the pattern, so the length of the piece is arithmetic.

A process that enumerates its own form

Take a twelve-step pattern, play it against itself, and move one copy along by one step at a time. The piece is over when the copy returns to where it started, so its length is twelve — arithmetic, not a decision. What is heard at each stage is the union of the two parts, and nobody composed any of it.

The third pattern nobody played

Two players play two even pulses and a third rhythm arrives that neither of them played. It has a + b − 1 onsets, its gaps read the same forwards and backwards, and it uses exactly min(a, b) different lengths — which is a better account of why 3:2 is a figure and 7:5 is weather than the number used for the last three essays.

One pattern, four metres. The same 6-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are the 3 is the beat 8, the 2 is the beat 4, every step is a beat 8, so the 3 is the beat and every step is a beat tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

Which of the two is the beat

A polyrhythm is notated as a bar of two with three laid across it. Run the metre rules used here over the composite and they choose the three — at every ratio tried, without exception. Add the one other rule the rules have, and the answer changes hands at a bar of 1,013 milliseconds, which is 118 to the minute.

The bell against every division of its cycle. A 12-step pattern of 7 onsets drawn round a circle, with 4 equal divisions of the same cycle as inner rings. Each ring's beats are filled where the pattern strikes them: 1 of the 2, 2 of the 3, 2 of the 4, 3 of the 6. No division has all of its beats struck and none has none of them, so the pattern belongs to no one of them and can be played against any of them. That is what a cross-rhythm is here: one pattern against the beat, and against more than one beat at once, rather than two patterns against each other.

The bell is not a polyrhythm

Five earlier essays have set one pattern against another. In the practice the word cross-rhythm was borrowed from, there is one pattern and it is played against the beat — against several beats at once. Counted against the four divisions of a twelve-cycle, the Ewe bell marks half of the two, two-thirds of the three, half of the four and half of the six, and all of none of them.

Where the two accounts part company. The spread of the error between the hands, in milliseconds, against bar number, averaged over 120 seeded runs of each model at 100 bars a minute. With one timekeeper it is flat at about 9 ms after 24 bars; with two it reaches 106 ms and is still climbing, because a random walk has nothing to return to. Measured players hold 3 against 2 inside about 25 ms indefinitely.

One player is not two clocks

Two accounts of a pianist playing three against two, simulated from the same noise. With a timekeeper in each hand the hands drift apart by a hundred milliseconds inside two dozen bars. With one timekeeper they never drift at all. The measurement everybody cites as evidence for a timekeeper turns out to be the same number under both accounts.

One pattern, four metres. The same 8-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are downbeat on step 1 -1, downbeat on step 2 -7, downbeat on step 3 -1, downbeat on step 4 -1, so downbeat on step 1 and downbeat on step 3 and downbeat on step 4 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

The right period at the wrong phase

Finding the beat is two problems, not one. How far apart the beats are, and where the first one is. This site's rule set answers the first confidently on a tresillo and returns a three-way tie on the second — the same score for the downbeat on step one, step four and step seven — which is not a near miss but no answer at all.

Syncopation against a bar of 8. An 8-step pattern with 3 onsets, against the metrical weights of its bar. A position's weight is zero on the downbeat and one lower at each level down the subdivision tree, drawn here as the depth of the bar hanging beneath it. A note on a weak position followed by a rest on a stronger one costs the difference. the tresillo, in a bar of eight scores 2 — the note at step 4 against the rest at step 5, costing 2.

Syncopation is a number about the metre

Longuet-Higgins and Lee price a syncopation at the metrical weight a note skips over. That makes it computable, and it makes it a property of a pair rather than of a rhythm — the son clave scores 4 read from step one, 2 from step three and 8 from step four, and not one onset has moved. Worse, the induction rules pick very nearly the reading that scores lowest.

Two metres as trees. Two metres — 9/8 as 3 × 3, 3 + 3 + 3; 9/8 as 2+2+2+3, 2 + 2 + 2 + 3 — each drawn as the bar dividing into beats and the beats dividing again. Simple and compound metres have the same number of beats and differ in the second division; an additive metre has beats of unequal length, which no single division produces.

No term for an unequal beat

A Balkan bar of nine is four beats, three short and one long. The induction rules lay their strong positions at a fixed period, so the only readings of nine it can offer are nine, three and one — and for a bar of seven, where seven is prime, the only readings are seven and one. The right answer is not among the candidates.

The deviations are not noise

Two published timing profiles, split into the quantities they actually carry. A jazz soloist thirty milliseconds behind the ride is almost pure offset and has no pattern at all. A Viennese second beat is almost pure pattern and has no offset. They are different things, they are reported under one word, and no figure of total deviation tells them apart.

The gap that decides whether two clocks are two. The closest approach of the two streams in each ratio, at 100 to the minute in 4-time — a bar of 2.4 seconds. It is the bar divided by the product of the two numbers, which is an identity and is checked here against the measured minimum. 9:5 is the last ratio whose onsets are securely separate at this tempo; past it the two streams' events fall inside the window in which the ear cannot put two onsets in order, and what is heard is one irregular pattern rather than two clocks. The bound has a product in it, which is why 3:2 and 4:3 are everywhere and 11:7 is a notation.

The ratio that stops being two

Eight earlier essays have taken two clocks at a rational ratio to be a thing a listener can hold. There is a ratio past which it is not, and the bound is not where anyone would look for it: the cycle is exactly one bar long at every ratio, so the length of the pattern separates nothing. What separates them is the closest the two streams ever come, which is one part in their least common multiple — 400 milliseconds for three against two, and seventeen for thirteen against eleven, which is not two events at all.

The rotation the necklace cannot see

Ask Bjorklund's algorithm for the world's timelines and the usual answer is that it produces them. Search every rotation of each Euclidean pattern for a match and the answer is more interesting: the tresillo is E(3,8) exactly, the bossa-nova and the standard bell pattern are rotations of theirs, and the son and rumba claves — the two best-known timelines in the world — are not Euclidean at any rotation whatever. Where the match does hold it holds up to a starting position, and a starting position is the one thing a timeline is.

One pattern, four metres. The same 12-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 3/4 -4, 6/8 12, 2/4 -4, so 6/8 wins. The page prints 3/4, which the model does not prefer — it ranks 6/8 above it. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

The time signature is a claim

A bar line is not a measurement. It is a claim about where the accents are, made before the sound exists, and there is a metre-induction model here that can be handed the same onsets and asked whether it agrees. On a hemiola it does not: the page says three and the model says six, by a margin of twelve against minus four. And on a bar of seven the signature is not even a candidate — what decides the reading is the beaming, which the signature does not contain.

The note that has a length

Every melodic figure so far reads a table where each note is a pair — a pitch and a duration — and throws the second number away. A step between two minims and a step between two quavers have been one event in every histogram it has drawn. Weighting the same statistics by time moves the step distribution by up to seven points, changes forty-four of a hundred and one contour signs, and turns up an off-by-one in the one figure that did use the durations: it took the length of the note arrived at where the time between two onsets is the length of the note left.

How unequal a beat is allowed to be

A Balkan bar of nine is four beats, three short and one long, and the long one is always three subdivisions against two. Never four against two, never five against three. The usual explanation is that this is what the tradition does; there is a better one and it needs no perceptual measurement at all. A group of four subdivisions decomposes into two twos and a group of five into a two and a three, so neither is a beat — it is two. The only indivisible lengths are two and three, which fixes the ratio at three to two before anybody listens, and leaves the tempo window to decide only how fast the bar can go.

5 onsets in 16, by evenness against locatability. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how much of the cycle has to be heard on average before its position is known. The two objectives are opposed — the perfectly even pattern is the hardest to locate and the most clustered ones are the easiest — so there is no best pattern, only a frontier, and 22 of the 273 are on it. son clave and the bossa-nova pattern are among them. The named timelines are marked.

What the clave buys with its unevenness

The essay before this one found that the two best-known timelines in the world are not Euclidean at any rotation, and asked whether they maximise something else. They do: how quickly a fragment of the cycle says where in the cycle it is. The son clave locates itself in nine of its sixteen steps where the even pattern needs fifteen — and of two hundred and seventy-three patterns, twenty-two are on the frontier between the two objectives and the son clave is one of them.

A duration category has a tempo range of its own. Each simple ratio's short note is the beat divided by one more than the ratio, so at a high enough tempo it falls under the fastest interval that can be a beat at all — 100 milliseconds. Each bar here runs from the slowest tempo at which the ratio's long note still belongs to a beat to the fastest at which its short note is still a note: 1:1 ends at 300 bpm, 2:1 ends at 200 bpm, 3:1 ends at 150 bpm, 4:1 ends at 120 bpm. The line is this site's swing curve, and where it crosses a category boundary the category it is leaving has already ceased to exist.

The short note is sitting on the floor

Swing is modelled here as a short note of constant absolute length, which was measured from drummers and left as a fitted parameter. The tempo window's fast edge — the shortest interval a series of events can be a beat at — was measured from listeners tapping. Both are a hundred milliseconds, and if that is not a coincidence then the swing ratio has no free parameter in it at all: the short note is not held constant, it is resting on the floor.

One envelope, and the three places a listener might be said to hear it. The amplitude envelope of a note with a 90 millisecond exponential attack, with the three criteria the literature offers drawn across it. The heard moment is 8.0 ms at the detection criterion, 28 ms at the perceptual-onset criterion and 94 ms at the perceptual-attack criterion. The physical onset is at zero on this axis and no criterion puts the heard moment there. The buttons play this attack against a two-millisecond one, started at the same instant.

A note is heard after it starts

Every rhythm essay until now has treated a note's onset as the moment it happens. It is not: the instant a listener aligns a note with a beat is later than its physical start by an amount the note's own attack decides, and for a sung or bowed note that amount is about thirty milliseconds — the size of the whole quantity six essays on microtiming set out to measure.

The distribution that can be measured is not the one the player has. A player aiming at a short note of 100 milliseconds with a standard deviation of 15, against a floor at 100. The pale curve is what the player is doing and the heavy one is what can be recorded, because the 50 per cent of the parent below the floor arrives at the floor instead. The measured mean is 112.0 milliseconds rather than 100 and the measured standard deviation is 9.0 rather than 15. At 160 beats per minute that turns an intended swing ratio of 2.75 into a measured 2.35.

A quantity resting against a wall

The short note of a swung pair was found sitting exactly on the fast edge of the tempo window. If that edge is a floor rather than a fitted number, then every swing statistic computed until now was computed on a censored sample — and a censored sample has a mean that is 0.80 standard deviations too high, a spread that is 40 per cent too low, and a correction gain that can come out twice what the players actually have.

The heard moment against the pitch, on an instrument whose own attack is 8 ms. A note cannot establish an amplitude in less than 4 of its own cycles, so the attack has a floor of 4 periods — 145 milliseconds at A0 and 1.9 at C7. Below A4 the floor is longer than the instrument's own attack and the pitch decides the heard moment; above it the instrument does. The lag runs from 46.0 milliseconds at the bottom to 2.5 at the top, a spread of 44 milliseconds that no player can play their way out of.

A low note cannot start on time

Three earlier essays have held the pitch at one value. A note cannot establish an amplitude in less than a few of its own cycles, so the attack has a floor that rises as the pitch falls — 146 milliseconds at the bottom of a piano and three at the top. On an instrument whose action takes eight milliseconds everywhere, that is a forty-three millisecond spread across the keyboard from the period alone, and no player can do anything about it.

How far a wall moves the mean, in spreads. The bias a floor puts into an observed mean, in units of the parent's own spread, against how far the parent sits above the floor. With the mean exactly on the wall the truncated bias is 0.797 spreads and the censored bias is 0.399 — half of it, exactly, because half the mass sits at one point and the other half is an upper half-normal. The earlier essay used the upper figure for a process that produces the lower one, so every bias it quoted is twice what a floor on execution actually causes.

The shape a wall leaves behind

A floor on execution biases every statistic computed on swing timing, and the bias was priced with Pearson's truncated-normal formulas. Those are the formulas for a sample with everything below the wall thrown away. A player who cannot execute a short gap does not throw the attempt away — it comes out at the floor. That is a censored sample, its bias is exactly half, and its skew is two thirds larger.

Which notes of a scored chord have to be played early. Four parts of one chord, each with its own instrument, its own pitch and its own dynamic, and the perceptual centre that comes out of all three. piano, sforzando on E1: an attack family of 8 milliseconds against a pitch floor of 97, so the pitch is what limits it, shortened by the dynamic to 65, heard 20.5 after it starts and needing to be played 12.0 early; flute, quiet on A5: an attack family of 60 milliseconds against a pitch floor of 5, so the instrument is, shortened by the dynamic to 69, heard 21.8 after it starts and needing to be played 13.2 early; violin, mezzo forte on E4: an attack family of 90 milliseconds against a pitch floor of 12, so the instrument is, shortened by the dynamic to 90, heard 28.5 after it starts and needing to be played 19.9 early; trumpet, forte on A3: an attack family of 30 milliseconds against a pitch floor of 18, so the instrument is, shortened by the dynamic to 27, heard 8.6 after it starts and needing to be played 0.0 early. The spread is 19.9 milliseconds, which is well above the two or three a listener resolves, so a conductor asking for these four to sound together is asking for four different physical onsets.

Which notes have to be played early

There are three separate contributions to one quantity — the instrument's attack family, the dynamic it is played at, and the note's own period — and every figure so far varies one and holds the others. Added together for a real scoring they do not add: a sforzando low piano note is pitch-limited to a hundred-millisecond attack and the sforzando shortens it back to sixty-five, so flattening the dynamics makes the ensemble's spread larger rather than smaller.

A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players.

A cycle that says where it is

Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

$The violin's attack times, string by string. Every playable cell of the earlier map, with its bow-force window turned into a time. A player aiming at the geometric centre of the window has to wait until the accelerating bow's maximum force rises to that value, which is a fraction one over the square root of the window width of the way through the bow's ramp — so a narrow window is a late note and the exponent is a half. The latest cell is C♯4 on the G3 string at 21.9 milliseconds, against 4.1 at C6 on the A4: a factor of 5.3 in time out of a factor of 29 in window width. The darkest cell is the same hardest place, unchanged — the map is the same map under a monotone change of units, and what is new is that the units are milliseconds, which a player and a listener both have access to.$

The hardest place is also the latest

A bow-force window is a ratio of forces, which nobody can hear. An attack time is milliseconds, which a player and a listener both have. The map of the violin's windows becomes a map of its attack times under a change of units, and the exponent turns out to be a half — so a window twenty-nine times narrower is only five times later.

The same tune read at six widths of the psychological present. A later essay made the detector's smoothing width a function of position, which removed its last free parameter but one — and the one it cannot remove is the width of the psychological present, because that is a fact about listeners rather than a choice. So the honest object is not a reading but a family of them, one per width. A short present finds 6 boundaries and a long one finds 2, and the family agrees on 0 of them. The fixed-width control, at its own best width, scores 0.67 against the adaptive readings' 0.67, 0.75, 0.33, 0.33, 0.40, 0.40 — so the adaptation does not win, which is what that essay reported too. What the family adds is the ordering: a boundary in every row is a different claim from one in a single row, and a single reading has no way to say so.

A family of readings

Removing the detector's free parameter, and then its constant tempo, cost persistence both times — the property that made its boundaries ordered rather than merely found. Recovering it means a family of adaptive readings rather than one, indexed by the width of the psychological present, which is the one parameter that cannot be removed, because it is a fact about listeners.

An ensemble finding an asynchrony nobody told it about. An earlier essay produced a map of required leads — which notes of a scoring have to be played early, and by how much — and nothing tells the players those numbers, because they are a property of the instruments' attacks rather than of the music. So an ensemble has to find them, and the mechanism is already here: each player hears sounds rather than onsets and moves their next onset toward the mean of the others'. The spread of arrival times starts at 20 milliseconds and settles at 4, crossing 5 milliseconds after 5 beats — about 1.3 bars of four. The leads it converges on match that map to within 0.2 milliseconds, which is what makes this a convergence rather than a coincidence: the fixed point of players listening to each other is every player leading by their own attack.

How many bars an ensemble needs

The map of required leads is something nobody tells the players, because the leads are a property of the instruments' attacks. So an ensemble has to find them, and the mechanism is the one the microtiming essays describe: each player hears sounds rather than onsets and moves toward the others. It converges on the map to within a fifth of a millisecond, in five beats, and there is a best correction gain.

Settling and being heard are not the same quantity. Across, how long the instrument takes to reach its steady amplitude, computed from its own physics — a resonance's Q, a bow's capture, an exciter's contact. Up, how long after its physical onset a listener places the note, computed from the measured shape of its envelope. Five instruments both accounts hold. The diagonal is where they would agree and nothing is on it. The ratio between them runs from 0.10 to 6.6, a factor of 69, and it sorts perfectly by mechanism: about 6.6 for a struck or plucked string, 2.2 for a bowed one, and about 0.15 for a wind. Ordering the five by each measure changes the place of 3 of them, and the one that moves furthest is the violin — third slowest to settle and the last to be heard.

A note starts twice

One account computes how long an instrument takes to settle, from its own physics. Another computes how long after its onset a listener places a note, from the shape of its envelope. Both come out in milliseconds and neither has ever been shown the other. Paired on the five instruments they share, the ratio between them spans a factor of sixty-nine and sorts perfectly by mechanism — and the violin is third slowest to settle and the last to be heard.

The passage that separates them, and a listener cannot hear it. A scoring changes at the halfway bar, and the two maps of required leads differ by 18.1 milliseconds at their widest. An ensemble that has internalised the map applies the new one on the first note of it and its spread never leaves zero. An ensemble that is listening to each other has to re-converge: its spread jumps to 11.8 milliseconds and takes 3 beats to get back under 5. The dashed line is twenty milliseconds, which is what a listener notices — and the disagreement never reaches it. So the two accounts are separable on a recording and very nearly not separable by ear, which is why nobody has noticed the distinction and why the measurement is worth making.

The passage that separates two players

An ensemble that has learnt where the asynchronies are applies them; one that is listening discovers them. In steady state the two are identical, which is why nobody has separated them. Change the scoring mid-phrase and they are not: one ensemble is wrong by twelve milliseconds for three beats and the other is not wrong at all — and twelve milliseconds is under what a listener notices and far above what a microphone resolves.

How long a bar of unequal beats can be, at a present of 3.5 seconds. Two quantities against the number of subdivisions in the bar. The bars are how many genuinely distinct unequal metres that length admits — groupings of twos and threes, up to rotation, discarding any that repeats a shorter grouping — and they run from one at 5 to 28 at 23. The line is how good a beat the best of those metres can manage once the whole bar is required to fit inside a psychological present of 3.5 seconds. It is flat at 0.865 up to a bar of 16 units, which is where the bar at the best subdivision first overruns the present, and falls after it: 17 at 0.830, 18 at 0.797, 19 at 0.765, 20 at 0.735. The supply of metres is still growing where the quality has begun to fall, so the lengths a tradition can use are a bounded prefix of an unbounded list.

How long a limping bar can be

The bound on an unequal beat turned out to be arithmetic, and the tempo window was left with only the tempo to decide. It decides nothing: every metre built from twos and threes gets the same answer, because the window is asked a yes-or-no question. Graded instead, an unequal beat costs 0.135 of the window's own preference at every bar length — and the constraint that does depend on length is the one nobody applied, that the whole bar has to fit inside the psychological present. At the subdivision that suits both beats best, a bar of sixteen units just fits and a bar of seventeen does not, which is where the supply of distinct metres has only started to grow.

Where a cycle of 16 at 16, 8, 4 outruns the listener's memory. The residual uncertainty a listener is left with once the evidence has stopped accumulating, against how long one turn of a 16-step cycle takes. The listener's memory of a step halves after 3.5 seconds throughout; what changes is how many steps that is. At a cycle of 1.6 seconds it is 35 steps and every design reaches certainty, which is the regime a clave is played in. At 60 seconds it is 0.93 steps and none of them does: layers at 16, 8, 4 settles at 1.89 bits, son clave settles at 2.27 bits, the bossa-nova pattern settles at 2.38 bits, the best single line of 7 settles at 1.85 bits. That is the range a gong cycle occupies, and it is the design that wins there.

The cycle that outruns the memory

A timeline and a colotomy were compared at equal strokes and the comparison had no clock in it. A memory span is a number of seconds and a cycle is a number of steps, so the two only meet through a tempo — and at a clave's two seconds a listener's memory covers twenty-eight steps and forgets nothing, while at a gong cycle's forty it covers 1.4 and forgets almost everything. The single line is the better locator up to twenty-three seconds a cycle and the layered code is better after it, which is very close to where each is actually used.

Five rulers on 5 onsets in 16, and what each frontier keeps. The same 273 rotation classes and the same locating cost, with the frontier recomputed against five different definitions of unevenness. The frontier's size varies from 7 to 22 of 273, so how exclusive membership is depends on the ruler. Of the named timelines of this size, son clave is on 4 of the five, rumba clave is on 0 of the five, the bossa-nova pattern is on 4 of the five. The four rulers that measure how large a pattern's departure from even is rank the whole census together, at no worse than 0.89 between any two of them; the one that counts how many distinct gap lengths a pattern uses rather than how large its departures are is anti-correlated with them at -0.26, and its frontier holds none of the named timelines and does not hold the perfectly even pattern.

The frontier and the ruler

A Pareto frontier is a claim about two quantities and only one of them was measured. The locating cost is exact; the unevenness it was traded against was one choice among several. Under four different rulers the son clave stays on the frontier and the frontier shrinks from twenty-two of 273 patterns to seven, so its membership means more rather than less — but the bembe stops being dominated, the two claves stop being equally uneven, and one plausible measure destroys the result entirely.

The bias a wall puts in a mean, against the parent it came from. The bias a wall puts into an observed mean, in spreads, for each of eight standardised parents, with the wall exactly on the parent's mean. The censored values run from 0.354 to 0.433, a range of 0.079; the truncated from 0.582 to 1.000, a range of 0.418. The formula now used is the one that hardly depends on a distribution nobody has measured, and the formula it replaced is the one that depends on it a great deal.

The parent nobody measured

Every figure drawn behind a wall so far assumed a normal parent, and the assumption turns out to matter in exactly the wrong place. The censored bias is half the parent's mean absolute deviation — a theorem, not a coincidence — so it lands between 0.35 and 0.43 spreads for every distribution tried, and the truncated one runs from 0.58 to 1.00. But the third moment proposed as the test moves 2.9 across parents against 0.65 between the two rules, and a censored sample from a slightly left-skewed parent has a skew of 1.007 where a truncated normal has 0.995. The statistic that does work is a count of ties.

What a slow start costs a trumpet, in its own settling time. The amplitude of a trumpet's A4 resonance — the 4th impedance peak, of Q 37, time constant 27.1 milliseconds — driven from rest by a pressure that rises over 10, 40, 80, 140 milliseconds, against the step every earlier figure has assumed. The step reaches 90 per cent of its final amplitude in 62.5 milliseconds. A ramp of 140 takes 157.8, which is 95.3 more — and that excess is 68 per cent of the ramp's own length. Across the whole range a player works in the excess is a little over half the ramp: 0.52, 0.56, 0.61, 0.68 at 10, 40, 80, 140 milliseconds. So the tongued attack is not something added to the note. It is the step, which is what has been computed all along, and what has a price is its absence.

What the tongue actually removes

The question this essay was written against asked for an impulse: a tongued attack, a martelé stroke and a struck key all deliver one before the steady drive begins. The arithmetic refuses the framing. A tongue release does carry energy at the note's own frequency, and it is worth 0.17 milliseconds on a trumpet against a settling time of 62 — capped at about 1.13 over the resonance's Q. What articulation is worth is the ramp it removes, and that is about half the ramp's own length: 44 milliseconds, and very nearly the same 44 on every wind instrument in the collection.

$14 players summed, against the one at their average onset. Each thin line is one player's rising envelope, started at its own moment, with a spread of 30 milliseconds about the beat and a 90-millisecond attack. The heavy line is the section: nominally identical sources add incoherently, so their powers add and the sum is the root of the mean of their squares, drawn here as a fraction of the section's own peak. The dashed line is the single player who started at the section's average onset. The section reaches the 6 dB below peak criterion at 18.1 milliseconds and that player at 27.6, a difference of 9.5. The section is early because the players who started first are already sounding while the average one is still building, and nothing a late player does can make the sum quieter.$

Twelve violins are more punctual than one

Every essay until now treats a part as one player, and an orchestral part is a dozen. Sectioning does two things at once and only one of them was expected: it pulls the part's heard moment forward, by four milliseconds against a map spanning twenty-six, and it makes the part's arrival more accurate by very nearly the root of the number of players. So the map of required leads applies to an orchestra better than it applies to a quartet, and the case where it fails is three trumpets rather than fourteen violins.

Where a bar of 25 and a bar of 9 first disagree. The additive metre 2+2+2+3+2+2+2+3+2+2+3 — 25 units, an onset at the head of every group — with the accents it predicts drawn above the accents predicted by reading it as a repeating bar of 9, which is the cut of it that agrees longest. The two rows are identical for 24 consecutive steps and differ for the first time at step 25, where the shorter reading expects an accent and the metre does not supply one. Nothing before that step distinguishes the two hypotheses, so a listener who has not heard 25 consecutive steps has no evidence either way — whatever they are disposed to hear.

A twenty-five is a nine until its last unit

Every account of long additive metres says they are heard as groups of shorter ones, and the metre-induction model had never been pointed at the claim. Pointed at it, the model does not prefer the group — it prefers the long bar outright, and would go on preferring it more the longer anybody listened. What it cannot do is start: the evidence that separates a bar of twenty-five from a bar of nine does not exist until the whole bar has been heard, and at the tempo an unequal metre is best played at the psychological present holds sixteen units.

The longest silence against the spread of the gaps. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.94, so the longest silence is not a second thing to know about a timeline; it is the same thing measured more crudely. Against the other axis on the frontier — how much of the cycle has to be heard before a listener knows where in it they are — it correlates at -0.02, which is nothing. The named timelines are marked, and all of them sit at 4, the shortest longest-gap any pattern of this size can have.

The longest silence is not a third axis

The frontier between evenness and locatability was drawn twice and both times against two quantities, with the third job a timeline does left uncomputed. Computed, neither candidate for it is a third quantity: the longest silence ranks the whole census with the spread of the gaps at 0.94, and syncopation is not a property of a cycle at all — every one of the 273 patterns changes its count when the bar line moves. What the request was actually asking for is a cap rather than an axis, and under the tightest one the census admits thirteen patterns and every named timeline is among them.

A note on the downbeat costs 1.89 bits and one on the offbeat 5.70. What each position in a bar of 4/4 asks of a reader, by two routes. The solid bar counts where the notes of this collection's own three tunes actually fall — 28, 2, 26, 4, 28, 0, 16, 0 notes at the 8 positions — and takes minus the log of the frequency. The rule across each bar is the same quantity from the stated metrical weights, 1, 0.15, 0.5, 0.15, 0.85, 0.15, 0.5, 0.15, normalised and logged the same way. Nothing makes the two agree. They put the eight positions in the same order, and they price the tunes' own rhythm a fifth of a bit apart — while differing by more than a whole bit about the quaver after the downbeat, which two notes in a hundred and four ever use. The two positions these tunes never touch at all are drawn at the floor, which is the same floor the melodic measure gives an interval nobody plays. A weight was always a probability waiting to be read as one.

Where the note is costs more than which note it is

Thirteen earlier essays measure a page, and the three that price a reader price only its pitches — every line they measure is a run of equal notes. A metrical weight normalised by its own sum is a probability, and minus its logarithm is bits — the same substitution made earlier for intervals. Measured over the tunes used throughout it comes out at 2.23 bits a note against the pitches' 1.89, so the larger half of a reader's load is where the note is.

5 censuses: the floor of the longest gap, the ceiling of the syncopation total, and how few patterns reach both. For each census of k onsets in n steps, counted by rotation class: 3 in 8: patterns 7, floor of longest gap 3, at the floor 1, ceiling of total 14, at the ceiling 2, at both 1; 5 in 8: patterns 7, floor of longest gap 2, at the floor 2, ceiling of total 12, at the ceiling 2, at both 2; 5 in 12: patterns 66, floor of longest gap 3, at the floor 6, ceiling of total 28, at the ceiling 6, at both 6; 7 in 12: patterns 66, floor of longest gap 2, at the floor 3, ceiling of total 20, at the ceiling 38, at both 3; 5 in 16: patterns 273, floor of longest gap 4, at the floor 13, ceiling of total 79, at the ceiling 9, at both 9. The rotation-total of syncopation is taken against a metre of 2 × 2 × 2 for 8 steps, 2 × 2 × 3 for 12 steps, 2 × 2 × 2 × 2 for 16 steps. In every census drawn the evenly spaced pattern is at the floor and at the ceiling.

The other censuses keep evenness, not locating

Five onsets in sixteen suggested two rules: a named timeline's longest gap is the shortest its census allows, and its syncopation totalled over every rotation is the largest. Nine of ten named timelines across five censuses meet both — but so does the evenly spaced pattern, in all twenty-one censuses from two onsets in eight to eight in sixteen, and in four of those five censuses the named timeline is the evenly spaced pattern. What the rules describe is evenness. The quantity the frontier was built on points the other way: the evenly spaced pattern is the slowest of its census to locate in nineteen of twenty-one, the standard bell pattern is the slowest of all sixty-six patterns of seven in twelve, and among the orders of its own gaps a named timeline is usually both the most even and the slowest to locate. The son clave, whose most even order is also the fastest, is the exception the locating story was told about.

An accent moves the longest separable bar from 16 units to 18, and no further. For every bar length from 9 to 25 units, over all 1820 arrangements of twos and threes that are not a repeat of a shorter bar, the fewest and the most consecutive steps before the whole bar beats every shorter cut of it. 9: onsets alone 9 to 14, long beat predicted 7 to 12, downbeat predicted 7 to 8; 10: onsets alone 10 to 13, long beat predicted 8 to 11, downbeat predicted 8 to 9; 11: onsets alone 11 to 18, long beat predicted 9 to 16, downbeat predicted 9 to 10; 12: onsets alone 12 to 17, long beat predicted 10 to 15, downbeat predicted 10 to 11; 13: onsets alone 13 to 22, long beat predicted 11 to 20, downbeat predicted 11 to 12; 14: onsets alone 14 to 23, long beat predicted 12 to 21, downbeat predicted 12 to 13; 15: onsets alone 15 to 26, long beat predicted 13 to 24, downbeat predicted 13 to 14; 16: onsets alone 16 to 25, long beat predicted 14 to 23, downbeat predicted 14 to 15; 17: onsets alone 17 to 30, long beat predicted 15 to 28, downbeat predicted 15 to 16; 18: onsets alone 18 to 29, long beat predicted 16 to 27, downbeat predicted 16 to 17; 19: onsets alone 19 to 34, long beat predicted 17 to 32, downbeat predicted 17 to 18; 20: onsets alone 20 to 35, long beat predicted 18 to 33, downbeat predicted 18 to 19; 21: onsets alone 21 to 38, long beat predicted 19 to 36, downbeat predicted 19 to 20; 22: onsets alone 22 to 37, long beat predicted 20 to 35, downbeat predicted 20 to 21; 23: onsets alone 23 to 42, long beat predicted 21 to 40, downbeat predicted 21 to 22; 24: onsets alone 24 to 41, long beat predicted 22 to 39, downbeat predicted 22 to 23; 25: onsets alone 25 to 46, long beat predicted 23 to 44, downbeat predicted 23 to 24. A present of 3.5 seconds holds 16.0 steps at 218 milliseconds a step, so the longest bar some arrangement of which separates inside it is 16 units on onsets alone, 18 with the long beat predicted and 18 with the downbeat predicted.

The accent buys two units, however loud it is

A twenty-five cannot be told from a group of shorter bars on its onsets until more steps have gone by than a listener's present holds, and the obvious objection is that nobody plays an aksak bar as bare onsets: the long beat is louder, and the bar's first beat is marked. So how loud does an accent have to be? The question has a surprising answer. Loudness is not the variable. The existing accent cue changes nothing, and delays the answer where it changes anything. An accent that a reading has to predict works at any strength at all, and at no strength does more than a fixed amount: on the long beat it buys the two steps of a short beat, and on the downbeat it takes every arrangement to one floor — the bar less its last beat — which no cue carried by the notes can break. The longest bar that can be heard as one moves from sixteen units to eighteen.

A listener who knows every metre recognises none of them inside the present. For every bar length from nine units to twenty-five, the fewest and the most steps from the downbeat before every other one of the 1820 arrangements of twos and threes has been contradicted by the stream, on onsets alone, with the long beats accented and with the downbeat accented, against the 16 steps a present of 3.5 seconds holds. onsets alone: recognised within the present for 0 of 1820; long-beat accent: recognised within the present for 0 of 1820; downbeat accent: recognised within the present for 85 of 1820. The dashed line is the present.

Knowing every metre is slower than knowing none

A long aksak bar cannot be told from its shorter cuts by induction before one step into its last beat, and no accent carried by the notes moves that floor. The obvious escape is a listener who knows the repertoire and recognises the metre instead. Recognition among all 1,820 arrangements of twos and threes never beats the floor, is never quicker than induction, and is slower for half the metres: a nine induced in 9 steps is recognised in 27. What breaks the floor is a small repertoire that leaves out the metre's own longest cut — with the cut known, no repertoire of any size does.

With a fading memory the slowest order of the gaps 1 1 2 2 2 2 2 is not the one a perfect memory finds. For each of the 3 cyclic orders of the gaps 1, 1, 2, 2, 2, 2, 2 in 12 steps, the bits of position still unknown once listening has settled, against how many steps a listener's memory of a step takes to halve, with a mismatch costing 6. 2 2 2 1 2 1 2: 12 → 4e-9, 6 → 4e-5, 4 → 0.007, 3 → 0.070, 2 → 0.540, 1.5 → 1.130, 1 → 1.786; 2 2 2 2 1 1 2: 12 → 2e-5, 6 → 0.035, 4 → 0.335, 3 → 0.795, 2 → 1.405, 1.5 → 1.694, 1 → 2.005; 2 2 1 2 2 1 2 (the standard bell pattern): 12 → 2e-5, 6 → 0.027, 4 → 0.231, 3 → 0.535, 2 → 1.028, 1.5 → 1.368, 1 → 1.819. With perfect memory the slowest to locate is the standard bell pattern; at a half-life of 6 steps the highest floor is the order 2 2 2 2 1 1 2, and at 1 it is the order 2 2 2 2 1 1 2.

The bell pattern is slowest only to a perfect memory

Among the orders of its own gaps, a named timeline is usually both the most even and the slowest to locate — for a listener who never forgets. Give the listener a memory that halves and the result comes apart. Of six timelines slowest among their orders with perfect memory, only the fume-fume stays slowest for every forgetting listener, and the standard bell pattern, which is the fume-fume with onsets and rests exchanged and settles at exactly the same floors, is second of its three orders for every memory of half its cycle or less. The census ranking survives better, and in fourteen of twenty-one censuses it was the arithmetic of a pattern that repeats.

Come in part-way with the downbeat accented, and no bar of sixteen units or more is recognised inside the present. For every bar length from nine units to twenty-five, the fewest and the most steps a listener who knows every arrangement of twos and threes needs to recognise the metre and where its bar begins, with the downbeat accented: coming in at a sample of steps inside the bar, against hearing it from its written downbeat. On onsets alone, or with the long beats accented, a metre entered part-way is never told from its rotations. 9: from inside the bar 10 to 17, 16 of 20 inside the present; from the downbeat 10 to 10; 10: from inside the bar 11 to 19, 12 of 20 inside the present; from the downbeat 11 to 11; 11: from inside the bar 12 to 21, 9 of 18 inside the present; from the downbeat 12 to 12; 12: from inside the bar 13 to 23, 8 of 24 inside the present; from the downbeat 13 to 13; 13: from inside the bar 14 to 25, 8 of 28 inside the present; from the downbeat 14 to 14; 14: from inside the bar 15 to 27, 4 of 28 inside the present; from the downbeat 15 to 15; 15: from inside the bar 16 to 29, 4 of 32 inside the present; from the downbeat 16 to 16; 16: from inside the bar 17 to 31, 0 of 32 inside the present; from the downbeat 17 to 17; 17: from inside the bar 18 to 32, 0 of 36 inside the present; from the downbeat 18 to 18; 18: from inside the bar 19 to 32, 0 of 36 inside the present; from the downbeat 19 to 19; 19: from inside the bar 20 to 37, 0 of 40 inside the present; from the downbeat 20 to 20; 20: from inside the bar 21 to 35, 0 of 40 inside the present; from the downbeat 21 to 21; 21: from inside the bar 22 to 40, 0 of 44 inside the present; from the downbeat 22 to 22; 22: from inside the bar 23 to 39, 0 of 44 inside the present; from the downbeat 23 to 23; 23: from inside the bar 24 to 44, 0 of 48 inside the present; from the downbeat 23 to 24; 24: from inside the bar 23 to 39, 0 of 48 inside the present; from the downbeat 23 to 25; 25: from inside the bar 22 to 44, 0 of 52 inside the present; from the downbeat 23 to 25. In all, 61 of 590 entries are recognised within the 16 steps of a 3.5-second present.

A dancer who comes in late needs the downbeat marked

Every window for recognising an aksak metre so far started at its written downbeat. A dancer joining a dance already going has not heard the downbeat, and the arithmetic of that is blunt: a metre entered part-way is, onset for onset, each of its own rotations heard from their downbeats, and the rotations are metres too — 2+2+3 and 3+2+2 are counted differently. So on onsets, and with the long beats accented, no metre is ever told from its rotations. Only an accented downbeat tells them apart, and with it a listener who knows thirty metres recognises 54 per cent of them inside the present from a random entry, against 1 per cent without.

A cycle already known, against a cycle just arrived at. How many bits of uncertainty about position a listener has, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, once as a listener arriving and once as a listener who has been hearing it long enough to settle. At 1.6 seconds a cycle the timeline goes 0.94 bits arriving and 0.00 settled, and the colotomy 0.71 and 0.00; At 16 seconds a cycle the timeline goes 1.10 bits arriving and 0.08 settled, and the colotomy 0.93 and 0.23; At 60 seconds a cycle the timeline goes 2.47 bits arriving and 2.27 settled, and the colotomy 2.14 and 1.89. The gap between each pair is what the repetitions are worth, and it narrows as the cycle slows. The two designs are drawn at their own step counts rather than at equal strokes, so the levels here are not the earlier ones and the gaps are.

Repetition buys least where it is needed most

Every locating figure so far is a listener arriving — the uncertainty averaged over the first cycle heard. Cyclic music comes round dozens of times, and the same model already carries the answer for a listener who has settled: a floor of uncertainty that nothing had read. At two seconds a cycle the repetitions close the whole gap. At sixty they close eight per cent for a single timeline and twelve for a layered code. A slow cycle is worse on the first hearing and gains less from the second, and the two disadvantages compound.

A late dancer needs the landmarks, not the rhythm

A listener who joins an additive-metre dance part-way recognises it far more reliably with the downbeat accented. Strip the stream down to its landmarks — the onsets that begin a bar or a long beat, with every other onset removed — and the listener does as well or better: knowing a hundred metres, 23 per cent are recognised within the present against 14 with every onset. Unmarked, the landmarks still beat every onset with the long beats accented. Neither half does it alone; what identifies a metre from a late entry is where its long beats sit relative to its bar.

Against a pulse, the bell pattern is the quickest of its orders to place. The bits of position a listener is still missing, averaged over the first cycle heard, for each cyclic order of the gaps 1 1 2 2 2 2 2 in 12 steps, heard alone, against a pulse every three steps and against a pulse every four, at perfect memory, half-life 3 steps, half-life 1.5 steps. 2 2 2 1 2 1 2: alone 1.08, 1.23, 1.73; against a pulse every 3 steps 0.69, 0.75, 0.96; against a pulse every 4 steps 0.63, 0.67, 0.85. 2 2 2 2 1 1 2: alone 1.22, 1.63, 2.02; against a pulse every 3 steps 0.60, 0.73, 0.92; against a pulse every 4 steps 0.83, 1.07, 1.36. 2 2 1 2 2 1 2 (the standard bell pattern): alone 1.25, 1.45, 1.82; against a pulse every 3 steps 0.54, 0.56, 0.67; against a pulse every 4 steps 0.55, 0.57, 0.68. Alone, the bell pattern is not the quickest order to place at any memory. Against either pulse it is the quickest at every memory.

Against a pulse the bell pattern is the easiest to place

Heard alone, the standard bell pattern is not the quickest order of its own gaps to place in its cycle, for a listener with any memory. Heard against a pulse every three steps or every four — which is how anyone hears it — it is the quickest, at every memory and at every alignment of pulse and bell, and by a wide margin: at a memory of a quarter of the cycle, 0.56 bits unplaced over the first cycle against 0.73 for either rival against a pulse in threes. Six of eight named timelines do the same. A timeline's order of gaps looks chosen for how it sits against the beat, not for how it sounds alone.

$A timed expectation would erase a slow cycle's cost, and a listener cannot time a slow cycle that well. Bits of position a listener with a 3.5-second memory is still missing over the first cycle of son clave, against how long the cycle takes, for a newcomer with no expectation, a listener timing the cycle with the Weber fraction a duration that long is judged with, and a listener timing it to ten per cent. a newcomer, no expectation: 2 s 0.94, 8 s 0.97, 24 s 1.41, 40 s 2.02, 60 s 2.47; timing as well as listeners do: 2 s 0.68 (w 0.150), 8 s 0.69 (w 0.150), 24 s 0.85 (w 0.150), 40 s 1.90 (w 0.375), 60 s 2.38 (w 0.375); timing the cycle to ten per cent: 2 s 0.47, 8 s 0.48, 24 s 0.55, 40 s 0.73, 60 s 1.02. At ten per cent even a sixty-second cycle is placed about as well as a newcomer places a two-second one. At the precision a listener actually has for durations of half a minute or more, the expectation is worth a tenth of a bit.$

An expectation cannot rescue a cycle too slow to time

A listener who knows a piece arrives with an expectation of where in the cycle they are, and the size of that expectation was the number the last essay said nobody had. It can be given one: a listener who has been timing the cycle carries a spread of their Weber fraction times the cycle, which is the same number of steps at any tempo. Timed to ten per cent, a forty-second cycle would be placed better than a newcomer places a two-second one. But forty seconds is judged in the band where the Weber fraction is nearer forty per cent, and there the expectation is worth a tenth of a bit.

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