Syncopation — where it appears
Named by 21 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
The beat that is never sounded
A listener who has heard four bars of a groove and then hears two bars with the downbeats taken out does not move the downbeat. This site's rule set does, every time, on every pattern tried — and the direction it moves in says exactly what kind of model would be needed instead.
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 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.
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.
What the onsets left out
Eight essays induce a metre from a list of ones and zeros, and every failure they recorded was argued about as a failure of the rules. Two of the three are not. Note length is already in that list and the scoring throws it away: put it back and the son clave's two-way tie resolves to the notated downbeat. But the groove with its beat removed cannot be repaired by any cue at any strength, and the reason is arithmetic rather than empirical — the true phase has no onset on any of its strong positions, so there is no credit for a cue to multiply and its line is exactly flat.
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.
Expectation is a curve, not a list
Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous, sharpening through a bar and collapsing when the change arrives — and the two ingredients for it were already here, in two other accounts. What comes out is a second surprise, for when a chord arrives rather than for which one it is.
Where the chord actually lands
Every timing surprise so far is evaluated on a downbeat: the curve spans a factor of 6.7 and every number is read at its peak. Charge the waiting as well as the arrival and the costs over a bar become a proper distribution, the spread between the best and worst beat rises to a factor of 19.5, and a third of the probability sits on a bar in which nothing changes at all.
The long note and the strong note
The segmentation that produces every object connected here has carried a free parameter since the day it was written: whether a note counts for its metrical weight or for how long it is held. Only the first has ever been drawn. The two name different chords on sixteen per cent of passages where the cues agree about the notes and forty-three per cent where they do not — and where they disagree most sharply a mixture of them picks a third chord neither one asks for.
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.
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.
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.
A syncopated melody needs its phrases to land
Put a melody whose long notes all fall on the half-bar over a harmony that changes once a bar and a walking bass, and a listener adding up the evidence hears the bar begin in the wrong place three times in four. The harmony knows where the bar is, but only through one witness: the bass's root votes for the barline in every passage, and the bass's motion never does — a walking bass lands on each new root by step, so it moves less at the barline than inside the bar. One syncopated voice ties the root, and the motion breaks the tie its way. What rescues the barline is not the harmony but the melody's own phrase ends: hold half of them from the downbeat and the reading finds the bar in 86 per cent of passages.
Two bars of drum hold the barline; a dozen bars of bass do not
A listener who settles where the bar begins and then holds it hears a syncopated tune against the bar only if the ground before it built enough confidence to spend. Read bar by bar, a walking bass under one chord a bar is worth less than half a vote a bar for the barline, and a syncopated bar takes an eighth of a vote back, so a tune needs about twelve bars of that ground to be heard against the bar nine times in ten. A drum on the downbeat makes each bar of ground worth a vote and a half, and two bars are then enough — but only for a listener who can bank more than four votes of confidence.
A listener who weights its witnesses needs eight bars of bass, not twelve
A syncopated tune over a walking bass needed about twelve bars of ground before a listener with a memory heard it against the bar nine times in ten, with every witness given one vote a bar. Let the listener weight each witness by how often it has agreed with the reading it holds and the root earns a second vote, the bass's motion loses half of one, and the syncopation loses almost all of its. Eight bars are then enough, a tune that syncopates every bar becomes hearable against the bar at all, and reading the syncopation backwards buys nothing that ignoring it does not.
A listener who knows the style needs no introduction
A listener who weights each witness to the barline by its record needed eight bars of walking bass before a syncopated tune was heard against the bar nine times in ten, and with no introduction at all it learned the syncopation's lesson instead. Finer records — one for each side a witness takes, one for each section — change nothing. What changes everything is arriving with the style's habits: a listener who already gives the bass's motion and the melody's long notes no vote hears the tune against the bar 93 per cent of the time from its first bar, and 94 per cent for a tune that syncopates every bar. The root need not be trusted more. The others need to be trusted less.
The habit worth bringing is to distrust the tune
A listener who arrives with a style's habits — the bass's motion and the melody's long notes credited at a coin toss — hears a syncopated tune against the bar from its first bar. But a listener does not know the style in advance. Give it two sets of habits, one for tunes that mark the bar and one for tunes that syncopate, and let it weigh them by how well each predicts what it hears: it does better than a listener with no habits and far worse than one who simply knows, and it becomes sure of the style in fewer than one passage in ten. The reason is that the evidence runs one way. A tune that marks the bar never testifies against the habit of distrusting tunes, so the habit that serves every style is the syncopating one, kept always.
A late root is believed because the record is kept by the reading
The habit worth bringing to a tune turned out to be trusting the bass's root and doubting everything else. A bass that anticipates the next chord does not break that habit — the root is still tied across the barline, and it is the bass's motion that lies. A bass that puts the root at the half-bar does break it, and nothing a listener learns can repair it, because each witness is judged by whether it agrees with the reading, and the reading was made by the liar.
Two honest bars convict a lying root
A bass that states its root late defeats every listener who learns from agreement, because the reading forms before any honest witness speaks. Give the passage a drum on its first downbeats and the same record that ratified the liar convicts it — but only a listener holding a habit that doubts the root can act on the verdict. Two strokes are enough for that listener, for as long as the tune lasts. One is not, and no amount of drum is enough for a listener that credits the root and the drum alike.
Named alongside it
The objects these essays reach for when they reach for this one.
MetreHarmonic rhythmDownbeatExpectationMetrical weightOnset patternClaveEuclidean rhythmEvidenceMaximal evennessMelodyRotation