Inference — where it appears
Named by 40 essays across 4 fields — each of them below, with the objects they name alongside it.
Three decisions that constrain each other
Every model here decides one thing at a time — the key from the pitch classes, the metre from the onsets, the chords from the metre — and an earlier essay ended by saying a listener does all three at once. Resolving them jointly costs a hundred and fifty-seven times the search and changes the reading of two passages in five. It never once changes the key, and the reason it cannot is the reason the whole account is built the way it is.
A key-finder that keeps the order
Every key-finding model until now begins by throwing order away — a histogram over a window, correlated against twenty-four profiles — and the thing that most obviously declares a key is an ordered pair of chords. A model that keeps the order costs 84 states and 7,056 transitions against 24 hypotheses and none, and what it buys is the one test a histogram was said not to pass: two keys in turn and two keys at once have histograms 98 per cent alike, and are two different sequences.
The cadence as evidence
Every model so far infers a key from a bag of notes, and the thing that most obviously declares a key is a cadence — an ordered pair of chords in which the order is the whole content. The essays on closure built a five-component cadence vector long before this and nobody has used it for this. Two passages differing in one note, one cadencing in C and one in G, get opposite answers from the ordered pairs and the same answer from the profile.
How much of the reading arrives late
Every margin reported earlier is two-sided: the best path through a key at a bar is the score into it plus the score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line. On a thirty-two-bar song it removes more than half the certainty, and on a passage built to be ambiguous it changes the key named at nine bars out of eleven.
The listener who forgets
Setting an analyst's reading of a key against a listener's measures what arrives late. Both passes assume perfect recall of their own half — which is as wrong going forward as knowing the future is going back. Put a decay on the forward pass and a listener with a memory of a few bars keeps a quarter of the confidence and nine tenths of the answers.
Two surprises and one event
A chord change is surprising twice over — in which chord it is, and in when it comes — and a listener meets one event. Adding two surprises needs to know how much they share, which is a fact about a repertoire nobody has. Sweeping it instead: the total moves by half, the ordering does not move at all, and the most surprising moment in a passage is the same one whatever the answer turns out to be.
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.
The bars a key is made of
A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.
The surprise of nothing happening
A hazard charges a listener twice — once when the chord changes and once, quietly, at every beat it does not. The second term was expected to dominate, because there are more beats than changes. It is a third of the bill at one chord a bar and never reaches a half at any rate a metre survives, because the cost of a beat where nothing happens is second-order small.
An attack time is not an attack
Eight earlier essays read a heard moment off an envelope, and every one of them used the same envelope shape without saying so: the source table records one curve for all nine of its families, and the map's own arithmetic does not carry the parameter at all. A published attack time fixes a ten-to-ninety time and nothing else. Under the two other shapes the same measurement admits, every millisecond computed so far doubles — and the constructed passage called inaudible earlier becomes three times a listener's threshold.
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.
Two cues meet in a corner
The profile finds a key's tonic and the bass finds a chord's degree, and until now each was swept with the other held at nothing. Swept together across 42 settings, the plane they make is not the ridge that was predicted. The tonic is a step in one direction, at a profile share of 0.55, and the bass cannot move it; the degree is a slope in the other, rising to 89 per cent as the bass is weighted, and the profile barely touches it. Every question is answered only in a corner of the plane — and the one place the two cues overlap is the one piece of music both can rescue.
A bass line is not a list of roots
Every bass note the key-finder has been given was its chord's root, and under that line a bass cue reads 89 per cent of scheme bars on the right degree. Give the same chords an economical bass that moves to the nearest chord tone, as a keyboard reduction would, and nearly half of them are inverted. The cue that rewards the triad rooted on the bass then reads 68 per cent at best and worse as it is trusted more; the cue that rewards any triad containing the bass cannot be fooled and stops at 61. The same inverted line does one thing the roots never did: it puts the leading note of each new key at the bottom, and finds the rondo's modulations.
The chords never move the barline
Every hypothesis the joint search had drawn was one bar long, and on one bar with a note in every slot the metre cannot choose a barline at all. Four bars with rests in them make the barline a decision the metre and the chords both have an opinion about, and the prediction was that the chords would move the barline more often than the barline moves the chords. It is the other way round, completely: whenever the two prefer different barlines the search takes the metre's, on up to 72 per cent of passages, and in fifteen hundred passages the chords never once move it. What the chords decide is the one thing the metre cannot see — whether the bar starts on the downbeat or half a bar later — and they decide it right a little over two times in three at best.
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.
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.
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.
The bass errs fast where the content errs slow
Asked how often the chords change, a reading built on pitch-class content names a slower multiple and never a faster rate. Give the passage a bass that states each new root and moves between chord tones inside a chord, and a reading built on the bass's moves errs the other way: it names a faster grid and never a slower one. At two chords a bar the bass is right every time; at a chord every two bars it is never right. Six ways of combining the two readings each trade one end of the range for the other.
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.
A bass that holds through a change marks the barline
At two chords a bar the chords change on the barline and on the half-bar alike, so a reading of where they change is at chance, and the metre ties the two. The bass has one more piece of evidence: a change inside the bar can be voiced over the note already sounding, and a change on the barline is voiced over its root. Hold the bass through one mid-bar change in six and the metre and bass together place the barline in 85 per cent of eight-bar passages; one in three, 97. The convention cannot be stronger than that, and the chord changes, asked first, only get in the way.
A walking bass marks the bar by the note it is on
The reading that finds a barline from where the bass moves collapses to twenty per cent when the bass moves on every crotchet, and the bass the last essay called busiest moved twice a bar, because its beat was the half-bar. Given one that really walks, the same computation with sounds the chord's root in place of moves rises from nothing to a whole note a bar exactly as the other falls, and the two are equal at a bass moving on three crotchets in eight. Neither places a barline alone: the root is still sounding on the quaver after it. The metre, which cannot tell a barline from a half-bar, tells it from the quaver after it at once.
A crowd costs a late dancer the silences, not the steps
A dancer who hears only the landmarks of an aksak rhythm — the heads of bars and long beats — recognises it from a late entry better than one who hears every onset, as long as everything played is heard. Let a share of what is played be lost in a crowd and the lead goes, but not for the reason expected. Most of the cost arrives before a single landmark is missed: a listener who allows that one might be can no longer treat a silence as proof, and the landmark stream is built almost entirely of silences. At one lost onset in ten, the caution alone takes 25 points from the landmarks and 5 from the full stream.
A second slot hears every modulation as a tonicization
A key-finder that names one key per bar reads a chain of applied dominants by moving the key with every chord. Give its state a second slot — the key a bar is in, and the key its dominant points at — and the thirty-two-bar song's bridge is read exactly as the scheme labels it, every bar, for the first time in these essays, and the blues's opening stays in its tonic. At every setting that does that, the rondo's two eight-bar episodes are heard as tonicizations inside C, and no setting of the three costs reads both. A modulation to a closely related key and a long tonicization are the same chords; what separates them is how long they last, and a cost for entering a state cannot count.
A modulation is a dominant that keeps coming back
A key-finder with a slot for an applied dominant read the thirty-two-bar song's bridge as its scheme labels it and heard the rondo's two episodes as tonicizations inside C, and no setting read both. The obvious repair — charge a tonicization by the bar, so a long one costs more than a short one — makes it worse: the reading gives up the bridge before it gives back a single episode, because the bridge spends six bars in applied chords and the episodes three and two. By the bar, the chain is the longest tonicization in the listing. What the episodes have and the bridge does not is a dominant that resolves to the same tonic again and again.
A crowd that hides bars is kinder than one that hides onsets
A dancer joining an additive-metre dance in a noisy room loses some of what is played, and the essay before this one lost it one onset at a time. A crowd does not. It hides stretches — a burst of laughter covers a bar. The prediction was that a stretch as long as a bar would hurt the stream of landmarks most, since it could remove every landmark in a cycle, and that long bars would survive best. Holding the share of time lost fixed, longer stretches help every stream, help the full stream more than the landmarks, and leave long bars as hard to place as ever.
A return is cheaper to avoid than to pay for
The essay before this one found what separates a modulation from a tonicization: an episode's dominant keeps resolving to the same target, and the bridge's dominants never do. So the key-finder was given a memory of where its last dominant went, and charged for a return — the prediction being that one setting would at last hear the episodes as keys and the bridge as a chain. None does. Every return the charge is aimed at is avoided for less than a nat, by hearing a D7 as a minor chord or by moving the passage into another key, while the reading the charge was meant to force costs twelve.
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.
The leading note is already inside the chord that hides the key
A key-finder kept hearing a rondo's episodes as passages inside C, and the escape it used was to hear a D7 as a D minor chord, throwing away its F♯. Charge a chromatic note by how strongly it points at another key, and that escape closes. Nothing else moves, because every F♯ and G♯ in the episodes was already inside the chord the reading heard — as the third of V of V. The charge takes a chromatic note seriously everywhere except where it was needed, and in the blues it mistakes a flattened seventh for a leading note.
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.
A dancer who hears the crowd can trust the quiet
A late dancer in a noisy room who cannot tell noise from silence has to treat every silence as weak evidence, and that caution was the largest cost in every figure before this one. A dancer who hears the laugh that covers a bar knows which silences to discount and can trust the rest. Losing a quarter of the music, that dancer recognises more metres than one who loses nothing but must allow for losses — and the landmarks, which a noisy room had made the worse stream to follow, lead again once the noise comes in bursts. The quiet has to be trusted to one onset in nineteen.
An honest drum has to come first
Two strokes of an honest drum at the very start of a passage protect a listener from a bass whose root arrives late in every bar. The same two strokes a bar later do nothing at all, and neither does a drum that then plays every downbeat to the end. A listener that keeps a record of its witnesses scores the late drum against the reading the root has already made, and convicts it within two bars. Every bar the lie is believed first multiplies the trust the drum would need, and by the seventh bar no trust short of certainty is enough.
In a loud room the accent has to be loud
A late dancer who hears the crowd as noise ignores everything played under it. Real noise is not that thorough. It hides the soft strokes and lets the loud ones through, and an accented downbeat is a loud one. Let the noise pass loud strokes and some soft ones, and marking the downbeat is worth more to the dancer than in a quiet room: with half the time under noise, 29 points of recognition against 23. Noise that swallows the accent halves its worth. In a quiet room no amount of loudness bought an accent anything extra; in a loud one, loudness is the whole of what the accent is for.
A break rescues a late drum only by ending the reading
A drum that enters after a late root has made the reading needs odds in the hundreds to undo it. Bring it in after a break in the texture, and the obvious guess is that the break helps by making the listener forget whom it trusted. It barely does. What rescues the drum is the listener forgetting what it heard: with the old reading gone, a drum on bar 9 needs odds of nine to one, and with the record gone too it needs less than the three to one it has always had. A break that leaves even a hundredth of the old reading behind rescues nothing at all.
In a crowd the downbeat belongs to the drum
In a loud room a marked downbeat is worth more to a late dancer than in a quiet one, so long as the mark gets through. Whether it does depends on where its sound lies, because talk is not loud everywhere: it is loudest from 250 to 400 hertz, exactly where a melody's notes are, and fifteen decibels quieter at the bottom of a drum. A louder melody note is heard half the time through talk ten decibels over the band; a low drum survives a crowd twelve decibels louder than that. With the talk fifteen decibels over the band, a downbeat marked in the melody is worth two-fifths less to the dancer than one marked by the drum.
A silent break has to outlast the reading's lead
A listener waiting through a break goes on expecting downbeats at the reading it holds, and each one that fails to come can be charged against that reading. Charged at a witness's vote apiece, the silence ends the reading only after as many bars as the reading has votes in hand: three bars after two bars of tune, eight after four, twenty-one after eight. A one-bar break would need a missing downbeat to cost twenty votes. And a silence that does end the reading leaves the record standing, which convicts the drum anyway.
A fill cannot end a reading, but it can win the next one
A drum fill in the break before a late drum is a witness that speaks while the old reading still stands, so every stroke it plays is scored as disagreeing, and its credit falls to a coin toss within two or three strokes: at three to one it can cast 1.37 votes in all, against a lead of twenty by the middle of a tune. A fill cannot end a reading. But once a silence has ended it, the same fill sets the next one, and the record that convicted the drum through every length of silence starts convicting the bass instead: after twelve bars of silence before bar 5 the drum goes from 7 per cent of passages to all of them.
A cadence has to empty the reading, not weaken it
A drum fill after a long enough silence wins the next section for an honest drum, because it speaks into a reading that is empty. A cadence might do the silence's work in a bar. It does, but only on one condition: before bar 9, a cadence that removes ninety-eight per cent of the listener's conviction leaves a two-stroke fill winning one passage in a hundred, and one that removes all of it lets the fill win every one. The fill turns the reading either way. What the last sliver decides is whether the fill's first stroke is recorded as a disagreement — and that record is what the late root then uses to win the reading back.
A cadence that votes must not vote as the drum
A cadence lands on a downbeat, so it could cast a vote for the barline at the moment it loosens the listener's grip. Whether that vote helps depends on whose it is taken to be. Heard as the tune's long note, it speaks with credit the tune spent long ago and is worth almost nothing. Heard as the drum, it is exactly one more stroke of fill, and is convicted as the fill's first stroke was. Heard as a witness of its own, it carries a third of a vote and no record to lose — and that moves the line from an emptied reading to a fiftieth of one.
A bell can mark a cycle without moving a stroke
A lead drummer's varied cycle helps a listener or misleads one, according to which stroke it moves, because a moved stroke can make the bell look as if it had started somewhere else. Give the varied cycle a different sound as well and the distinction goes. Heard nine times in ten and a misleading variation is no longer one; heard always, it no longer matters which stroke moved, or whether any did. A mark heard seven times in ten does almost nothing, and the arithmetic of that threshold is the same as a coin's.
Named alongside it
The objects these essays reach for when they reach for this one.
MetreHarmonic rhythmExpectationKey-findingAdditive metreAksakEvidencePerceptual presentDownbeatPriorCadenceModulation