Prior — where it appears
Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.
A modulation and a borrowing are one number apart
The key-finder that keeps the order has one tuned parameter, and said so. Sweep it and the parameter turns out to be the whole verdict: below a threshold the model hears two keys alternating, above it one key with borrowed chords. The threshold rises with how slowly the keys alternate — and the value it chose sits above every threshold in range, so its finding about fast alternation was a consequence of the tuning.
Surprise is a number
The chord that did not come was described rather than measured. Its measure is the information content of what did arrive, and a model of the probability has been to hand since the key-finding essays — eight root-motion weights, ordinal and stipulated. Reading them as a distribution prices a deceptive cadence at 2.71 bits against a perfect one's 1.85, and turns up the fact that the largest of the eight had never been read by anything.
A short note is heard more in tune than it is
Four earlier essays treat a key as something that reduces the noise in a pitch judgement. Treat it instead as a prior and the prediction changes kind: not a smaller error but a systematic bias, pulling a short note toward the nearest scale degree by an amount the Fourier bound sets. Thirty cents out of tune on an eighth-of-a-second note is heard as eight. And the part the debt got wrong is the part that matters — the bias does not vanish on a long note. It stops at 17 per cent at A4 and at 48 per cent at A2, because the likelihood's width has a floor that no duration removes.
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.
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.
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.
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.
An announced plan pays the cycle and punishes the blocks
A slow introduction, a title or a programme note can tell a first-time listener the plan of a piece before the first lapse — and be wrong. Trusted as far as it deserves, an announcement gives back half of what being told the plan is worth to four cycled functions only once it is right nine times in ten. But how far to trust it depends on the piece: a cycle rewards believing even a coin-toss announcement completely, while functions in blocks of two are made worse by an announcement right half the time, and a confident wrong one can cost a listener forty points of the first minute.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
ExpectationInferenceMetreHarmonic rhythmCadenceSyncopationTonal hierarchyBorrowed chordDeceptive cadenceDifference limenEntropyHarmonic analysis