Concept

Grouping — where it appears

The division of a stream of events into units, which a listener performs and the notation only partly specifies. Its boundaries are marked more reliably by lengthening than by any pitch feature, so a score without timing does not fully determine them.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

A 7-semitone sequence at 120 ms a tone. Tones drawn as pitch against time, one bar per tone. The events are the same in both readings of this pattern; what changes is whether a listener assigns them to one line that leaps back and forth or to two lines that each stay put. Nothing in the drawing decides which, and nothing in the sound does either.

The ear builds objects, and sometimes offers a choice

What arrives at an ear is one pressure signal. What a listener gets is a set of separate things — a violin, a voice, a car outside. The assignment is a construction, and the clearest evidence is that it can be flipped by changing nothing but the speed: one sequence of tones is a single line when slow and two lines when fast, with a wide region in between where the listener may choose.

perception · Auditory scene

A phrase is a number of seconds

Musical phrases are described in bars, and four is the number everybody names. But the constraint that fixes a phrase is a property of the listener and is measured in seconds, so the bar count is whatever the tempo makes it. Across seven ordinary tempos the bar count that lands inside the window moves by a factor of eight, while the window itself does not move at all.

form · Phrase

Two ways to fill eight bars, and only one of them accelerates. The period against the sentence, drawn as the lengths of their constituent units against position on a grid of eight bars. The ratio beside each row is the mean unit length in its second half divided by the mean in its first: the period at 1.00, the sentence at 0.67. A ratio below one is an acceleration — the unit shortening as the phrase approaches its arrival — and a ratio of one is a plan whose unit never changes length.

One of these eight-bar phrases accelerates

The sentence and the period both occupy eight bars, both end with a cadence, and both are recognised by ear rather than counted. What separates them is arithmetic. One halves its unit halfway through and the other does not, and the difference comes out as a single ratio — 0.67 against 1.00 — computed from nothing but the lengths of the parts.

form · Phrase

Three ways to arrive at the same final tempo. Tempo against position in the closing passage, ending at 35 per cent of the opening tempo, for curvature exponents 1, 2, 3. All three begin and end at the same tempo, so what separates them is the middle: at the halfway point they read 68 per cent for linear in score position, 75 per cent for constant deceleration, 80 per cent for q = 3. The straight line is the one nobody plays. Measured ritardandos fit the decelerating curves, which is the whole of Kronman and Sundberg's argument: a closing gesture has the shape of a body stopping rather than of a dial being turned, and the parameter that varies between performances is the final tempo rather than the shape.

An ending is a deceleration

Every performance slows down at the end and the slowing has a shape. Tempo read against score position is the velocity of a body stopping — a square root rather than a straight line — and the three candidate curves agree at both ends by construction, so the whole audible difference is in the middle, where they part by fifteen per cent of the passage's length.

form · Closure $A silence measured in two ways that were not chosen to agree. How far a carried beat drifts during a silence, in fractions of a beat, for beat periods of 200 ms, 550 ms, 2000 ms and a tempo estimate 5 per cent wrong — which is the published discrimination limen rather than a figure chosen here. Half a beat of drift is where the metre coming out of the silence is no longer the one that went in, and it is reached after 10 beats whatever the tempo, which is 2.0 seconds at 200 ms, 5.5 seconds at 550 ms, 20.0 seconds at 2000 ms. The shaded band is the psychological present, 2 to 8 seconds, measured by a completely different literature and used in this collection to bound a phrase. The two answers overlap: a silence under about 2 seconds is a rest inside something and one over about 5.5 is after it.$

A silence long enough to be an ending

Four of the five closure components are present or absent. Silence is the one with a continuous scale, so it is the one that can be given a threshold — and the threshold arrives from two literatures that were not chosen to agree, landing between three and a half and five and a half seconds. In a large hall, the room's own decay uses up most of it.

form · Closure

Where the page ends a phrase, and where the ear does. Twinkle, twinkle with two sets of phrase boundaries on it. The lower curve is a local boundary detector — a peak in how much the interval and the note length change from one to the next, with nothing in it about bar lines or harmony — and the marks above it are where the notation puts the phrase ends. It finds 100 per cent of them and 2 boundaries the page does not have. Where the two agree it is because a long note is sitting at the join; where they disagree the page is marking a grammatical unit and the detector is finding a perceptual one.

Where a phrase ends

Run a boundary detector over the three tunes used throughout and it agrees with the notated phrasing on one of them perfectly and on another almost not at all. The reason is which cue each tune uses: Twinkle's phrases all end on a long note, so a duration-weighted detector finds five of five with no false alarms; Ode to Joy's run on in crotchets and its phrasing is in the intervals, where a duration detector finds one of three and a pitch detector finds all three and eight others. No fixed weighting serves both, and the published one is worse on each tune than the single cue that tune uses.

form · Phrase

Twinkle, twinkle, phrased at the level each tempo selects. The number of boundaries the model finds when its smoothing scale is set by the psychological present rather than chosen, against the tempo the tune is taken at. The scale in notes is the present's 3.5 seconds divided by the mean note length, so a fast tempo puts more notes inside the present and smooths harder. The page's own phrasing has 5 boundaries, drawn as the flat line; the model matches it best at 160 beats per minute, where the present holds 8.2 notes. The same tune at two tempos is read at two levels, which is the prediction and is not a free parameter.

The level the tempo chooses

The boundary detector has a scale parameter and an earlier essay left it free, ending with the sentence that names this one: the scale is in notes and the psychological present is in seconds. The psychological present is two to eight seconds, a tempo converts one to the other, and the level a listener reads then stops being a parameter at all — which is a prediction with teeth, because the same tune at two tempos should be phrased differently at levels the arithmetic names in advance.

form · Phrase

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.

rhythm · Phrase

Named alongside it

The objects these essays reach for when they reach for this one.

Phrase Perceptual present Tempo Notation Segmentation Tactus Cadence Closure Hierarchy Scale-space Antecedent consequent Auditory scene analysis

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