Acoustics & Music Theory
why it sounds like that
Concept

Phrase — where it appears

The span of music heard as a single gesture, bounded above by how long a series of events can be held as one thing. Its boundaries are marked most reliably by a lengthening of the final note rather than by anything in the pitches.

Named by 18 essays across 3 fields — each of them below, with the objects they name alongside it.

A phrase is a number of seconds, and the bars follow the tempo. Phrase durations for 1, 2, 4, 8, 16-bar phrases at seven tempos, on a logarithmic seconds axis, with the 2 to 8 second window shaded. The window is a property of the listener and does not move; which bar count falls inside it is decided entirely by the tempo.

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 Five signals, computed separately, and no total. The five components of closure for 4 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted.

An ending that exists so a bigger one can

Half of the cadences in tonal music are built to fail. A phrase that stopped convincingly at bar four would be a piece four bars long, so the ending at bar four is engineered to arrive and not to settle — and the components it withholds are exactly the ones its partner at bar eight supplies. Closure is nested, and the nesting is what turns two phrases into one thing.

form · Closure The same induction, one level up. Bar-level onsets from thirty-two-bar AABA — a bar is marked where a section or a key begins — scored against hypermetres of 2, 3, 4, 6, 8 bars with the identical function the beat-level figures use. The best-fitting period is 8 bars, which at 108 beats a minute lasts 17.8 seconds.

The bar above the bar

A four-bar group is a bar whose beats are bars. That is not an analogy — it is the same computation, and the same metre-induction model produces one when it is handed bars instead of beats, unchanged. What decides where the hierarchy of levels stops is not in the arithmetic at all, and it is a number the phrase essay already measured.

form · Metre induction How often the chord changes, and what a room allows. Chord changes a second implied by each style's stated rate and tempo, on a logarithmic axis, with the rate above which a room leaves more than one earlier chord above 20 dB marked for six rooms. The style rates are conventions rather than corpus measurements and the figure says so; the room rates are arithmetic from the reverberation time.

How often the chord changes

Two pieces can use the same chords in the same order and be nothing alike, because a progression says which chords and not how fast. Harmonic rhythm is the second variable, it runs over a factor of thirty between the styles that use it, and both of its limits are set by things that are not harmony — a listener's memory at the slow end and a building at the fast one.

harmony · Progression How much of the rule a walk with no rule reproduces. Post-skip reversal in 20,000-note random walks with no melodic knowledge of any kind. An unbounded walk reverses after 50.0 per cent of leaps, which is the chance rate and is the check that the measurement is right. Confining it to 12 semitones raises that to 61.3 per cent. Reaching the 70 per cent that corpus studies report needs a central tendency of 0.95 — an almost deterministic pull back toward the middle at the edges of the range. A wall is not enough; there has to be a spring.

The leap that pays itself back

Every melody textbook teaches that a leap should be followed by a step in the opposite direction, and every corpus that has been counted agrees — around seven leaps in ten are answered that way. A random walk with two walls, no memory of the leap and no rule of any kind reverses after 61 per cent of them, and the residue is not a rule either. What is left when the walls are accounted for is a prediction the rule does not make, and it is the prediction that decides between them.

form · Melody Where a twelfth comes from. The range of a walk with no walls, against how many notes it runs for, at three settings of the one parameter it has. The parameter is fitted to the post-skip reversal rate and to nothing else; the range is then read off. With no central tendency at all the walk passes two octaves by 60 notes and keeps going. At the setting that reproduces 70 per cent reversal — κ = 0.78 — the range is 11.9 semitones at thirty notes and 17.9 at a hundred and twenty. It grows logarithmically, so over the whole plausible length of a tune it sits between an octave and a fifteenth, and a twelfth is the middle of that. The three tunes carried here are marked and all three fall below the curve.

The twelfth, and where it comes from

Melodies occupy about an octave and a fifth, and an earlier essay set out to explain that by the singer's register break and found that it does not: the chest mechanism alone spans two octaves and a semitone. The answer is in a parameter the essay on leaps fitted and then put down. A walk with no walls whose central tendency reproduces the post-skip reversal rate has a range that grows logarithmically — six semitones at eight notes, twelve at thirty, eighteen at a hundred and twenty — so across every length a tune plausibly has, the span is between an octave and a fifteenth.

form · Melody 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 The boundaries that survive each amount of smoothing. The local boundary strengths of Twinkle, twinkle read at every scale: the curve is smoothed with a Gaussian of the width on the horizontal axis and the peaks that survive are counted. Small scales give 11 boundaries and large ones give one, and the notation marks 5. The level with that many falls at a width of 2, where the model finds 100 per cent of the notated boundaries and 100 per cent of what it finds is notated — a comparison with no threshold in it, which is what the scale parameter buys.

A boundary at a stated level

A boundary detector run over three tunes agreed with the notation on one and barely at all on another, and left two things owing: a version with a scale parameter, and a version run on performance timings. Both are paid here, and they pay differently — the scale removes a free parameter from the comparison and does not rescue the hard case, while two per cent of rubato does.

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 tempo turns, and almost nothing moves. The earlier arrival reading — what is sounding at the final chord over what the listener has been hearing — swept over bar lengths from 0.5 to 5 seconds, which is 480 down to 48 beats a minute, at 3 closing lengths. Every curve is nearly flat. Across a tenfold change of tempo one gesture's reading moves by a factor of 1.201 and the other's by 1.098, while the gap between the two gestures — which is what that essay was measuring — is 1.228. The expectation was that the tempo would decide the answer, on the grounds that a two-second bar against a two-second release is a comparable pair. The premise is wrong in a way the sweep makes obvious: the thing being compared with the release is not a bar, it is the WHOLE ENDING, which is 2 to 8 bars long and is therefore far longer than the release at every tempo anybody plays. The running impression has caught up with the closing texture before the final chord arrives, at 0.5 seconds a bar and at 5, and what is left is the last bar's own jump.

The parameter that did not decide the answer

An earlier essay on closure ended by naming the tempo as the thing every number in it was resting on, and said it was the kind of parameter that had caused trouble before by turning out to decide the answer. Turned across a tenfold range at a closing gesture of fixed length it moves the reading by four per cent, against a twenty-three per cent gap between the gestures it is distinguishing. The parameter beside it in the same figure — how many bars the gesture occupies — moves it by twenty, and nobody had named that one at all.

form · Closure How far the detector looks, note by note. The number of notes that fit inside a 3.5-second present at each point of the tune, once the performance has lengthened its phrase-final notes by 30 per cent. It runs from 5 to 10 notes against a constant 7 for the unperformed version, and it dips exactly where a boundary is, because a boundary is where the performance slows. Reading the boundary-strength curve with that width at every point instead of one width everywhere gives an agreement of 0.55 with the notated phrasing, against 0.36 for the fixed width the present dictates and 0.71 for a fixed width fitted to this tune. The dips are marked, and the notated boundaries are the vertical lines: the detector narrows itself at the places it is supposed to find, which is the circularity this figure has to be honest about — the lengthening was put there by the notation.

A detector whose resolution the performance sets

The boundary detector lost its free parameter when the psychological present became a number of notes at a stated tempo, and what that held still was named at the time: a performance slows into a phrase end, so the number of notes inside the present is not the same everywhere in a tune — it falls exactly where a boundary is. Making the width follow the performance recovers half of what removing the parameter cost, and honestly leaves the other half.

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 Six named proportions, as blurred as the durations that make them. Six proportions between two parts of a piece — 1 : 1, 4 : 3, 3 : 2, golden section, 2 : 1, 3 : 1 — placed on one axis by the logarithm of the ratio of the longer part to the shorter, and drawn as bars one criterion wide (d′ = 1) for a listener timing both parts with a Weber fraction of 7%, 15%, 35%. Bars that overlap are proportions that listener cannot tell apart. At 7%, 4 of 5 neighbouring pairs stay apart; at 15%, 3 of 5 neighbouring pairs stay apart; at 35%, 0 of 5 neighbouring pairs stay apart.

A proportion is only as fine as its two durations

Analyses of form measure proportions in bars and report them to three figures — a climax at 0.618, a section in the ratio 3 : 2. A listener has each part only as an estimate of how long it lasted, and a ratio of two estimates is blurred by both. Timed as well as anyone times a single second, eleven proportions fit between 1 : 1 and 3 : 1; timed from memory over minutes, two do. The golden section is told from 3 : 2 only below a Weber fraction of 5.4 per cent.

form · Proportion A count is least reliable at both ends and best in the middle. How precisely a listener knows the length of a section they are counting, against how many units long it is, at three kinds of timing judgement. A slip — one unit miscounted, at 2% a unit — accumulates as a random walk, so its relative cost FALLS as the section lengthens. A lapse — the count lost altogether, at 1% a unit — compounds, so the chance of still having the count falls geometrically and a long enough section is certain to lose it. A listener who has lost the count is back to timing, so the two failures mix into a floor. Against a Weber fraction of 7.5% the count is worth most at 21 units, where it is 1.7 times finer than timing, and falls back under a quarter better by 98 units; Against a Weber fraction of 15% the count is worth most at 10 units, where it is 2.4 times finer than timing, and falls back under a quarter better by 101 units; Against a Weber fraction of 38% the count is worth most at 4 units, where it is 3.7 times finer than timing, and falls back under a quarter better by 102 units. The length at which it stops being worth much is nearly the same in all three, because it is set by the lapse rate alone.

A count is not an estimate

Both established routes to a proportion are estimates — a duration timed, blurred by a Weber fraction, and a duration stored, biased by what was new. A listener who has induced a hypermetre has a third, and it is exact until it fails. It fails two ways that pull opposite: a slip miscounts one unit and its relative cost falls as the section lengthens, while a lapse loses the count entirely and its chance compounds. The mixture has a floor at about four units, where counting is 3.7 times finer than timing, and it is worth almost nothing past a hundred.

form · Proportion Timing blurs a whole form evenly; counting sharpens it downward. A piece of 480 seconds divided 7 times, each level half the length of the one above, with how many proportions between 1 : 1 and 3 : 1 a listener can tell apart at each. Timed, the answer is 2.1 at the top and 5.2 at the bottom, a spread of 2.4 — because a timing judgement's Weber fraction is a step function of duration and almost every level of a piece falls in one step of it. Counted, in units of 2 seconds, the answer runs 2.2 to 10.3, a spread of 5.5. At no level does timing separate 3 : 2 from the golden section.

A form is sharp at the bottom and vague at the top

A movement is divided into sections, each into phrases, each into bars, and every level is a ratio of two estimates. Timed, the hierarchy is almost uniformly blunt — 2.1 distinguishable proportions at the top and 5.2 at the bottom, because a Weber fraction is a step function of duration and six of a piece's seven levels fall in one step of it. Counted, the same hierarchy runs from 2.2 to 12.2 and sharpens monotonically downward. At no level of either does timing separate 3 : 2 from the golden section.

form · Proportion The golden section and an equal division are one judgement. Where a boundary falls in a piece, as a share of its length, with the band a listener cannot tell from the golden section shaded. A stretch of minutes is judged with a Weber fraction of about 38%, so one criterion's worth of ratio spread around 0.618 covers everything between 0.492 and 0.730 — a quarter of the piece wide, and containing the halfway point. 1 : 1 and 4 : 3 and 3 : 2 and golden section and 2 : 1 are inside it. A claim that a climax falls at the golden section rather than at the middle is, at this resolution, not a claim about anything a listener could hear.

A golden section is a coin toss with six coins

An analysis that reports a climax at 0.618 of a piece has not tested one prediction; it has looked at a piece with several defensible boundaries and reported whichever landed nearest. The rate at which that happens under no hypothesis is one line of arithmetic, and the tolerance it needs is not a number chosen on the page — it is the blur a listener's own timing puts on the judgement. Over a stretch of minutes that blur covers everything from 0.492 to 0.730 of the piece, which contains the halfway point, and six candidate boundaries produce a hit eighty per cent of the time.

form · Proportion The breath is the looser ceiling nearly everywhere. How long a trained singer can hold a phrase on one breath, across a compass and at four dynamics, against the 8-second ceiling the psychological present puts on the same phrase. The flow through the folds rises with pitch and with loudness, so the breath ceiling falls both ways: at 60 decibels it runs 32.6 seconds at the bottom of the compass to 21.2 at the top; at 70 decibels it runs 23.1 seconds at the bottom of the compass to 15.0 at the top; at 80 decibels it runs 16.4 seconds at the bottom of the compass to 10.6 at the top; at 90 decibels it runs 11.6 seconds at the bottom of the compass to 7.5 at the top. The shaded line is the listener's ceiling and it does not move. The breath binds only where the two lines cross — 1 of the 40 cells drawn, all of them loud and high. So the constraint everybody names when asked why a phrase is the length it is, is almost never the constraint that decides it.

The ceiling everybody names is the loose one

Ask why phrases are the length they are and the answer given is the breath. It is arithmetic — usable lung volume over the air a note costs per second — and it comes out between fifteen and twenty-three seconds at a comfortable dynamic and between seven and twelve at a loud one. The ceiling the present moment imposes, the two-to-eight seconds inside which a stretch is heard as one thing rather than as a series, is two to three times tighter at almost every note and dynamic. A singer in an adagio is not running out of breath at the phrase end. They are running out of present.

form · Phrase

Named alongside it

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

Perceptual presentDurationGroupingHypermetreMusical formSegmentationTempoNotationProportionScale-spaceWeber fractionCadence

All concepts