Acoustics & Music Theory
why it sounds like that
The collection

Every essay — page 20

Page 20 of 27, continuing through the fields in the same order.

Pitch and tuning Intervals and chords Scales and modes Harmony and voice leading Rhythm and metre Timbre and acoustics Perception and the listener Instruments and their design Form and structure Series Objects Sounds Search

Perception and the listener

Loudness is not amplitude and a beat is not in the signal. What the ear adds, hides, merges and supplies — measured, with the numbers every other field has been assuming.

3 tones, one power, and the interval between them. 3 tones of fixed total power, spread symmetrically about 440 hertz, drawn against the interval between neighbours. Piled on one pitch they are one sound of that power; separated by more than a critical band — 4.5 semitones here — they are 3 sounds whose loudnesses add, and the same power reaches 2.08 times the loudness at 5 semitones. The two lines are two models of the same rule and they disagree about how abrupt the change is, not about where it goes.

A chord is not as loud as its notes

The first essay on loudness said what a tone's loudness is and recorded that it had said nothing about a chord's. Here is the missing rule, and it has a musical consequence nobody would predict from it: the same three notes, at the same power, are twice as loud in the treble as in the bass — because the critical band that makes a low triad five times rougher also makes it one sound instead of three.

8 figures Three ways a category boundary could move, and how far each moves it. The predicted shift of one boundary against how strong the context is, for three mechanisms. Expectation alone — a listener who thinks one category 20 times more likely than the other — moves the optimal boundary by σ²·ln(odds)/Δ, which with the eleven-cent noise used here is 2.8 cents at ten to one and 3.6 at 20. Re-learning the centres from a context 30 cents away moves it by half of that, 15 cents. Selective adaptation moves it the OTHER way. The two directions are what an experiment would separate, and no absolute calibration is needed to do it.

The boundary that barely moves

Every identification figure here has fixed category centres, and the essay before this one ended by admitting that real boundaries are supposed to move with context. Three mechanisms could move one, and their predictions are an order of magnitude apart and in two different directions. Expectation on its own — a listener who thinks one interval twenty times more likely than the other — is worth three and a half cents.

7 figures How much earlier an accent is heard, by mechanism. An accented note on an instrument with a 90 millisecond attack, drawn against how many decibels louder it is, with the three ways it can arrive early separated. A criterion tied to the note's own peak on an unchanging envelope gives exactly nothing. The same criterion on the shorter rise a harder-driven instrument has gives 5.3 milliseconds at 12 decibels. A criterion at a fixed level gives 23.0. Both together give 24.0, and the rise at that dynamic is 73 milliseconds rather than 90. The rise-shortening exponent is stipulated at 0.15 rather than measured, and the two upper curves would separate further if it were smaller.

Playing louder is playing earlier

An accent has two effects on when its note is heard and neither is a timing decision. A harder-driven instrument has a shorter attack, and a criterion set by the surrounding music is crossed sooner by a bigger rise — so a twelve-decibel accent on a bowed note is heard twenty-four milliseconds early with no change whatever in when the bow was put down. It is also the measurement that tells the two competing models apart.

8 figures Where a room stops sending the two ears the same sound. The correlation between the two ears' signals against frequency, for a seat 15 metres from the source in a 15,000 cubic metre room with a 2 second reverberation time. The pale curve is the diffuse field alone — sin(kd)/(kd) for an ear separation of 17.5 centimetres, which first crosses zero at 980 hertz. The heavy curve adds the direct sound, which is coherent and lifts the whole thing by an amount the direct-to-reverberant ratio sets. At 125 hertz the coherence is 0.98 and at 1000 it is 0.16.

Where the two ears stop agreeing

A room sends both ears versions of the same sound, alike at low frequencies and increasingly unlike at high ones. Where they stop resembling each other is 980 hertz, and it is set by the 17.5 centimetres between the ears rather than by anything about the room — which is within a quarter of a frequency found earlier for a completely different reason. One minus that correlation is spaciousness, and it is computable from a room's own reverberation.

6 figures Why a concert hall is narrow. The lateral energy fraction at the middle seat as the same hall is widened, everything else held. It peaks at 12 metres across at 0.235 and falls to 0.000 at 44. A wide hall's side walls are further away, so their reflections arrive later, weaker and — this is the part Sabine's model cannot say — from nearer the front, where the sideways weighting discounts them. The shoebox halls the orchestral repertoire was written for are all between about eighteen and twenty-five metres wide, and this is the arithmetic they are the answer to.

A room with directions in it

Every room until now has been a reservoir of energy that drains at a rate. That model has no directions in it at all, so it cannot say the one thing every published measure of spaciousness is about: how much of what arrives comes from the side. Mirror the source in six walls and every reflection acquires an angle and a time — and the answer to why a concert hall is narrow falls out at eighteen metres.

7 figures How much of each spectrum a listener can assemble into one note. Each partial of each spectrum at the harmonic number it is nearest, against the whole-number series that fuses the most of them, with anything more than 1 per cent out marked as heard separately. an ideal string keeps 10 of 10; a piano string keeps 9 of 10; a bell keeps 7 of 8; a bar keeps 2 of 6; a kettledrum keeps 3 of 5. The fundamental is capped at a tenth of the top partial, and the cap is load-bearing rather than tidy: a bell's ratios are all whole multiples of a tenth, so an unconstrained search finds a fundamental twenty-five harmonics down, calls every partial exact, and reports that a bell fuses perfectly. Nothing that high is resolved and the low harmonics of it are not there.

The spectrum that will not fuse

A partial about one per cent off its harmonic is heard as a sound of its own rather than as part of a note. Apply that criterion to a whole spectrum instead of to one mistuned component and it becomes a count: a piano string keeps nine of its ten partials, a bell keeps seven of eight, a bar keeps two of six. The physics of inharmonicity has had an essay here for a long time. This is what it sounds like.

6 figures Every arrival at one seat, against the delay at which it would be an echo. The echogram: each reflection at its delay after the direct sound and its level relative to it, in a 22 by 45 by 15 metre hall with 18 per cent absorption. The line is the published echo threshold for speech — 40 milliseconds at equal level and about 3.0 more for each decibel of attenuation — so anything to the RIGHT of it is late enough and loud enough to be heard separately. The once-reflected rear wall arrives at 210 milliseconds, 25 decibels down, against a threshold of 114 — well past it. Nothing here stands clear enough of its neighbours to be heard as an echo, and 181 arrivals are fused with the direct sound instead.

An echo is prevented by the crowd around it

The echogram says when every reflection arrives and how loud it is; the published echo threshold says when a reflection that late and that quiet is heard separately. Put one against the other and the rear wall of every hall anybody builds is past the threshold — a 45-metre hall puts it 210 milliseconds late and 25 decibels down against a threshold of 114. It is not heard as an echo, and what saves it is not the geometry. It is everything else arriving at the same time.

6 figures Two fusion cues, and they do not agree about a single spectrum. Each spectrum twice. Hollow is the harmonicity census — the fraction of partials near enough a whole multiple of one fundamental to fuse, which is harmonicity. Filled is the same fraction under common fate: how many partials decay at a rate within a factor of 2 of the strongest partial's. Ranked by harmonicity the order is an ideal string, a piano string, a bell, a kettledrum, a bar; ranked by common fate it is a kettledrum, a bar, an ideal string, a piano string, a bell. The two orderings are nearly reversed. An ideal string is perfect on the first cue and 20 per cent on the second, and a kettledrum — the worst spectrum in this collection for fitting a series — is the only one whose partials all die together.

The partials that do not die together

The fusion census is a still photograph: it asks whether a set of partials fits one harmonic series and has no term for time. Put time in and the ranking reverses. An ideal string is perfect on harmonicity and holds a fifth of its partials together by decay; a kettledrum — the worst spectrum in this collection for fitting a series — is the only one whose modes all die at one rate.

6 figures Where the sound is, and how wide. An earlier essay sorted a hall's arrivals into echoes and everything else. Everything else is not nothing: a reflection too early to be heard as a separate event still moves the apparent source, widens it and colours it, and all three come out of the same list of times, levels and angles. At this seat the direct sound arrives from 26.6 degrees off the front and the image is heard 8.5 degrees left of it, pulled by the near side wall. The apparent width is 33 degrees, from a lateral energy fraction of 0.23. And the strongest early reflection arrives 0.6 milliseconds behind off 1× the floor, which is a comb filter with notches every 1608 hertz and 25 decibels deep. The trading ratio and the discount on late arrivals are stipulated rather than measured, so the degrees are ordinal: what the figure claims is the direction and the shape, not the number.

A position and a width

Sorting a hall's arrivals into echoes and everything else settled that fusion is not a yes or a no. Everything else is not nothing: a reflection too early to be heard separately still moves the apparent source, widens it and colours it. All three come out of the same list of times, levels and angles, and none of them needed a new input.

6 figures What a competition decides when the two cues do not agree. Each spectrum with its two cue readings and the grouping the competition chooses. Harmonicity asks whether a partial is near enough a whole multiple to belong; common fate asks whether it decays at the same rate as the rest. Where they disagree there is no rule in this collection, so the published apparatus is used instead: every way of splitting the partials into one stream or two is scored for the partials each cue says it has wrongly grouped and wrongly separated, and the cheapest wins. an ideal string — harmonicity 100 per cent, common fate 20, and the competition says one stream; a piano string — harmonicity 90 per cent, common fate 20, and the competition says a cut after partial 2; a bell — harmonicity 88 per cent, common fate 13, and the competition says a cut after partial 1; a bar — harmonicity 33 per cent, common fate 67, and the competition says a cut after partial 4; a kettledrum — harmonicity 60 per cent, common fate 100, and the competition says one stream. The exchange rate between the two cues is the number nobody here can supply, so what is reported beside each is how many decades of it leave the answer unchanged.

The exchange rate nobody has

There are now two cues that disagree about how a spectrum divides, and every figure so far reports them separately because there is no principled way here to weigh one against the other. The published apparatus is a competition between grouping hypotheses with a cost per cue — and the useful thing it produces is not the winner but how much of the exchange rate the winner survives.

7 figures The hall, as two numbers a listener has. Every reflection at this seat, placed by the interaural delay it produces rather than by the direction it came from. Time runs down; the dot's size is its energy. The direct sound is at 0 microseconds and the reverberation spreads over the whole available range, with a root-mean-square width of 322 against a geometric maximum of 656. That is the position and width computed earlier, in the units a listener has instead of the vectors used until now. 10 of the 56 reflections arrive from behind and carry 9 per cent of the energy — and they are drawn where they are because the interaural delay of a reflection from 120 degrees is identical to one from 60.

The hall through a head

Every direction computed so far is a vector from a seat to an image source, and a listener has no vectors. Run the echogram through the head computed six essays ago and two things happen: the position and width become microseconds, and a third of the room disappears — because both cues fold at ninety degrees and a reflection from behind is identical to one in front.

7 figures Nothing at all until fifteen decibels, and then it depends on the tempo. The fraction of a line's partials that stay above threshold, over how fast the line moves and how much louder everything before each note is. Darker is more lost. The whole left-hand side is white: at equal levels a note cannot be masked by its predecessor at any tempo, and that is a proof rather than a measurement — forward masking leaves a threshold at most ten decibels below the masker, and a note's own partials mask each other from the same components at full level. The boundary is between twelve and eighteen decibels, and beyond it the loss grows with the tempo: at 280 to the crotchet and 36 decibels of contrast, 26 per cent of the line's partials are gone. Fifteen decibels is about the gap between a forte and a piano.

An equal note cannot be masked

Three earlier essays are about one instant, and forward masking lasts two hundred milliseconds — longer than a note at any brisk tempo. So a fast line should be a sequence of events hiding each other, and it is not: a note masks itself ten decibels harder than its predecessor can, at any speed. What does hide a line is dynamic contrast, and the boundary is fifteen decibels.

7 figures The cue that settles it. Every spectrum to hand, arbitrated by the earlier competition and then again with the onset cue added at equal weight. 3 of the 5 change their verdict, and all 3 change the same way — from splitting into two streams to staying as one: a piano string, a bell, a bar. Nothing changes the other way, because the onset cue on a struck source votes for fusion on every partial and can only ever push toward one stream. The bell is the case worth naming: its partials are wildly inharmonic and it is heard as one sound, which is a fact the harmonicity cue alone cannot produce.

The cue that settles it

Arbitrating between two grouping cues meant sweeping an exchange rate nobody could supply. The cue it had no term for at all is the one every account calls strongest, and its strength is computable: a struck string's partials start together to within a tenth of a millisecond against a threshold of twenty. Put that into the competition and three of five verdicts change, all the same way — and a bell becomes one sound.

7 figures A page has two decibels and a player has sixty. Across, parts added to a final chord one at a time, each at the same level; up, the loudness that results, on a logarithmic scale. Going from one part to eight moves the total by 1.8 decibels and does not move it monotonically — four parts are louder than five and than eight. The faint line is what a naive power sum would give: 9.0 decibels. The band down the right is the same chord played by people, from forty to a hundred decibels, which spans 62. So a texture that thins from eight parts to one is not a diminuendo. It is a change of colour at constant loudness, and everything the closure figures call a dynamic belongs to the performance.

A page has two decibels

The account of closure called its dynamic component a corpus debt: it had no model of dynamics in a form. Loudness supplies one, and applying it answers the debt by refusing it. Adding parts to a final chord one at a time, each at the same level, moves the loudness by under two decibels and not monotonically — while a player has sixty. A texture that thins is not a diminuendo.

7 figures A turn of 0.99 degrees tells front from back. A source 45 degrees off centre and its mirror image 135 degrees off, which produce the same interaural delay and are therefore the same signal to a listener who does not move. As the head turns the two predictions separate: the front source's delay falls and the rear source's rises, because the fold at ninety degrees puts them on opposite branches of the same curve. They differ by the 15-microsecond threshold after 0.99 degrees of turn — which is exactly half the 1.97 degrees a source would have to move for the same listener to notice it moving, and it is half for a reason: a turn displaces the two hypotheses from each other by twice what it displaces either of them from where it started.

The turn is half the angle

A stationary head cannot tell a sound in front from the same sound behind, and an earlier essay said so at length. The turn that breaks the confusion is 0.84 degrees — exactly half the angle a source would have to move for the same listener to notice it moving, and half for a reason. In a hall the same turn does something else: the source swings at 8.9 microseconds a degree and the room swings at 2.7, so a listener who moves is separating the soloist from the reverberation as well as the front from the back.

7 figures The same geometry at four sizes of head. Woodworth's interaural delay against direction, for 4 head radii from 5.8 to 9.8 centimetres. The whole range runs from 431 microseconds for a newborn to 735 for a large adult, and it scales exactly with the radius because the delay is (r/c)(θ + sin θ) and r is a multiplier. The detection threshold does not scale with the listener, so the number of distinguishable delays across the whole range falls from 98 to 57: a smaller head has the same directions in front of it and a shorter ruler to measure them with.

A smaller head in the same hall

Ten earlier essays draw one head. Every parameter belonging to the room has been varied by some figure and the one belonging to the listener never has, and it is the only one whose change the detection threshold does not follow: a six-year-old in the same seat receives the same fifty-four reflections at the same instants and reads them onto an axis with seventy distinguishable positions instead of eighty-seven. The speed of sound, swept over every temperature a hall is ever at, changes nothing at all — and the reason it cannot is the reason head size can.

7 figures A clarinet's partials, each on its own resonance. The 8 partials of the clarinet's chalumeau D that ride an impedance peak, each building toward its steady amplitude as 1 − exp(−t/τ) with τ = Q/πf from that peak's own Q. The time constants run from 11.9 milliseconds to 64.5, so the partials do not arrive at different times — they all begin the instant the reed does — and what differs is how fast each approaches its final level. The horizontal bars are how far apart the first and last are at three criteria: 5.5 ms at 10 per cent, 36.4 ms at 50 per cent, 121.0 ms at 90 per cent. A twenty-millisecond asynchrony is the threshold for hearing a partial out of a note, and this note crosses it at 32 per cent of steady amplitude — so whether a blown note's onset cue is unanimous or divided is decided entirely by how far along a partial has to be before it counts as having started.

A blown note does not start late, it starts slowly

Computing the onset cue removed a free parameter and turned out to be unanimous, and it predicted that a wind instrument would put it back, because a blown note's partials arrive over tens of milliseconds. They do — 121 on a clarinet — and it is not an asynchrony: every partial begins the instant the reed does and they differ in rate, not in time. Read at a tenth of the steady amplitude the spread is 5.5 milliseconds against a threshold of twenty, so the cue is still unanimous, and the missing number is no longer the exchange rate but the criterion.

7 figures Eleven partials is one partial too many. What fraction of a spectrum the harmonicity census finds fused, against how many partials it is asked to census. At ten a perfect harmonic series fuses 10 of 10 and the fundamental it finds is the right one. At eleven it fuses 5 of 11 and the fundamental jumps to exactly 2.00 — the octave above. The cause is the cap the census carries for a reason established earlier: without it a bell fuses perfectly at a fundamental nobody could hear, so the search refuses any fundamental more than about ten harmonics below the top partial. At eleven partials the first thing that cap excludes is the series' own fundamental, and the census then takes the octave and calls every odd partial inharmonic. So the number of partials and the cap are the same number, and nothing had ever said so, because every earlier figure censuses ten.

Eleven partials is one too many

Six earlier essays census exactly ten partials and no figure has ever passed another number. At eleven, the harmonicity census stops finding a perfect harmonic series' own fundamental, takes the octave above it, calls every odd partial inharmonic, and the competition cuts an ideal string in two. It is not the arbitration — the cost of a second stream was swept over a factor of fifty and every verdict came back identical — it is a cap that exists for a good reason and turns out to be the same number as the count.

7 figures The census with the criterion moved under it. Every instrument's speaking time in milliseconds, against the fraction of the steady amplitude counted as speaking. The criterion is in the wind instruments alone: a resonance takes −ln(1−p)·Q/(πf) to reach a fraction p, so those lines rise across the whole picture, while a bow's capture and an exciter's contact contain no criterion at all and are flat. Every earlier figure sits at 0.9, where the wind instruments are the slowest things in the collection by a factor of 20.5. At 0.05 they are the fastest: a violin's G3 string is the slowest at 15.5 milliseconds and a trumpet takes 1.4. The three clusters cross at a criterion between 0.18 and 0.39, which is inside the range the perceptual measurements work in — their three named criteria are 15 decibels below peak, 6 decibels below peak, and ninety per cent — and the settling figures have only ever used the third of them.

Read at two different heights

Ten placements of these figures, one value: the settling criterion is nine tenths in every one of them, and nothing is measured behind it. It is a multiplicative constant only inside the mechanism that has it — a bow's capture and an exciter's contact contain no criterion at all — so moving it rescales one of three clusters against two that stand still. The most-quoted number here, a factor of sixty-nine between the instrument's account and the listener's, is 5.9 at the criterion the listener's own measurements use, and the ordering an earlier essay was written about does not exist below a fifth.

7 figures An entering part is worth 0.9 phons, in the middle of its range. A texture of 5 parts at 62 decibels each, with one more part added at the same level, tried at every semitone from C2 to C7. The vertical axis is what the addition is worth in phons, and a phon is a decibel here; the shaded strip is the difference limen for loudness, so an entry inside it is not heard as a change of level. The median entry is 0.89 phons and only 29 of 61 clear the limen — the lowest of them at A♭4, 415 hertz. The best available, at B♭6, is worth 4.2. The two lines are the two loudness models to hand: they agree everywhere above the tenor register and part company below it, where the greedy critical-band grouping reports 24 entries that make the texture quieter and the excitation pattern reports none.

A part entering is not a change of level

Five earlier essays measure a sonority that is already sounding. The one thing an orchestrator actually controls is the entry, and priced at every semitone it comes to 0.89 phons — under the difference limen — with only 29 of 61 available entries clearing it and none below A♭4. What an entry is instead is an object arriving, and the reason is that its own rise time is faster than the listener's.

7 figures Articulation is worth 2.6 phons, and nobody counts it. A passage of notes at 80 decibels, 2 to the beat, at 7 tempi and 3 articulations, scored against the same level held continuously. The variable is the fraction of each inter-onset interval that is sounding — 0.95 is a legato, 0.4 a staccato — and the vertical axis is what that costs the passage's running loudness in phons. Nothing here is anybody playing harder or softer. At 40 to the beat the span from legato to staccato is 1.47 phons; at 200 it is 2.61, because a staccato note there lasts 60 milliseconds and no longer reaches its own loudness either.

A staccato is a dynamic mark

Every loudness figure in this collection is of a sound that has been going on long enough, and no note in music has. Run the running-loudness model on notes with lengths in them and an articulation turns out to command 1.5 phons at a slow tempo and 8.1 at a fast one — more than the 1.8 decibels a whole texture commands, on the same page, written down in the same ink, and counted by nobody.

7 figures The top voice arrives whole and the bottom one arrives as a sine. 4 parts sounding together, each of 8 partials, with every partial tested against the summed masked threshold of every component in the texture. A filled mark is a partial the listener receives and an open one is a partial the part would have had alone and does not have here. The bass at C3 keeps 1 of 8, the tenor at C4 keeps 2 of 8, the alto at E4 keeps 5 of 8, the soprano at G5 keeps 8 of 8, every one of them at 70 decibels. Every part is at the same level and the difference is entirely where each one sits: masking spreads upward, so the part at the top of the texture has nothing above it to be masked by and the part at the bottom has everything.

The listener is given the top voice, and the bass as a sine

Four earlier essays put the masker and the probe in the same voice. Put them in different voices — a four-part texture at one level — and the soprano arrives with all eight of its partials, the alto with five, the tenor with two and the bass with one. Balancing the loudness, which is the constraint a scoring is solved under, changes none of that: equal loudness is not equal spectrum and cannot be made so.

7 figures The roughest chord on the page is at C1 and the roughest one heard is at A♭2. One close triad at 70 decibels through 17 registers, with its roughness computed twice: over every partial in the score, and over only the partials that stand above what the chord itself masks. The written curve rises all the way down and its maximum is the lowest register drawn, C1, which is the low-interval rule as it has always been computed here. The delivered curve turns over at A♭2 and falls to nothing below A♭1: a close triad down there is not rough, because it is not arriving as a chord — 1 of its 24 partials survives at C1 and there is almost nothing left to beat against anything.

A low chord stops being rough by stopping being a chord

Every count of audible partials until now is of a chord at middle C, and the three registers it did compare span C3 to C5 — a third of the range a chord is written in. Move the same triad down and the count collapses: 79 per cent of its partials arrive at E3 and 4 per cent at C1. So the roughest chord on the page is the lowest one and the roughest chord a listener receives is at G2, and where that maximum sits moves nearly two octaves with the dynamic.

7 figures One attack time, three shapes, 48 ms of disagreement. Three amplitude envelopes with the same 90-millisecond attack time, which is the only quantity the published tables report. a resonator from a step rises as one minus a decaying exponential; an excitation ramping rises in a straight line; a ramp through a resonator is a raised cosine. Each is normalised so that its own 10-to-90 per cent rise takes exactly 90 milliseconds, so all three are the same measurement. The horizontal rules are the three criteria a heard moment is read off. At 6 dB below peak the three shapes put the heard moment at 28.5, 56.4, 76.3 milliseconds — a spread of 48, on one attack time, from a property nothing in the table records.

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.

7 figures