The dissonance arrives and the dynamic does not

A subito piano is a rate, not a level made this anchor’s objective a functional over time: a scoring is a succession, a written dynamic is a target for a listener’s running impression rather than for the instantaneous sound, and the loudness ladder’s smoother has a two-second release. What came out was that a marking asking for a fall is often unproducible.

Its last line names the other half:

What is owed next is the roughness over time.

A scoring decides two things. How loud each part is, which is the loudness ladder’s, and how rough the result is, which is this one’s. The fourth rung put a window under the first and the second has never had one.

The roughness has its own window and it is short #

A roughness is not an instantaneous quantity any more than a loudness is. It is a fluctuation — partials beating inside a critical band — and a listener needs several cycles of that fluctuation before there is anything to judge. That is the argument the ninth rung of the voice ladder makes, and it uses the same construction: a window of several cycles of the roughness’s own rate.

The rate is computable and it is not slow. For the five chords of the passage this anchor has been using, the median fluctuation rate of the roughest pair in each is between 79 and 134 hertz. Four cycles of that is thirty-seven milliseconds.

The loudness release is two seconds.

Fifty to one, and that ratio is the whole essay.

What it does to a passage #

At a second and a bit per chord — a slow harmonic rhythm, a chorale — the roughness curve and its smoothed version are indistinguishable. Every chord’s roughness is delivered whole and the transitions are sharp.

The loudness curves are not. The raw one steps; the smoothed one rounds every corner, arrives late at every rise, and never gets down at all after a fall — which is the fourth rung’s finding and is why a subito piano is not producible.

So on the same passage, with the same players, one of the two quantities the orchestrator is choosing survives to the listener intact and the other is averaged.

And the pace decides how badly #

Sweeping the pace puts a number on it.

The roughness contrast is kept entirely from four and a half seconds a chord down to about forty-five milliseconds. Below that it collapses, and the collapse is abrupt because a boxcar window either fits inside a chord or does not.

Forty-five milliseconds a chord is thirteen hundred chords a minute. A demisemiquaver at a fast tempo is about sixty milliseconds. Nothing anybody plays is fast enough to blur a dissonance.

The loudness contrast is a different story at every pace on the figure. At two and a half seconds a chord it keeps 83 per cent — and that is its best, because the release does not finish even then. At three hundred milliseconds it keeps 82; at eighty, 67; at fifteen, 44.

A passage moving at four chords a second has lost a third of its dynamic contrast before a listener sees it, and none of its harmonic contrast.

The sizes are not comparable either, and that is a separate point #

The two contrasts are also different sizes to begin with, and the fourth rung’s sibling already knew why.

Across the five chords of this passage the roughness varies by a factor of 4,146 and the loudness by a factor of 2.8. That is the second rung’s finding in a passage rather than in a chord: roughness is quadratic in pressure and loudness is compressive, so a change of scoring that moves the loudness by a factor of three moves the roughness by three orders of magnitude.

Put the two together and a listener’s picture of a passage is a dynamic contour that is flattened and slow, over a dissonance contour that is enormous and instantaneous. The thing the ear tracks most sharply in a succession of chords is not how loud they are.

That is not what a score looks like. A score has dynamics written on it in eight steps and has nothing written on it about roughness at all — roughness is a consequence of the notes and the scoring and is never marked. So the quantity a composer controls explicitly is the one the ear smears, and the quantity nobody writes down is the one that arrives whole.

Which of the two a composer can hear while writing #

There is a consequence for the act of composing that is worth separating from the consequence for listening.

A composer at a desk hears a passage in imagination, chord by chord, slowly. At that pace both quantities are delivered whole: the loudness curve on the figure above keeps 83 per cent of its contrast at two and a half seconds a chord and the roughness keeps all of it. The dynamics a composer imagines are the dynamics a slow reading gives.

Played at tempo they are not. A passage at four chords a second delivers two-thirds of its written dynamic contrast and all of its harmonic one, which means the balance between the two shifts as the tempo rises — the faster the music, the more of what a listener receives is dissonance and the less of it is dynamics.

That is a prediction about rehearsal that anybody could test and it points the wrong way from the usual advice. A conductor who finds a fast passage dynamically flat is not being disobeyed; the passage is flat, at that speed, by an amount this figure gives. What would fix it is not more dynamic range but a slower tempo, and the exchange rate is on the curve.

What this does to the objective #

The fourth rung’s objective was a functional over time in one quantity. This makes it two, and they behave so differently that the joint problem simplifies rather than getting harder.

The loudness term has to be solved over the whole passage, because each chord’s level affects what the listener has at the next one. That is a coupled optimisation and the fourth rung had to iterate it.

The roughness term does not. At any musical pace the window is shorter than a chord, so each chord’s roughness is delivered on its own and the passage’s total roughness is a sum of independent terms. The roughness half of the objective is separable and the loudness half is not.

That is a useful thing for the anchor to know, and it says something about how orchestration is actually done. A scoring decision about balance has consequences three chords later; a scoring decision about which instrument takes which note does not.

Why the roughness window is short, and it is not a choice #

It is worth saying where thirty-seven milliseconds comes from, because it is not a constant anybody measured and it is not free.

The rate at which a roughness fluctuates is the frequency difference between the partials that are beating. In a chord of two to six parts spread over two or three octaves, the pairs that fall inside a critical band are high partials of widely separated notes — the sixth partial of one against the fifth of another, and so on — and those are tens of hertz apart. A hundred hertz of fluctuation is four cycles in forty milliseconds.

The loudness release is long for a completely different reason. It is not a number of cycles of anything; it is a measured property of how a listener’s impression of level decays, and it is two seconds because that is what was measured.

So one window is set by the signal and the other by the listener, which is why they are free to differ by a factor of fifty and why the difference is stable: change the music and the roughness window changes with it, staying inside the notes, while the loudness release does not move at all.

That also means the roughness window is shorter in the treble and in dense chords, where the beating pairs are further apart in hertz, and longer in the bass. A low, close-spaced chord has a slow fluctuation and therefore a long window — and it is exactly the arrangement this collection has been calling rough in the bass since the consonance ladder.

What the loudness correction does to the roughness #

There is one place the two do interact, and it is the fourth rung’s failure case.

A chord whose written dynamic cannot be produced drives the correction toward silence — the fourth rung’s subito piano, whose levels bottom out and still leave the impression 1.14 sones above the target. Roughness is quadratic in pressure, so a correction of ten decibels downward divides the roughness by a hundred.

So a passage scored to make its dynamics come out right is a passage whose quiet chords are far quieter in roughness than in loudness. The corrections the fourth rung computed are of the order of a few decibels on most chords and thirty on the one it calls unproducible, which is a roughness ratio of a thousand on that chord alone — larger than any change of scoring could produce and produced accidentally, as a side effect of chasing a dynamic marking. The correction that fails to deliver the dynamic delivers an enormous change in the other quantity instead — and the listener, who has the roughness whole and the loudness smeared, receives most of it.

A subito piano that cannot be heard as a drop in level can be heard as a drop in dissonance. That is a mechanism for a device the model called impossible, and it costs nothing to produce because it is the same physical act.

Which computation produced the numbers #

The passage is the fourth rung’s, unchanged: five chords of two to six parts with a written dynamic in sones for each, at a stated seconds per chord.

The loudness track is the loudness ladder’s pair of one-pole smoothers — a short one and a long one in series, with the long release at two seconds — applied to the instantaneous sone value of each chord at its own level.

The roughness of each chord is scoringRoughness at the same levels: every partial of every part against every partial of every other, through the Plomp–Levelt curve scaled to the critical bandwidth, and the whole thing quadratic in pressure.

The roughness window is four cycles of the median fluctuation rate over the chord’s own rough pairs, computed per chord and averaged. Four is the voice ladder’s number and is asserted there; the rate is computed. The window is a causal boxcar over the preceding interval, because a listener integrates the past and because a centred window straddles every chord boundary by construction and would make the answer independent of the pace.

Both contrasts are read at the end of each chord, when a listener has had the whole of it, and both smoothers are causal — so the only thing separating the two numbers is the length of the window.

Where the model stops #

Four cycles is asserted. Everything about the roughness window’s length rests on it. Two cycles halves the window and moves the cliff to twenty milliseconds; eight doubles it to seventy-five, which is still faster than most notes. The ordering against the loudness release survives any number in that range, which is what the argument uses.

A boxcar is not an ear. The abruptness of the roughness cliff is the window’s shape rather than a fact about hearing, and a smooth window would give a smooth roll-off in the same place.

The passage is five chords. The contrast figures are max over min across those five, so a different passage gives different numbers. The ratio of the two windows does not depend on the passage at all.

The roughness is computed on one timbre. Every part in the passage is given the same partial list, so the differences between chords are differences of voicing and register rather than of scoring — which is the one variable this anchor exists to study and which the first rung puts at nearly a factor of two on its own.

And roughness is not dissonance. Everything here is sensory roughness, which this collection has been careful about since the consonance ladder: a listener’s judgement of a chord contains expectation, context and familiarity, and none of those has a thirty-seven-millisecond window.

What the picture cannot show #

It cannot show the notes changing. Every chord here is a block, and real voice-leading moves some parts and holds others — so the roughness changes continuously through a transition rather than stepping, and the window would then be measuring a slope rather than a step.

Nor can it show forward masking. A loud chord makes its successor partly inaudible for a couple of hundred milliseconds, which is a longer time than the roughness window and would eat into exactly the contrast this essay says survives.

It cannot show the room. Two seconds of reverberation puts each chord’s spectrum into the next one, which is an acoustic version of the smearing this figure attributes to the ear.

It cannot show a single instrument. Everything here is a scoring of several parts, and a solo line has roughness only between its own partials, which is a property of the instrument rather than of the passage. The whole contrast this essay measures exists because more than one thing is sounding.

And it cannot show what a listener does with either. Having a quantity is not the same as using it, and no experiment here asks whether a listener who receives a large roughness contrast notices one. That is the standing limit on every rung of this anchor: the objective is stated in units nobody has shown a listener minimises, and the case for it is that its two halves are the two things an orchestrator manifestly does control.

Whose music, and when #

The passage is a generic tonal one and the dynamics are generic. Nothing here is about a repertoire.

The observation that fits a repertoire is the last one about the score. Written dynamics are a nineteenth-century preoccupation — the number of distinct markings in an orchestral score roughly quintuples between 1800 and 1900 — and they are markings on the quantity a listener smears. Over the same century harmonic rhythm slows and orchestration gets much denser, both of which raise the roughness contrast the listener receives whole.

That is a correlation with a plausible direction and no evidence, and it is stated here only so that the temptation to make it an explanation is visible.

Where this ladder goes next #

Five rungs. Who plays what and how loud is one question; the ranking survives the dynamic and the chord does not; a dynamic mark changes what a note is; a subito piano is a rate rather than a level; and now the other quantity that succession has, which needs no correction and arrives whole.

What is owed after this is the mask. Both windows on this page are integration times, and there is a third temporal constant in this collection that is neither: forward masking, whose couple of hundred milliseconds sits squarely between the thirty-seven and the two thousand. A loud chord does not merely raise the running loudness; it removes part of the next chord from the listener altogether, and it does so for a time longer than a roughness window and shorter than a loudness release. That is the term that would say whether the dissonance contrast this rung says survives is actually received — and it is the one piece of machinery this anchor needs that already exists on another ladder.