Acoustics & Music Theory
why it sounds like that
Timbre and acoustics

A dynamic mark changes what a note is

Every spectrum until now is a shape with a level in front of it, so that playing ten decibels louder raises every partial by ten. That is true of exactly one instrument in an orchestra. Everybody else steepens their own spectrum as they lean on it, and a trumpet's centre of gravity moves from the second partial to the sixth across a dynamic range while an organ flue pipe's does not move at all.
15 min read 7 figures The instrument is not the modelThe design was forcedNotation hides things

Assumes: The ranking survives the dynamic and the chord does not · The mark that is not a level

There is an assumption running under every spectrum figure in this collection, and it has been there since the first one. A timbre is a list of amplitudes, and a level is a number in front of the list. Play ten decibels louder and every partial goes up by ten.

That assumption is what makes a spectrum a shape — something that can be normalised, compared with another shape, and reasoned about without a dynamic. It is the reason the first rung of this anchor could search over levels at all, since the search assumed each player’s spectrum stayed put while its gain moved.

It is also false for every instrument in an orchestra except one, and the exception is worth naming first because it is the control the rest of the argument needs. An organ flue pipe has a fixed dynamic. Its voicing — the wind pressure, the mouth cut-up, the nicking — is set at a bench by a voicer and the player has no dial at all; a stop is loud or it is not. A flue pipe really is a shape with a level in front of it. Nobody else is.

Trumpet at three dynamics, as a spectrum rather than a level. The radiated partials of a trumpet at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 2.19 to 6.41, a factor of 2.92, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 3 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument.
Fig. 1 A trumpet at three dynamics, with each spectrum normalised to its own strongest partial so that only the shape is being compared. A linear source would give three identical pictures. The centre of gravity moves from about the second partial to about the sixth — and the loudest of the three has almost nothing in its fundamental, which is what makes a fortissimo brass note carry through a texture that would swallow it played softly.

What a player is actually doing

Blowing harder into a brass instrument does not scale the pressure waveform. The lips are a valve, and a valve driven harder shuts faster and opens later; the pressure pulse acquires a steeper front, and a steeper front is upper partials. The reed is a valve is the essay about the mechanism, and blowing harder is playing sharper is about a different consequence of the same nonlinearity.

A bowed string does something similar for a different reason: more bow force moves the stick-slip corner toward the ideal sawtooth and away from the rounded one, so the high partials come up faster than the low ones. A voice does it because a higher subglottal pressure closes the folds more abruptly, which the glottal pulse is the shape of. A piano does it because a harder blow shortens the hammer’s contact time and a shorter contact low-passes less, which the mark that is not a level computed in full.

That last one is worth pausing on, because this collection has already done this calculation once, from the inside, for one instrument.

A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there.
Fig. 2 The piano’s version of the same fact, computed from the felt rather than stipulated: each dynamic marking, the level it produces, and the partial at which the contact pulse first has a null. The scale of marks is ordinal and the scale of nulls is not, so the marks are an ordering laid over a continuous quantity that moves for a physical reason.

The model, stated as a tilt

What this rung needs is not another instrument-specific derivation. It needs the same variable for every radiator in the orchestration solver, so that the joint problem can be run with the nonlinearity and without and the two answers compared.

So the model is a tilt. Amplitude n is multiplied by n raised to a power, which is exactly a straight line of a stated slope in decibels per octave of partial number, and the slope grows linearly with how far the level is above a reference. One number per instrument, in decibels per octave per ten decibels of level:

trumpet   3.0        clarinet  1.6        violin    0.9
voice     1.8        oboe      1.2        flue pipe 0.0

These are asserted, ordinal numbers and not a measurement of any instrument. This collection has recorded the same warning against a good many of its own figures and it applies here as squarely as anywhere: the ordering is the result and the magnitude is the illustration. What is published, repeatedly and by everyone who has measured it, is that a brass spectral centroid moves a great deal across a dynamic range and a string’s moves much less. These tilts reproduce that ordering by construction, which means the figure below is a way of seeing the ordering rather than evidence for it.

The flue pipe’s zero is the exception to that caution and is a fact rather than an assertion. It is there so the figure has a line that is exactly flat, and so that a reader can tell the difference between a model in which nothing happens and a model in which the tilt is small.

Where the centre of a spectrum goes as the player leans on it. The spectral centroid — the power-weighted mean partial number — for 6 radiators over a sixty-decibel range, all on the same written note. trumpet moves from 1.81 to 6.59; voice on “hod” moves from 1.61 to 6.71; clarinet moves from 1.12 to 3.75; oboe moves from 1.27 to 2.91; violin moves from 1.22 to 1.55; flue pipe moves from 1.81 to 1.81. Flue pipe is the control and is exactly flat, because an organ flue pipe's dynamic is fixed at the voicing bench and a player has no dial to turn. The ordering is the result and the magnitudes are the illustration: what is published is that a brass centroid moves much further across a dynamic range than a string's, and these tilts reproduce that ordering by construction.
Fig. 3 The spectral centroid — the power-weighted mean partial number — for six radiators across sixty decibels on one written note. The flue pipe is flat by construction and the others are not. The gap between the trumpet’s line and the violin’s is the whole reason a brass section can be asked to play quietly and still be a brass section, and the reason a string section cannot be asked to sound like one.

What it does to the answer

Run the joint solve twice — once with fixed spectra, once with the tilt — and the interesting result is that the ordering survives and the cost of getting it wrong does not.

With fixed spectra the joint answer for a C major triad on a clarinet, a violin and an oboe is violin, oboe, clarinet, at nought, minus seven and a half, and minus one and a half decibels. With the nonlinearity it is the same three players in the same places at the same levels. The roughness of that answer moves by four per cent, because the parts that were pushed down are now also duller and the parts that were left up are brighter.

What changes is the penalty for solving in stages. It was nine per cent; with the nonlinearity it is nearly eleven. Making the spectra depend on the levels makes the two halves of the problem more entangled, which is the direction anybody would guess and is worth having computed rather than guessed — the guess could have gone the other way, since a brighter loud part might have been expected to make the equal-level ranking a better predictor of the balanced one.

The same problem, with the excitation nonlinearity and without. The joint solve run twice: once with each instrument's spectrum a fixed shape, which is what every earlier figure assumes, and once with the shape a function of the level it is played at. With fixed spectra the answer is violin · oboe · clarinet at 0, -7.5, -1.5 dB; with the nonlinearity it is violin · oboe · clarinet at 0, -7.5, -1.5. The ordering survives, and the roughness of the answer moves by 4 per cent. The two-stage penalty is 9.0 per cent with fixed spectra and 10.7 with the nonlinearity.
Fig. 4 The same problem solved twice. The upper pair is the fixed-spectrum world every earlier figure lives in; the lower pair has each instrument’s shape depending on the level it is played at. The winner does not move. The gap between the joint answer and the assignment-first answer opens by about two percentage points, which is the entanglement the nonlinearity adds.

One number in that comparison is worth reading against the sweep two sections below, because the two together say something neither says alone. The joint answer at middle C is the same three players at the same levels with the tilt and without — and the register sweep finds that this is true at seven of the eight roots tried. The stability is not a property of middle C; it is a property of the problem, and the one exception is the place where the answer was about to change anyway.

Why the winner does not move, and what would move it

The ordering surviving is not an accident and it is not a reassurance. It follows from what the tilt does to a set of spectra: it rotates all of them in the same direction, and it rotates them by amounts that depend on the instrument rather than on which note the instrument is playing. So an arrangement that was smoothest because a particular pair of spectra kept its partials apart still has that pair apart afterwards — the pair is brighter, and so is every other pair.

What would move the winner is a tilt that depended on the note, and there is a plain physical reason for one: an instrument’s filter does not move with its pitch. A trumpet’s bell cuts at a fixed frequency, so brightening the source by three decibels an octave puts far more energy past the cutoff on a low note than on a high one, and the amount of brightening that reaches the air is therefore pitch-dependent even when the amount at the source is not.

That effect is in these figures, because the tilt is applied to the source and the filter is applied afterwards — which is the right order and is the only reason the trumpet’s fundamental disappears at the loudest level rather than merely shrinking. What was missing was a sweep over pitch with the tilt on, to say at which register the nonlinearity starts changing assignments rather than only changing costs.

The register at which it does

Running the joint solve at eight roots from C2 to G5, with the tilt and without, gives an answer narrow enough to state as a note.

root fixed spectra with the tilt
C2, C3, G3, C4 violin, oboe, clarinet the same
G4, 392 Hz violin, oboe, clarinet clarinet, oboe, violin
C5, G5 clarinet, oboe, violin the same

The assignment moves in exactly one place, and it moves because the fixed-spectrum answer already changes somewhere near there. Bisecting for the crossing gives 395.9 hertz without the tilt and 378.2 with it, so the nonlinearity does not invent a new arrangement anywhere — it shifts an existing boundary down by seventy-nine cents, three quarters of a semitone, and G4 happens to fall between the two.

That is a small answer and it is the useful kind, because it separates two things the rung had run together. The tilt raises the roughness of the best scoring at every register, by between four and eleven per cent, so it changes what an arrangement costs everywhere. It changes what an arrangement is over a window less than a semitone wide. A player asking whether the nonlinearity matters to their part can therefore be told: it matters to how rough the chord is at every pitch, and to who plays which note at one pitch only.

One further thing fell out of the sweep and is worth recording because it is a property of the constraint rather than of the tilt. At G2 there is no feasible answer at all: no distribution of levels on the grid brings the three parts’ loudness shares inside the four-and-a-half per cent tolerance, the closest being 6.4 per cent. Low chords on these three instruments cannot be balanced in the sense the solver means, which is a fact about a clarinet, a violin and an oboe at 98 hertz rather than about a model. With the tilt on, the closest feasible error falls to 4.54 per cent — just outside — so brightening the loud parts nearly rescues it, which is the direction a real orchestrator would expect and is the one place in this rung where the nonlinearity helps.

The same intervals, a clarinet under and over. Each interval scored twice: once with a clarinet on the lower note and a violin above, once the other way round. With one spectrum for both notes these bars would be identical, because the roughness sum is symmetric; with two spectra they are not. The largest disagreement is at the major sixth, where one arrangement is 5.1 times rougher than the other with the same two notes in it.
Fig. 5 The asymmetry the whole argument rests on, from an earlier essay on spectra: the same written interval is a different amount of rough depending on which instrument is underneath. A tilt that brightens one of the two players and not the other moves every bar in this figure by a different amount, which is the mechanism by which a dynamic can in principle reorder an assignment — and the reason it does not do so here is that the tilt brightens both.

The thing a score cannot say

A dynamic marking is one symbol and it is now doing two jobs. It sets how much a part contributes and it sets what the part contributes, and the second is not derivable from the first without knowing the instrument.

That has a consequence for the notation that the notation ladder named and could not quantify. Two parts marked ff are not at one level, because the mark is ordinal within an instrument rather than absolute across instruments — and now they are not at one timbre either, and the divergence is instrument-specific. A trumpet at ff and a violin at ff have moved their centroids by very different amounts from where they were at pp, so the blend of a passage changes with its dynamic in a way no transposition of the marks can undo.

This is the arithmetic under a piece of orchestration advice that is usually given as taste: that a soft brass chord and a loud one are different colours rather than one colour at two sizes, and that a doubling which works at one dynamic has to be checked at the other. The advice is old and the reason for it is one exponent.

Violin at three dynamics, as a spectrum rather than a level. The radiated partials of a violin at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 1.24 to 1.52, a factor of 1.22, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 0.9 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument.
Fig. 6 The same three levels on a violin, which is the comparison the trumpet figure needs. The shape does move, and it moves far less — the centroid goes from about the fifth of a partial above one to about half of one. A string section’s fortissimo is mostly louder; a brass section’s is mostly different.

There is a harder version of the same claim, and it is the one that makes “what the part contributes” more than a figure of speech.

The level at which each partial becomes audible at all. Each point is one partial of one note, drawn at the whole-note level below which that partial is under the threshold of hearing and contributes nothing — not a small amount, nothing. C2 needs 39 dB before its FUNDAMENTAL is audible and 24 before its eighth partial is; E4 needs 10 dB before its FUNDAMENTAL is audible and 15 before its eighth partial is. In the treble the highest partials go first, which is expected. In the bass they do not: the threshold of hearing rises about fifty decibels between 500 and 30 hertz, far faster than a 1/n spectrum falls, so a quiet bass note loses its fundamental and keeps the partials that were supplying the roughness. At 20 dB the bass has 0 of 16 partials and the treble note has 12.
Fig. 7 The level below which each partial of each note is under the threshold of hearing and contributes nothing at all — not a small amount, nothing. A C2 needs 39 dB before its own fundamental is audible and 24 before its eighth partial is; an E4 needs 10 before its fundamental is.

So a dynamic mark does not only tilt a spectrum, it decides how many partials there are. Below a note’s own cutoffs the upper partials are not quiet, they are absent, and the number present is a function of the level and the register together. That is why the two jobs the mark is doing cannot be separated by any amount of care in the notation: one of them changes the size of the object the other is scaling.

Which computation produced the numbers

The base spectra are the collection’s own radiatedPartials, unchanged: a source through a filter, with nothing fitted. The trumpet is new and is built the same way — the 1/n sawtooth a lip valve produces to first order, through the bell as a high-pass filter at eleven hundred and fifty hertz.

That cutoff is the bore ladder’s own number and it is quoted here as a model output rather than a measurement, for the reason that ladder states: published trumpet cutoffs are nearer fifteen hundred, and the gap between the two is what the bore anchor’s next rung is about. Using the site’s own figure rather than the published one keeps the collection consistent with itself, and every claim here is a comparison between two levels of one instrument rather than a claim about an absolute frequency.

The tilt is applied by multiplying amplitude n by n to the power of the per-octave slope divided by 6.02, which is the number of decibels an octave is, and the slope is scaled linearly by the level’s distance above seventy decibels. Passing a tilt of zero recovers the old function exactly, and the joint solver takes the choice as a flag rather than by calling a different function — which is what makes the two runs in the fourth figure comparable rather than merely similar.

The centroid is the power-weighted mean partial number, computed on the normalised spectrum so that it is a property of the shape and not of the level.

Where the model stops

A tilt is a one-parameter caricature of a nonlinearity. A real excitation does not shift every partial by a straight line on a log-log plot; brass spectra develop a shock front whose effect is closer to a moving corner frequency than to a rotation, and reed spectra move their formant-like features rather than tilting. The tilt gets the direction and the ordering right and nothing else, and it is used only where those two are what the argument turns on.

The tilt itself has no dependence on pitch, though its effect does, because the filter it passes through is fixed. A trumpet’s brightness at ff is not the same story at the bottom of its range as at the top, and an instrument is not one timbre is the essay about that. What the sweep above establishes is only that the assignment is insensitive to it outside one narrow window; the costs are not, and a model with a pitch-dependent slope at the source would move them again.

And the reference level is asserted. Seventy decibels is where the tilt is zero, so every spectrum in every figure here is exact at seventy and a caricature away from it. Nothing sets that number except the convenience of having the old figures unchanged.

What the picture cannot show

It cannot show that the tilt is the right shape. The comparison that would settle it is a recorded spectrum of one instrument at several dynamics against this model at the same levels, and this collection has no recordings — every figure in it is a model evaluated at published parameters, which is its normal state and is worth restating on a rung whose central object is asserted.

Nor can it separate the player from the instrument. A fortissimo is a thing a person does, and how much brighter a trumpet gets at ff depends on the trumpeter as well as on the trumpet. The tilt has one number where there are plainly two.

And it cannot say what any of it sounds like. The sound buttons on the first figure play the three shapes at one loudness, which is the honest way to hear a change of spectrum and is not what a change of dynamic sounds like — that would need the level to move as well, and then the shape and the size would be confounded exactly as they are in the music.

Whose instruments, and when

The flue pipe’s zero is a fact about a mechanism and holds wherever there are organs. Everything else here is a claim about instruments with a continuously variable dynamic, which is most but not all of them: a harpsichord has none, a clavichord has a very small one, and both belong to repertoires whose expressive apparatus is arranged around the absence.

The orchestration consequence belongs to the repertoire in which the dynamic range is large and written down, which is the same nineteenth-century boundary the previous rung ran into. A Classical orchestral forte and a late-Romantic one differ by enough decibels that the tilt between them is audible as a change of colour, and the scoring practice on either side of that boundary differs in exactly the way the model predicts: earlier scoring doubles for weight and later scoring doubles for colour, because later scoring has more colour to work with.

Where this ladder goes next

Three rungs, and the anchor has what it was opened for: the two halves joined, the level let move, and the spectra let move with it.

What the ladder owes now is the variable none of the three has touched. Every scoring here is one chord held steadily, and an orchestration is a succession — the loudness ladder’s third rung is entirely about a running impression with a two-second release, so a balance that is correct at an instant is not necessarily correct in a passage, and a part that enters is not the same as a part that was already there. Joining this anchor to that smoother would make the objective a functional over time rather than a number, which is a different kind of optimisation and needs nothing this collection has not got.

Part 3 of 14

One essay in the series on orchestration. The essays either side of this one:

Roughness and loudness do not rise together. One four-note chord, played at levels from 35 to 95 decibels, with both quantities drawn as multiples of what they are at the quietest. Roughness is quadratic in pressure, so 60 decibels multiply it by 1.0e+6. Loudness is compressive — about ten phons to a doubling of sones — so the same range multiplies it by 96. The gap between the two lines is the quantity: roughness per sone rises by a factor of 1.0e+4 between a pianissimo and a fortissimo of the same chord. The ranking survives the dynamic and the chord does not Part 2 — Roughness is quadratic in pressure and loudness is compressive, so sixty decibels multiply a chord's roughness by a million and its loudness by ninety-six. Roughness per sone therefore rises ten thousandfold between a pianissimo and a fortissimo of the same four notes — and yet the ranking of which doubling is smoothest, over four hundred and eighty voicings, does not move by a single place. A written dynamic is an instruction to the listener's impression. Every earlier scoring holds one chord still. A passage is a succession, and the running impression of loudness carries a chord into the one after it, so what a marking asks for and what playing the marking produces are different things. Here is a five-chord passage with a written shape. Playing each chord at its own written loudness gives the running impression 2.4, 3.0, 4.2, 5.6, 4.0 sones against the 2.4, 3.0, 4.2, 5.6, 2.0 that were asked for — right until the last chord, where it misses by 2.0. Solving for levels that make the impression arrive at the marking does not fix it: the last chord's target is I, two parts, and it is unreachable — the correction runs to silence and the impression still sits 1.1 sones above. A subito piano after a full chord is not a level a player can produce. It is a rate of change, and the smoother's two-second release is what refuses it. A subito piano is a rate, not a level Part 4 — All three earlier essays score one chord held still. An orchestration is a succession, and the running impression carries a chord into the one after it — so a written dynamic is an instruction to the listener's impression rather than to the instantaneous sound, and there are markings that cannot be produced at all. The correction runs to silence and the impression still sits above the target.

Where this goes next

Roughness and loudness do not rise together. One four-note chord, played at levels from 35 to 95 decibels, with both quantities drawn as multiples of what they are at the quietest. Roughness is quadratic in pressure, so 60 decibels multiply it by 1.0e+6. Loudness is compressive — about ten phons to a doubling of sones — so the same range multiplies it by 96. The gap between the two lines is the quantity: roughness per sone rises by a factor of 1.0e+4 between a pianissimo and a fortissimo of the same chord. The ranking survives the dynamic and the chord does not Roughness is quadratic in pressure and loudness is compressive, so sixty decibels multiply a chord's roughness by a million and its loudness by ninety-six. Roughness per sone therefore rises ten thousandfold between a pianissimo and a fortissimo of the same four notes — and yet the ranking of which doubling is smoothest, over four hundred and eighty voicings, does not move by a single place. A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there. The mark that is not a level There are six of them, they carry no units, and a performer has to turn one into a number before it means anything. What they instruct is not loudness. On a struck string a harder blow shortens the hammer's contact from 2.26 milliseconds to 0.95, which moves the first null of its own pulse from the third partial to the sixth: the partials between those are not quieter at pianissimo, they are gone. A fortissimo is a different sound, and the page has one word for both things it changes. One instrument, five spectra. The first 12 partials of a bowed string at 5 pitches, each passed through the same fixed body response and normalised to its own loudest partial. The resonances stay where they are and the partials slide under them, so the pattern is different at every note: the power-weighted centroid runs from 1.02 to 1.71 across the compass and is not monotonic in pitch. A source with no body at all would give 2.35 at every pitch, which is the single number every roughness figure here uses for a string. An instrument is not one timbre Every roughness verdict so far uses one partial list per instrument, and a violin does not have one. Its body's resonances stay where they are while the fundamental moves, so the radiated spectrum is different at every pitch: the power-weighted centroid runs from 1.02 to 1.64 across the compass, and it is not monotonic. Which means the result that a spectrum chooses its own scale gives a different scale at every note on the same instrument — three minima in the dissonance curve at the bottom of a violin's range, two in the middle and four at the top.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

BrightnessDynamicsExcitation pointOrchestrationRoughnessSpectral balanceSpectrumTimbre