The mark that is not a level

The page has three ladders on it. The vertical one counts letters, the horizontal one counts subdivisions of an asserted bar, and the third is written underneath in italics and has six rungs: pp, p, mp, mf, f, ff.

The third is the only one with no units at all. A crotchet is a defined fraction of a bar and a space on the staff is a defined step of a scale; a mezzo-forte is louder than a piano and quieter than a forte and that is the whole of what the system says. Somebody has to turn it into a number, and this essay is about what happens when they do.

An ordinal ladder is not a scale #

The first thing to notice is a mathematical property rather than a musical one. The marks are ordinal: they have an order and no distances. Nothing in the notation says that mf to f is the same step as p to mp, and nothing says that either of them is the same step in a string quartet and in a symphony.

The consequences of that are measurable and they have been measured. When performances of the same passage are recorded and their sound levels tabulated, the level associated with a given mark spans about twenty decibels across players and halls, and the ranges for neighbouring marks overlap substantially — a forte from one player is quieter than a mezzo-forte from another often enough that the two distributions are hard to separate. Those figures are quoted here; this site has no recordings and measures nothing of the kind.

What it does have is the physics of what happens to an instrument when a player follows the instruction, and that turns out to be the more interesting half.

It is worth being clear about why an ordinal ladder is a reasonable thing to write. The alternative — a number — would have to be a number of something, and every candidate is wrong. A sound pressure level is a property of a point in a hall. An amplitude is a property of a recording. A hammer velocity is a property of one instrument’s action. There is no quantity a composer has access to that survives being carried to another room, so the notation names a position in a range and leaves the range to be supplied. The mistake is not the ladder; it is reading the ladder as though the quantity it orders were loudness.

The hammer decides which partials exist #

A piano string is struck by a felt hammer. Felt is a nonlinear spring: the harder it is compressed the stiffer it gets, so a faster hammer is in contact with the string for a shorter time.

The single number that carries this is the first null of the contact pulse. A force pulse of duration τ has its first zero at a frequency of about 1.5/τ, and every partial above that is suppressed. At the softest dynamic that null is at partial 2.5; at the loudest it is at partial 6.

The partials above the null are not quieter at pianissimo so much as removed. That is a different claim from “the note is softer”, and it is why a recording of a soft note played back loudly does not sound like a loud note. Nobody who has tried to fake a fortissimo with a volume control has been convinced.

Three consequences of that are worth having as numbers, and each of them says something the ordinal ladder cannot.

How much of the thirty-five decibels is the spectrum #

The level rises 35.1 decibels from pp to ff. The velocity ladder is a factor of thirty-two, and thirty-two in amplitude is 30.1 decibels, so five of the thirty-five come from somewhere other than the string moving further. They come from the widening spectrum: at ff the note has significant energy in partials that at pp are below the null and contribute almost nothing to the total, and the extra partials add power.

One seventh of the dynamic range is therefore not a change in how hard the string is driven but a change in how many partials are being driven at all. The fundamental on its own rises 32.6 decibels — more than the 30.1 the velocity alone accounts for, because at pp the contact pulse is long enough to be attenuating even the first partial, and shortening it lets the fundamental through as well as everything above it.

The invariant the choice of ladder cannot move #

The computation note below says the velocity ladder is stated rather than measured, and that choosing a different one moves every number together. That is true, and there is something stronger to say: the relationship between the two is fixed by the felt exponent and nothing else.

Contact time falls as velocity to the power minus a quarter, so the null moves as velocity to the power a quarter, while the level rises as twenty log of the velocity ratio. Eliminating the ratio between them gives the first null moving by a factor of 10 to the power D over 80, where D is the velocity-only dynamic range in decibels. At the ladder used here that is 10^(30.1/80) = 2.378, which is exactly the factor the figure reports.

So the spectral consequence of a dynamic range is a quarter power of it, and that is a much weaker dependence than the prose around it suggests:

Doubling the movement of the null costs twenty-four decibels of extra range. An instrument cannot buy much spectral change with loudness, which is the quantitative version of why sul ponticello had to be a separate word rather than another rung on the same ladder.

A partial does not simply get louder #

The most useful correction is to the picture of a band of partials switching on as the player plays harder. Tracking one partial through the whole range shows something else.

The fourth partial, relative to the fundamental, starts at −35 decibels at pp, climbs to about −25 by p, and then falls to −41 in the middle of the range before climbing steadily to −13 at ff. The dip is at a hammer velocity of about 1.7, and the reason is exactly the mechanism this essay is about: that is where the first null, sweeping upward through the series as the blow hardens, crosses the fourth partial and annihilates it.

Every partial has such a notch and they are at different dynamics, because the null passes each one at a different velocity. So a crescendo is not a spectrum filling in from the bottom; it is a moving zero sweeping up through the series, and each partial in turn brightens, vanishes, and returns. Nothing in the notation can express a non-monotonic partial, and no listener would describe a crescendo that way — but it is what the model says the string is doing, and it is a sharper reason than loudness for why a fortissimo is a different sound.

Struck softly and struck hard, the same note has almost the same envelope: the attack is a few milliseconds either way and the decay is set by the string and the bridge rather than by the hammer. What a dynamic mark changes is the spectrum and not the shape, which is why a recording played quietly does not sound like a piano played quietly.

It is not only the piano #

The piano is the clearest case because the mechanism is simple and this site already had the model. It is not the exception.

The site’s own voice ladder reached the same conclusion from the other end: a note is never at its pitch because the singer is doing several things at once, and pressure is one of the things being varied. A dynamic mark handed to a singer is an instruction about the glottis, and what comes out of the glottis is a spectrum.

Brass is the strongest case of all and this site cannot compute it. At high amplitude the pressure wave inside a long cylindrical bore steepens as it travels, because the crests move faster than the troughs, and by the time it reaches the bell it has grown a shock front and a spectrum with far more high-frequency energy. A trumpet at fff is not a trumpet at mf turned up: it is a different waveform. The effect is quoted here and not modelled, because everything else in this collection is a linear acoustics model and this is the one place where linearity fails outright.

Three fixed partial lists are what every other figure on this site computes with, and they are exactly what a piano does not have: its list is a function of how hard the key was struck, so a “string spectrum” is a spectrum at one dynamic.

And the ear does not measure decibels either #

Suppose the mark did specify a level exactly. It still would not specify a loudness, because the map from one to the other is not a straight line and is not the same at every frequency.

That is a real compositional fact and it is why the quietest thing audible is a frequency-dependent quantity rather than a number. A dynamic mark applied to a whole texture is applied to instruments in different registers, and the same instruction produces different perceptual changes in each.

There is a sharper case, and this collection has already measured it: roughness is level-dependent, so the same chord is harsher when it is louder.

And roughness rises with level for reasons that have nothing to do with the hammer — more partials clear the threshold of hearing — so a loud chord is harsher partly because it is brighter and partly because it is loud, and the two contributions are separable only by computing both.

Where the marks are least misleading #

None of this makes the notation foolish, and it is worth saying what the marks are good at, because it is a real thing and it explains why six ordinal symbols have survived four centuries.

They are relative and local. A crescendo over four bars is a well-specified instruction — get louder — and it does not require anybody to know how loud. A subito piano is a discontinuity, and a discontinuity is defined by the thing it follows. The marks work as a differential notation and fail as an absolute one, and almost all their use is differential.

They also scale with the forces. An orchestral ff is not one instrument’s ff; it is a texture, and adding players raises the level in a way this site has computed.

And a long-term average spectrum of an orchestra against a solo voice is the same variable one level up: what a forte marking buys an ensemble is a change of spectral balance, and the band it changes it in is the one the voice has to compete in.

The one place the page does specify a spectrum, and it is not a dynamic #

There is an instructive exception, and it is worth putting beside the marks because it shows what the notation does when it really wants a timbre.

Where a composer needs a particular sound rather than a particular level, the page abandons the dynamic system entirely and names a technique: sul ponticello, con sordino, flatterzunge, senza vibrato, col legno. Every one of those is an instruction about the spectrum, every one is a word rather than a symbol, and none of them is on a ladder. The notation has, in effect, two vocabularies for timbre — one ordinal and implicit, one categorical and explicit — and the ordinal one is the one everybody reads as loudness.

Two of those words name things this collection has measured. Sul ponticello is a change of bowing point, which is the same comb the next rung is about; con sordino is a change to the body’s filter, which is what a violin’s resonances do to everything the string produces. Both are spectral instructions given as names because there is no axis to put them on, and the contrast with pp to ff is exact: where the effect could be ordered, the page made a ladder and lost the spectrum; where it could not, the page named the thing and kept it.

What a page would need to say instead #

It is tempting to conclude that the notation should specify decibels, and it should not. A decibel figure is wrong for a different hall, a different instrument, a different edition of the same instrument, and a different number of players; it is a measurement of a performance and the page is not a record of a performance.

What the page is doing is naming a position in the range of the forces present, which is genuinely what the instruction needs to be, and then relying on the performer to know what the physical consequence is. On a piano the consequence is a shorter hammer contact. On a voice it is a higher subglottal pressure. On a trumpet it is a shock front. The mark abstracts over all three, and the abstraction is exactly right for a system that has to serve every instrument.

The cost is that the abstraction is invisible. A reader who has not thought about it will take ff to be a loudness instruction, and will then be surprised that a fortissimo passage recorded quietly does not sound like a fortissimo passage.

Which computation produced the numbers #

The contact time is hammerContactMs, which is τ proportional to velocity to the power −0.25 — the exponent is from the measurement literature on piano hammers, and it is a model of felt rather than a fit to a particular instrument. The partial envelope is the spectrum of a half-sine force pulse of that duration, whose first null is at 1.5/τ.

The six velocities are a stated ladder in factors of two, from 0.25 to 8. They are not a measurement and no claim here depends on them being right, because everything the figure says is a comparison between the ends of the ladder: contact falls by a factor of 2.4, the null moves by a factor of 2.4, and the level rises by 35 decibels because amplitude is proportional to velocity over a factor of 32. Choose a different ladder and every number moves together.

The spectral centroid is the power-weighted mean partial number over the first twenty-four partials, computed from the same amplitudes the figure draws.

The decomposition of the 35.1 decibels is that same total against 20 log of the velocity ratio, which is what the level would be if the spectrum did not change shape. The quarter-power invariant is algebra on the two exponents rather than a fit, so it holds for any velocity ladder and for any instrument whose contact time follows the same power law; what it does not survive is a different exponent, and the exponent is the one quantity here taken from the measurement literature rather than derived.

The partial-by-partial sweep runs the same function at quarter-octave steps in velocity and reports each partial’s level relative to the fundamental of the same blow, so what it shows is a change in balance rather than a change in level — every partial is louder at ff in absolute terms, including the ones whose relative level dips.

The 35-decibel figure is a property of the model and it happens to agree with the usual quoted range for a single piano note, which is thirty to thirty-five decibels. That agreement is a check rather than a result; nothing in the model was set to produce it.

What the picture cannot show #

It has no room in it. Every level here is at the instrument. What reaches a listener has passed through a hall whose reverberation adds energy and whose distance removes it, and beyond a certain distance the room is louder than the instrument. A dynamic mark that is right in a rehearsal room is not right in a cathedral.

The hammer model is a single half-sine pulse, which is the standard first approximation and is known to be wrong in detail: a real hammer separates from the string and can strike it again, and the felt’s stiffness depends on how it has been voiced. The direction of every effect here survives that; the exact null positions do not.

Nothing here is measured on a performance. The twenty-decibel spread across players is quoted from the literature, and the overlap between neighbouring marks is quoted with it. This site has no corpus of recordings and cannot check either.

And the brass case is stated and not computed, which is the largest gap in the essay. Nonlinear steepening in a bore is the most dramatic instance of the phenomenon this rung is about, and it needs a nonlinear propagation model that nothing else in this collection would use.

The ladder from here #

This rung found a spectrum inside an ordinal ladder. The next stays with the spectrum and changes instruments: on a fingerboard the same written note can be stopped in four places, the speaking lengths differ by a factor of two, and a player’s hand meets a different fraction of the string at each — so the page names one note and a tablature names one sound, and neither contains the other.