A page has twelve phons, and its entries always said so

This essay was meant to pay a small debt and pays a large one instead.

The small debt was named at the end of a subito piano is four seconds longer in the bass. Two of the loudness figures the essays on loudness lean on hardest — the two decibels a whole page of orchestration commands and the one phon an entering part is worth — were described there as absolute quantities computed at a kilohertz, and what was owed was their balance: each part given its own register and its own level, the loudness summed across the ear rather than read at one frequency, and the question asked again. The prediction was that the answer could go either way, because the parts that dominate a texture’s loudness are not the parts that dominate its energy.

That question gets its answer below, and the answer is small. On the way to it, the arithmetic turned up something much larger: the two decibels were never a fact about the ear. They were a fact about one of the two ways these essays sum loudness, and the other way — the one the entry essay used and preferred — gives a page twelve.

The debt, as it was stated, and why it was not quite the debt #

The first thing to check was the premise, and it did not hold.

The page’s figure was computed in a page has two decibels by putting every partial of every part into critical bands and converting each band to loudness with that band’s own equal-loudness contour. Nothing in it was read at a kilohertz. The kilohertz convention belongs to the running-loudness smoother the later essays built for notes with time in them, and the subito piano essay was right to correct that. The static page count had the contours in it all along.

What it did hold fixed is the second half of the debt: every part played at the same sound pressure level, 62 decibels, whatever its register. That is not what an ensemble does. Players and conductors balance by ear, and a part balanced by ear in the bass is carrying far more power than one in the treble, because the ear is less sensitive down there. So the balance question is real even though the kilohertz framing was not. It asks whether an ending built up by ear gains more than one built up by meter.

To answer it the loudness has to be counted in a way that can see a bass part doing work — and that is where the trouble began.

Two ways of counting, and the one that sorted sixty-four partials into two bands #

There are two loudness models in these essays, and a chord is not as loud as its notes built both. The first groups partials along the ear’s frequency axis: walk up the spectrum, and any partial within one critical band of the last joins its group; then each group is converted to loudness once. The second integrates an excitation pattern: every partial casts a skirt along the ear, the skirts are added, and the loudness is the area under the result. The second is the standard way of computing loudness from a spectrum. The first is a shortcut, and its own description records the defect: a chain of components each nine-tenths of a band from the next becomes one group however long the chain.

A string tone is such a chain. Its upper harmonics are closer together than a critical band from about the fifth upward, so they link into one group already. An eight-part chord is a dense thicket of such chains, and there is nowhere in it for the linking to stop.

One string note at middle C has eight partials and the grouping finds seven bands. Two notes find seven, three find five, five find three, and eight notes — sixty-four partials spanning two and a half octaves of fundamentals — find two. Each of those two groups is then converted to loudness as though it were one sound at its combined intensity, and one very intense band is worth much less than the six or seven moderate ones it has swallowed. So as parts are added, the model loses bands faster than it gains intensity, and the total goes down.

That is the non-monotone curve the page essay reported — four parts are louder than five, and than eight — and it is the reason its span came out at 1.8 decibels. A part entering is not a change of level had already found the grouping at fault for the dips inside the curve, and said so plainly. It concluded that the headline survived, because a span between two ends is barely affected by what happens between them. That was the step that did not hold: the ends are affected most of all, because the eight-part end is the densest spectrum in the whole comparison.

Counted along the ear #

The excitation pattern has no groups and nothing to chain.

The picture is what a reader would expect of an orchestra before any theory. One middle C lights a stretch of the ear from near the bottom of the Bark scale to about the sixteenth critical band. Four parts, reaching up an octave, extend it to about the twentieth and fill in the valleys between one note’s partials. Eight parts, reaching down to C3 and up to G5, extend it further at both ends. Nowhere does adding a part lower the curve, because a pattern is a sum of skirts and a skirt is never negative.

To turn that area into something comparable with the old figure, it has to be expressed as a loudness level — and here there is a second correction, smaller than the first. The earlier essays converted a ratio of sones to phons by the textbook rule that a doubling of loudness is ten phons. That rule describes the sone scale, and this particular excitation model does not obey it exactly: a single tone grows by 2.3 times per ten decibels rather than 2, which its own documentation records. Read through the textbook rule, every level change it reports is overstated by about a fifth. So every loudness level here is found the way the phon is defined: as the level of a 1 kHz tone that the same model judges equally loud. The model’s error sits on both sides of that match and cancels.

On that scale the build-up runs 73.2, 77.9, 79.5, 81.6, 82.5, 83.9, 84.5, 85.6 phons — rising at every step, 12.4 phons from one part to eight. The eight parts together carry nine decibels more power than one, and they are louder than nine decibels would make them, because they are spread over the ear rather than piled into one place. That is the chord essay’s own central finding — spread is loud — arriving at the scale of a whole ending.

The number the entries always implied #

What makes this more than a quarrel between two models is that the pieces had already been priced, and the pieces never added up to two.

The entry essay priced a part entering a five-part texture at every semitone from C2 to C7, on the excitation model, and found a median of 0.89 phons — about the smallest change of loudness a listener can notice. Recounted on the calibrated scale, the median is 0.81; that essay’s conclusion stands almost untouched. An ending built up from one part to eight is seven entries. The later ones, into four to seven parts already sounding, add 0.6 to 1.5 phons each, exactly where the census says an entry into a texture of that size should sit. The first two add 4.7 and 1.6, because entering a texture of one note is entering a mostly empty ear.

Seven entries worth between half a phon and five cannot sum to under two, and they do not: they sum to 12.4. The two numbers the collection published — an entry worth one phon, a page worth two — were never compatible. The first was computed on the excitation model and the second on the grouping, and the discrepancy sat in plain sight across two essays for as long as both have been up.

That is the surprising connection in this essay, and it is a connection between two of the collection’s own results rather than to anything outside it. Each was checked where it was written. What nobody did was multiply one by seven.

And the balance, which is what was actually asked #

With a model that can see a part doing work, the debt’s own question can be answered.

Balancing by ear asks for almost nothing above middle C: every part from C4 to C7 needs between 60.6 and 63.2 decibels to match. It asks for a great deal in the bass: 66 at C3, 73 at C2, 81 at C1. That curve is the equal-loudness contours seen through a string tone’s partials — the fundamental of a low note sits where the ear is deaf, but its upper partials do not, so the extra level needed is well short of what the contour for a pure tone would demand.

The ending’s own eight parts reach down only to C3, where balancing asks for four extra decibels, so balancing them changes almost nothing: 12.2 phons against 12.4. The debt’s prediction that balance could move the page’s number in either direction was right in principle and nearly empty in practice, for this texture.

Give the ending a real bass — C2 and then C1, as a close with double basses and a contrabassoon can have — and the answer changes. At 62 decibels each, the two bass octaves add 0.4 phons: almost nothing, which is the old intuition that a bass part is cheap. Balanced by ear, they add 2.2 — five times as much — while raising the chord’s power by 10.4 decibels against 1.0. That is the sentence the debt was written for, now with numbers in it: the bass octaves of a balanced tutti carry most of its energy and about a sixth of its loudness gain. The parts that dominate the energy are not the parts that dominate the loudness, and a model that meters every part at one level misses both halves of that.

What a page commands, against a player #

The comparison the page essay was built around was two decibels for the page against sixty for a player. Both halves change when they are put on one scale.

Sixty decibels of level on a four-part chord spread from C4 to C5 is 46.9 phons of loudness level, not sixty, because loudness level grows more slowly than level for a spread spectrum at loud levels. The page is 12.4. A player commands about four times what a page does, not thirty.

That does not rescue the older claim that thinning a texture is a change of colour rather than a change of dynamic. It inverts its size. Going from eight parts to one takes away twelve phons, which by the standard doubling is a factor of more than two in loudness — a quarter of everything the player’s own range covers. A texture that thins to a single line at the end of a piece is a diminuendo, and a large one, whether or not a hairpin is written over it.

Whatever the spectrum #

The last question is whether the recount depends on the string tone the ending was drawn with.

It does not, on the model that has no grouping: pure tones, strings, clarinets, reeds, bells and organ pipes all give between 12.4 and 17.9 phons. The grouping gives anything from −1.8 to +24.4 — negative for strings and reeds, whose many close partials chain into one group, twenty-four for pure tones, which have no partials to chain and so keep every note in a band of its own. That spread is the clearest single statement of what went wrong: the grouping’s answer is a property of partial spacing, and the ear’s is not.

How the numbers were produced #

The voicings are the page essay’s, unchanged: middle C, then G4, E4, C5, G3, E5, C3 and G5 added in that order and never removed. Every part is the collection’s string spectrum — eight partials — at 62 decibels per part, or at its balanced level where stated.

Both loudness models are the second loudness essay’s, unchanged. The band model groups components single-linkage along the Bark axis and converts each group with the ISO 226 contour at its own centre. The excitation model spreads every component along the Bark axis with the masking skirts every essay on masking uses and integrates Zwicker’s specific loudness from 0.5 to 24 Bark in steps of a twentieth.

Every loudness level is the level of a 1 kHz tone with the same total loudness in the same model, interpolated from a table of that tone at every half decibel. A balanced level is found by bisection: the level at which a note, alone, is as loud as middle C at 62 decibels on the excitation model.

Where the model stops #

Neither model is a listener. The excitation model is the standard computation and has no defect of the kind found here, but it is a model of steady tones heard from the front. It knows nothing of the room, of how the sound of eight real players arrives, or of attention; and for chords of instruments whose spectra change with dynamic, it holds each spectrum fixed.

The balance is one listener’s balance. Balancing each part to match middle C alone is a stated convention. Orchestras balance whole sonorities rather than parts one at a time, and a bass part in a real tutti is often balanced to be felt more than heard. Whatever convention is used, the bass octaves are where it matters.

And the recount does not reach the rest of the collection by itself. Several earlier essays quote the two decibels, including the closure essay on final chords and the staccato essay, which ranked a page’s texture below its articulation on the strength of it. The articulation’s two to eight phons were computed on the running-loudness smoother and are not affected; the ranking is. On the recount, texture is at least as large a dynamic as articulation, and for a full thinning to one line it is larger.

What the picture cannot show #

Whether twelve phons of thinning is heard as quieter or as emptier. Loudness level is one number and a thinning texture changes many things at once — its colour, its roughness, how many objects it contains. The recount says the loudness drops by a factor of two; it cannot say that a listener attends to that drop rather than to the loss of the objects, and the older reading, that a thinning is heard as a change of colour, may still describe what listeners report even though it misdescribes the loudness.

And the chord is always entire. Every figure here is a sonority that has been sounding long enough to reach its full loudness. An ending arrives in time, over the seven seconds a loudness impression takes to come down and the fifth of a second it takes to go up, and the dynamic shape of a real thinning is that smoother applied to these levels, which has not been done here.

Still open: whether the ending’s dynamic belongs to the page after all #

The page essay concluded that an ending’s dynamic belongs entirely to the performance, because a page could produce only two decibels. With twelve, that conclusion is open again. An ending that thins from a tutti to one line drops more loudness than most players take off for a written diminuendo, and an ending that fills from four parts to eight with balanced bass octaves gains more than a hairpin usually buys. The dynamics are in the score already computed a passage’s dynamic curve from its texture alone and passed it through the running smoother, on the grouped model; the same computation on the excitation pattern, applied to the four ending textures the closure essays compare, would say how much of a written ending’s dynamic shape is already on the page before anybody plays it — and whether the closure component that was declared unreachable from a score is, after all, mostly there.