A clarinet keeps what a string loses

The register rung ended by naming what every figure on this anchor holds fixed, and it is not a number a placement could have changed:

Every masker, every probe, every chord, every line, every part of every texture is a string-like spectrum of eight partials falling as one over n, and it is not even an option a placement can pass.

Two rungs had reserved the same doubt from different directions. The proof that an equally loud predecessor takes nothing depends on the predecessor’s components coinciding with the successor’s, which is true of one instrument and false of an ensemble; the texture rung made the reservation from the other side. And a clarinet is the obvious counterexample to the whole picture: its even partials are twenty-eight decibels below its odd ones, so its components stand a twelfth apart at the bottom rather than an octave, which puts them in different filters and ought to let it keep far more of itself.

The sweep is one line and the answer is no.

On the ladder’s own score the clarinet is the worst of the six. It reaches 54 per cent of itself at its best register against a string’s 79, and it is below the string at every register above the bottom octave. A reed is 71 per cent, an organ rank 72, and the two spectra with their even partials removed — the clarinet and the bell — are the two at the bottom of the table.

That is a real number and it is the wrong question, and finding out why is the whole of this rung.

What the count was counting #

The mechanism the reservation appealed to is right, and it is worth drawing before the score is argued with, because the two are separate facts.

A string spectrum puts its components an octave, a fifth, a fourth, a third apart and so on down; at middle C its closest audible pair is 0.87 Bark, which is inside one filter. A clarinet, whose audible components are partials one, three, five and seven, puts its closest pair at 2.25 Bark at the same register — two and a half filters apart, and comfortably resolved. At C2 the same comparison is 0.57 against 1.19. The clarinet’s components are twice as far apart in the ear’s own units, everywhere.

So the reservation’s physics is correct and its prediction still fails, which means the failure is in the counting. It is: the clarinet’s four even partials are twenty-eight decibels down, and a component twenty-eight decibels below its immediate neighbour is inaudible beside that neighbour before any chord is built. Put a single clarinet-spectrum note in silence and four of its eight components are already gone. The share of written partials therefore charges the clarinet for four components it never had, and charges the string for none, because a string’s are all audible alone.

The score confounds what a chord takes with what a spectrum never possessed. Change the denominator and the answer inverts.

On a root at A♭2 the clarinet triad’s twelve audible components are twelve of the twelve those three notes would have had separately: the chord takes nothing. At middle C it keeps twelve of fourteen, 86 per cent. The string triad at middle C has nineteen of twenty-four, and every one of the five it loses is lost to the chord rather than to itself. The bell spectrum, whose components are also odd-only, keeps 93 per cent across almost the whole compass.

So the debt’s prediction is confirmed and the ladder’s own headline statistic said the opposite of it. The lesson is not about clarinets. A share whose denominator is what the score contains rather than what the ear could have had is a statistic that punishes any spectrum with weak components in it, and this anchor has been quoting one since its third rung.

The register result survives every spectrum #

Which raises the question the register rung asked and could not answer: is a masking result on this site a result about the ear or a result about a violin?

The shape holds. Every spectrum has the same arch — almost nothing at the bottom of the compass, a maximum somewhere between the bass and the treble clefs, a decline above it — and every spectrum has the same cause, which is the critical band being thirty-seven semitones wide at C1 and 2.6 at E6.

At C1 the numbers are 4 per cent for a string, 8 for a clarinet, 11 for a bell, and zero for a reed and for an organ rank. Scored the other way they are 7, 17, 20, 0 and 0. A close triad at the bottom of the piano is not arriving as a chord on any instrument in the collection, which is the register rung’s headline and which the ladder about how a chord is spaced reaches from the composer’s side, and the two spectra that do best down there are the two whose components are furthest apart — the same mechanism, at the register where it has the most work to do.

A single note in the bass is already masking itself #

There is a second finding hiding in the alone baseline, and it is the reason the bottom of the arch is so low.

The baseline is what a note keeps with nothing else sounding, and at the bottom of the compass it is already small. A string-spectrum note at C1 keeps one of its eight components in silence; a reed keeps one of seven, an organ rank three of six, a clarinet four of eight and a bell five of nine. At C3 and above every one of them keeps all of its components, or all but one.

The mechanism is the one the register rung named and it operates a level lower than that rung applied it. A critical band at C1 is thirty-seven semitones wide — three octaves — and a harmonic series puts its first four partials inside two of them. Those partials are not a chord masking a chord; they are one note masking itself, and the louder low ones take the quieter high ones inside their own filter. What is left is whatever stands clear at the top of the series.

Two things follow. The first is that the spectra keeping most of themselves alone down there are the odd-only ones, whose components are furthest apart — the same mechanism this whole rung is about, arriving before any second note is added.

The second is a caution about reading the arch, and it needs the two effects separated. A string triad on C1 keeps fifteen of its twenty-four components when its three notes are played one at a time — one from the root, six from the third, eight from the fifth — and one of twenty-four when they sound together. So nine components are lost to self-masking and fourteen more to the chord, and the note at the bottom has already lost seven eighths of itself before the chord exists. A low chord stops being a chord partly because a low note has stopped being a note, and every figure this anchor has drawn has measured the two together.

And the register rung’s sharpest claim — that the roughest chord on the page is the lowest one and the roughest chord a listener receives sits nearly two octaves above it — is 16 to 20 semitones for every spectrum drawn. The written maximum is the lowest register in every case, because the low-interval rule is monotone and has no spectrum in it. The heard maximum is at A♭2 for a string, a reed and an organ rank, and four semitones lower — E2 — for a clarinet and a bell. The rung that priced a third against its register is where the written half of that comparison came from, and nothing in it changes here.

That is the answer to the question. The register result is about the ear. The level of the arch moves with the spectrum, by as much as a factor of two; the position of its maximum does not, and neither does the two-octave gap between what is written and what arrives. Those are the two things the low-interval rule is a claim about, and both of them survive.

The reservation two rungs recorded, priced #

That leaves the reservation itself, which is about a line rather than a chord and which two rungs recorded independently.

The proof it doubts is a proof rather than a computation. Forward masking leaves at most ten decibels below the level of the sound that has stopped; a note’s own partials already mask each other at that note’s own level; ten decibels less than something already present cannot take anything away. So a legato line of equally loud notes on one instrument loses exactly nothing to itself — which is what the figures show, to machine precision, for every spectrum.

The reservation is that the proof compares totals while the masking happens frequency by frequency, and that an ensemble breaks the correspondence: a clarinet’s third partial sits where a flute has nothing of its own to defend the place. If the two instruments alternate, each note’s predecessors are components in gaps rather than components on top of components, and the argument’s bookkeeping no longer holds.

The reservation is correct about the mechanism and wrong about the size. The worst pairing anywhere — a string alternating with a clarinet — takes 0.80 per cent, six pairings out of ten take nothing measurable at all, and the components taken in the three that do are the clarinet’s own even partials, which were twenty-eight decibels down before anything else sounded. A component that far below its neighbour is on the edge of audibility in isolation and is the first thing any additional masker removes.

So the honest reading is that the proof is more robust than its authors expected rather than that it survived by luck. The ten-decibel floor on forward masking is a large margin, and it takes a component that is already marginal to be pushed over by a foreign spectrum. What would break the proof is a level difference between the two instruments — which is a different parameter, and is the one the rung that swept it priced at twenty-four decibels.

This is a claim about a scoring practice as much as about the ear, and the practice it is a claim about is the one where a line is deliberately handed between instruments mid-phrase: the Klangfarbenmelodie of Schoenberg and Webern, and its much older ordinary form in the Classical orchestra, where a melody passes from oboe to flute at a phrase joint. The arithmetic says that whatever such a handover costs, it does not cost the listener partials.

Which computation produced the numbers #

The masking pattern is the codec spreading function on the Bark axis, the same one this anchor has used since its first rung: a lower slope of about twenty-seven decibels a Bark, an upper slope that shallows as the masker gets louder, summed in intensity over every other component present, with the threshold of hearing as a floor. A component is audible when its own level exceeds that sum.

The six spectra are the ones this collection already carries and are used unchanged: a single component; eight partials falling as 1/n; a clarinet with its even partials 28 decibels down; a seven-partial reed falling more slowly; an odd-only bell of nine; and a six-partial organ rank with a weak third. The whole tone’s level is defined as the root-sum-square of its components, so a spectrum with more energy in its upper partials has a quieter fundamental at the same nominal level.

The alone baseline is the same function run on one note with nothing else sounding, which is what makes the second score a score of the chord rather than of the spectrum. Roughness is the Plomp–Levelt kernel over every pair of components belonging to different notes, computed twice over the same chord: once over everything written and once over the surviving set.

The line figure is the forward-masking computation the fourth rung used, with a spectrum per note rather than one for the line, sampled twelve times through each note’s own life and looking back three notes. Each figure asserts that a uniform line keeps every component it had alone, which is the published proof restated as a check — if any single-instrument line lost anything the mixture result would be measuring the arithmetic rather than the ensemble.

Where the model stops #

Two of the three spectra the debt named are not in this collection. A bell’s partials are genuinely inharmonic and the bell in this table is not: it is an odd-harmonic caricature, nine components at whole multiples with the even ones zeroed, which is the right shape for the resolvability question and the wrong object for a real bell. The machinery here places a component at n times the fundamental by construction, so a struck spectrum with partials at 2.0, 2.4, 3.0 and 4.2 times the prime cannot be passed through it without an extension. This collection does carry those ratios, in the ladder about what a bell is tuned for, and joining them to this anchor is real work rather than an option.

And there is no flute here. The debt said a flute’s partials are three, and the nearest thing in the table is a six-partial organ rank. A three-partial spectrum would be the extreme case of the mechanism this rung is about — nothing at all inside a critical band above the fundamental — and the prediction, which is untested, is that it keeps essentially all of itself at every register above the bass clef.

A spectrum is not a constant of an instrument. Every spectrum here is one fixed set of amplitudes, and a real clarinet’s even partials are twenty-eight decibels down in the chalumeau and much closer in the clarion; a string’s slope changes with the bowing point and with the dynamic, which is a parameter the rung about how loud a chord is swept and this one holds fixed. The rung about what a loud note’s own air does to it is the same point from the instrument’s side. Every number here is a number about one playing condition.

What the picture cannot show is the fusion. Everything counted here is a component standing above a threshold, and a listener does not hear components; they hear notes. Two partials in one filter are not “one lost partial” to a listener, they are a note with a slightly different timbre, and whether the chord is heard as three notes is a question this anchor has never asked and the auditory-scene ladder asks about different objects. A clarinet keeping 100 per cent of its components in a triad does not mean a listener hears three clarinets.

Where this ladder goes next #

Seven rungs. One sound hides another, and it hides upward; a sound hides what came before it; a chord hides itself, by an amount that depends on how loud it is and how it is spaced; a line hides nothing without a dynamic contrast; a texture hides its own bass; a register decides more than any of them; and now a spectrum, which decides the level of every one of those answers and the position of none of them.

What is owed now is the inharmonicity, and the previous section says exactly why it is owed and what it costs. Every component on this anchor sits at a whole multiple of a fundamental, in the arithmetic as well as in the tables, so the two spectra that would test this rung hardest cannot be passed through it: a bell, whose partials are at 2.00, 2.40, 3.00 and 4.20 times its prime, and a stiff piano string, whose n-th partial is displaced by a factor that grows as n squared. Both are already computed in this collection, in the ladders about what a bell is tuned for and about how a piano’s wire departs from the series. Placing an inharmonic spectrum on the Bark axis and asking how much of it survives its own masking is the same computation with the frequencies read from a table instead of from a multiplication, and the prediction it makes is sharp: a bell’s partials are unevenly spaced, so a bell keeps almost all of itself in the treble and loses its two lowest components in the bass, where they converge into one filter that a harmonic spectrum’s never enter. It needs no corpus and no listener, and what would come out is whether the arch this rung found in five spectra is a property of harmonic series or of hearing.