A low chord stops being rough by stopping being a chord

The listener is given the top voice ended by pointing at what every figure on this ladder has held still:

Every count of audible partials on this ladder — the chord, the voicings, the line, and the texture above — is computed on a chord built on middle C or a line that starts there.

That is not quite the whole of it. A loud chord is a smaller chord did compare three close voicings at three registers, and they span C3 to C5. Over that range the share of a triad’s partials that survive its own masking goes 67, 79 and 71 per cent — a modest arch, easily read as noise inside a table whose other rows were about spacing.

A chord is written over about six octaves. Draw the same triad across all of them and the arch is not modest.

Four per cent at the bottom and seventy-nine in the middle #

Take a close major triad — root, third, fifth, ten partials a note, twenty-four components in all — hold it at seventy decibels, and move it from C1 to E6, counting at each register how many of its components stand above the masked threshold the rest of them put up.

At C1 the answer is one of twenty-four. At E3 it is nineteen. At E6 it is thirteen.

Four per cent, seventy-nine per cent, fifty-four per cent. The same chord, the same level, the same spectrum, moved.

The mechanism is one number and it is the width of a critical band in the unit the music is written in. The ear’s analysis bands are evenly spaced on the Bark scale, which is nearly linear in hertz below about five hundred and roughly logarithmic above it, so a band that is thirty-seven semitones wide at C1 is seven at middle C and 2.6 at E6 — a factor of fourteen across the keyboard.

Thirty-seven semitones is three octaves. A close triad at C1 has its three fundamentals inside seven semitones and its first several partials inside those three octaves, so most of the chord is analysed by one filter. The ear is not being offered a chord down there; it is being offered one band with a great deal of energy in it, and the components inside a band add their intensities and are converted to loudness once.

Which moves the roughest chord two octaves up the keyboard #

The count is interesting; what it does to the roughness is why this rung reaches outside its own ladder.

A third is rougher in the bass is one of this collection’s most-used results, and a chord is a register derives the whole low-interval rule from it: the same three pitch classes are five times rougher at the bottom of a piano than in the middle, because a critical band is wide down there and the notes of a low triad fall inside it and beat.

Computed over every partial in the score, that curve is monotone. The roughest register is the lowest one drawn, at every dynamic tried, without exception. The rule is exactly as stated.

Computed over the partials the chord actually delivers, it is not monotone at all. The delivered roughness rises going down, turns over at G2, and falls to essentially nothing across the bottom octave.

At C1 the ratio is stark: the score says a close triad there is the roughest object on the keyboard and the listener receives one component of twenty-four. One component cannot beat against anything. The roughness delivered is zero, not because the chord is smooth but because it has stopped arriving as a chord.

Eighty-two per cent of the roughness this collection computes for a triad at C2 is between partials the listener does not have. The third rung of this ladder found that share to be about a third at a loud dynamic in the middle register and called it a correction that bent the curve without overturning anything. In the bottom two octaves it overturns the ordering.

The share delivered, register by register at seventy decibels, is a clean arch: nothing at all at A♭1 and below, 18 per cent at C2, 72 at C3, 90 at E3, 87 at middle C, 67 at C5 and 33 at E6. So the third rung’s “about two thirds arrives” is a middle-register number that happens to sit near the middle of a curve spanning zero to ninety, and reading it as a constant is what let four rungs pass without anybody varying the register.

The far end of the arch turns down too, and it is worth checking why rather than assuming. It is not the threshold of hearing: at E6 the masked threshold is above the quiet threshold for every one of the twenty-four components, so nothing there is lost to silence. It is the same self-masking as at the bottom, arriving from the other side. A chord at E6 has its upper partials between four and sixteen kilohertz, where the Bark scale flattens out again — a fifth spans 2.6 Bark at middle C’s sixth partials and 1.9 at E6’s — so the three notes’ high partials crowd back together, and their own levels have fallen as one over n and have little margin left. The register where a chord delivers most of itself is around E3 to middle C, which is where a chord is usually put.

And the maximum is not at a register, it is at a register per dynamic #

The result nobody was looking for is what happens when the same computation is run at other levels.

The written column never moves and the delivered column climbs nearly two octaves over the ordinary dynamic range of an ensemble.

The direction is the surprise. Playing louder makes a chord rougher — that is the consonance ladder’s own level result, and the amplitudes enter the beating kernel quadratically — and it also makes it mask itself harder, because the spreading function’s upward slope flattens as the masker gets louder. The second term bites first in the bass, where the components are closest in Bark, so the loud tutti chord loses its bottom end from the ranking while everything above it gets rougher.

So “the roughest register” is not a property of the ear or of the chord. It is a property of the ear, the chord and the dynamic together, and a scoring decision made at one dynamic is being made about a different maximum from one made at another.

The reach, in the unit the score is written in #

It is worth drawing the spreading function itself in semitones, because that is where the register dependence is hiding and it is invisible on the Bark axis the first rung used.

On Bark the pattern is a pair of straight lines and looks the same everywhere, which is what makes it drawable. Converted to semitones it is a different object at every register. A tone at C2 at eighty decibels raises the threshold over sixty-one semitones — five octaves — and the same tone five octaves higher, at C7, covers thirty-two. Downward it reaches seventeen semitones at C2 and eight at C6.

Five octaves is not a detail about a chord. A single loud low note takes most of the audible spectrum out of service, and that is the arithmetic behind a fact orchestrators state as a warning about pedal points and low brass.

Quieter, the same tone reaches half as far: thirty-two semitones at C2 at fifty decibels rather than sixty-one. That is the level dependence of the upward slope, and it means a texture’s masking is not a fixed property of its spacing.

Two mechanisms for one rule, and where they hand over #

The low-interval rule now has two derivations in this collection and they are not the same derivation.

Above about G2 it is roughness. The triad’s notes and their low partials fall inside one critical band and beat, the beating is audible, and the chord sounds muddy in the ordinary sense. That is the consonance ladder’s account and it is unaffected by anything here.

Below about G2 it is absence. The chord no longer delivers enough components to beat with. It is not muddy; it is a single loud thing with a pitch that is hard to place, and the reason to avoid it is that nothing arrives rather than that too much does.

The crossover is not sharp and it moves with the dynamic, but the two ends are qualitatively different events and a rule stated as “low thirds are muddy” describes only one of them. That the practice is the same at both ends is not a coincidence — both are reasons to spread a texture at the bottom — but the account is not one account.

The spacing rule falls out with the right sign at both ends #

The third rung compared five spacings at one register. Running the register and the spacing together turns the comparison into an instruction, and the instruction changes sign halfway up the keyboard.

At C2, seventy decibels, the close triad delivers seven of its twenty-four components. Open it out and it recovers:

Spreading a chord in the bass doubles how much of it arrives. The last row is root, fifth-plus-octave, third-plus-two-octaves — partials one, three and five of the harmonic series, which is exactly the arrangement where to put the third found smoothest by an exhaustive search over roughness. Two quantities computed from two literatures pick the same voicing.

Run the same four spacings at middle C and the ordering reverses: the close triad delivers nineteen of twenty-four and every spread arrangement between sixteen and eighteen. Closing a chord up in the middle register delivers more of it, and opening one out in the bass delivers more of it, which is the rule every treatise states as “spread at the bottom, close at the top” and which nothing here was told about.

The reason is the band again. Down there the components of a close chord share a filter and mask each other from inside it, and spreading them puts them in separate filters. Up here the filters are narrow enough that a close chord is already in several of them, and spreading it only moves the upper notes into a register where the threshold of hearing is rising.

The loudness side of the same question runs the other way and is worth putting beside it, because the two are usually conflated.

That comparison is about spacing at one register. The last figure is the same register axis on the other quantity this collection computes from a critical band, and the two run in opposite directions in a way worth holding together: the low chord that delivers least of its own spectrum is also the one that buys least loudness from the power put into it.

The low chord is quiet, spectrally impoverished and computed as maximally rough, and only the third of those is what a score says about it. A chord is not as loud as its notes is the first of the three and this rung is the second, and they have the same cause: one critical band, several components, counted once.

Which computation produced the numbers #

The chord is a close major triad — root, major third, perfect fifth — with ten partials asked of each note and eight available, since the string spectrum this ladder uses has eight. So every figure is over twenty-four components, and the 10 every placement on this anchor passes has been doing nothing for four rungs, which is worth saying and is not a finding about anything.

Each component is tested against the summed masked threshold of every other component in the chord, with the threshold of hearing in quiet as the floor. The spreading function is the collection’s own: twenty-seven decibels per Bark downward and 24 + 0.23/fₖ − 0.2L upward.

The roughness is the Plomp–Levelt kernel summed over pairs of partials belonging to different notes, weighted by both partials’ pressures — the same sum roughness can be computed uses, with the level-aware amplitudes the third rung introduced. It is computed twice, over all the components and over the surviving ones, and nothing else differs between the two.

The registers are drawn every four semitones from C1 to E6. The maxima quoted in the table are from a search at every semitone over the same range, which is why the table names G2 where the figure’s grid names A♭2.

Where the model stops #

The Bark scale below 200 hertz is the weakest part of it. The published bandwidth functions disagree by a factor of three down there — this collection has said so since the third is rougher in the bass — and everything in this essay’s bottom two octaves depends on which one is used. The direction of the result does not, because every candidate function makes the band wider in semitones at the bottom; the crossover register does.

The spectrum is a string’s. Eight partials falling as one over n, so the low components carry most of the energy and the model is generous to a spectrum whose energy is higher up.

The threshold model is all-or-nothing. A partial one decibel below the masked threshold contributes zero here and a partial one decibel above contributes fully. Real masking is graded, and a graded version would smooth the delivered curve without moving its maximum.

And a masked partial contributing no roughness is an assumption. The roughness kernel and the spreading function come from different literatures and neither was fitted for the other. That is recorded on the third rung and it is load-bearing here too.

What the picture cannot show #

It cannot show what a low chord sounds like. A sonority delivering one component of twenty-four is not silent, and this figure says nothing about what it is instead. The candidates are a pitch, a rumble and a felt vibration, and none of them is a count.

Nor can it show the fundamentals. Every partial that survives at C1 is a fundamental, and a chord heard as three sine tones is still a chord in the sense a listener would use.

It cannot show the room. A hall’s reverberant field is much flatter in the spectrum than the direct sound and its low-frequency decay is usually the longest, so a real low chord is masked by its own reverberation as well as by itself.

Nor a real bass instrument. A double bass, a contrabassoon and a pedal organ pipe have quite different spectra, and the ear hears the list of what is there — which for a low note is a list this model says is almost empty and which every player of those instruments would dispute.

And it cannot show a listener told what to listen for. Every threshold here is a detection threshold, measured on listeners asked to find a tone, and a listener following a bass line is doing something else.

Whose music, and when #

The chord is a generic triad and the levels are generic. Nothing here is a measurement of a repertoire.

The observation that fits one is the shape of the bottom of a score. European orchestral practice puts a single line in the bottom two octaves and spreads its harmony above — the bass has a note, the harmony starts a tenth or a twelfth higher — and it has done so since long before anybody could have measured a critical band. This essay’s numbers say the practice is right for two reasons that swap over at about G2, and neither of them is the reason usually given.

The repertoire that does the opposite is worth naming. Organ registration builds sixteen- and thirty-two-foot stops into thick low sonorities on purpose, and so does a good deal of twentieth-century orchestral writing. On this arithmetic those are not chords being heard as harmony; they are single objects with a great deal of weight, which is a fair description of what they are used for.

Where this ladder goes next #

Six 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; and now a register, which decides more than any of them and had been held fixed throughout.

What the ladder owes now is the timbre, and the criterion that found this rung finds that one. Every figure on this anchor — 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. A clarinet’s even partials are twenty-five decibels below its odd ones, so its components are an octave and a fifth apart at the bottom rather than an octave, which puts them in different filters and should let it keep far more of itself; a bell’s are inharmonic and a flute’s are three. And the case that matters is the mixed one: an equal note cannot be masked recorded that its proof depends on the predecessor’s components coinciding with the successor’s, which is true of one instrument and false of an ensemble, and the texture rung made the same reservation from the other side. Two rungs have now named the same missing parameter. It needs no corpus and no listener — this collection carries six spectra already — and what would come out is whether a masking result on this site is a result about the ear or a result about a violin.