A rough arrival is rough because of its spacing

Eighty-one chords, one number decided that the roughness of a voicing belongs beside a chord’s surprise rather than inside it, and reported the two as a pair. It ended by naming a third quantity the pair had been computed without. Every roughness in those essays is a number about the shape of a spectrum normalised to one, so it contains no level at all, and a chord in the bass and the same chord in the treble are not equally loud at the same written dynamic.

The question it put was whether a rough arrival is rough because of its spacing or because of its level. The two are separable in the model even though a player cannot separate them at the instrument, and the arithmetic to separate them already exists: a roughness summed in pressures rather than in a normalised spectrum, with partials below the threshold of hearing dropped, and a loudness integrated over the excitation pattern.

Separated, the answer is plain. It is the spacing — and the way level enters is not the way the question expected.

Three ways to read one voicing #

The progression and its voicings are the previous essay’s, unchanged: the four chords after the tonic, each in the voicing nearest the hands’ last position inside four choral compasses, and the whole passage moved bodily by octaves to change register without changing any spacing.

Each arrival is then read three ways.

The shape is the roughness the expectation essays use: Plomp and Levelt’s model summed over every pair of partials of every pair of notes, with a string-like spectrum whose amplitudes are relative. It says how rough the voicing’s arrangement of partials is, and nothing about how loudly they sound.

At one level is the same double sum with every note played at 70 decibels. Amplitudes become pressures, so each pair’s contribution is the product of two pressures, as Plomp and Levelt’s formula says it must be, and any partial below the threshold of hearing at its frequency is dropped rather than made small. That is the roughness the ranking survives the dynamic and the chord does not used to find that sixty decibels multiply a chord’s roughness by a million and its loudness by ninety-six, and that the ranking of doublings does not move by a single place.

At equal loudness is the same sum again, with every note of each register played at whatever level makes that register’s chords as loud, in sones, as the written register’s chords are at 70 decibels. This is the reading a performance actually asks for, because nobody plays a bass line quieter than the melody above it merely because the page says the same dynamic.

At one dynamic, level only rescales #

The first comparison is between the shape and the roughness at one level, and it comes out exactly even: the lowest register is 8.6 times rougher than the highest both ways, to the first decimal.

The reason is in the arithmetic. With every note at one level, every pair of partials that is audible has its product multiplied by the same factor, and at 70 decibels nearly every partial of a string spectrum in these registers is above threshold. A uniform factor cannot change a ratio. At a single written dynamic, the register’s roughness is exactly the voicing’s spacing, scaled.

The same thing seen across dynamics is a test that could have failed. If level mattered to roughness beyond a common factor, then somewhere between pianissimo and fortissimo the four arrivals would change order, because a partial dropping below threshold removes pairs from one voicing and not from another. At the written register, an octave down and an octave up, they never do: IV to V is roughest at every level from 40 to 90 decibels, and the other three keep their places.

The exception is informative about where the exception could come from. Two octaves down, at 40 decibels a note, the two smoothest arrivals — the tonic moving to the submediant and the submediant moving to the subdominant — change places, and at 50 decibels and above they change back. That is the threshold at work: the threshold of hearing rises steeply below a few hundred hertz, so a quiet bass chord loses low partials that a louder one keeps, and the pairs they would have made go with them. A chord is a register found roughness falling fifteenfold across the piano from exactly the crowding of low partials into critical bands, and the threshold is the one place the level can reach in and edit that crowding.

Level can reorder arrivals only where the music is quiet and low, and even there it reorders the two that are nearly equal already. Everywhere else, which arrival is rough is a property of how its notes are spaced.

Equal loudness is not one level #

The previous essay’s registers held every note at one written dynamic, and that is not what a listener hears as one dynamic.

At a single level of 70 decibels a note, the passage two octaves down is less than half as loud as it is at the written register — 16.1 sones against 36.5 — and an octave up it is louder, at 44.3. That is the equal-loudness contours doing what a subito piano is four seconds longer in the bass found them doing to every loudness figure on the site: the ear is least sensitive at the bottom, so a bass sound at a given level is quieter than a treble one.

To play the low passage as loud as the written one, every note has to go up by 12.8 decibels. An octave up, it has to come down by 3.2. One loudness is four levels, spread over sixteen decibels, and the spread is largest exactly where the roughness is largest.

Equal loudness multiplies the roughness #

The third reading follows from those two facts. Roughness, summed in pressures, grows with the square of the pressure; the bass needs more pressure than the treble to be as loud; so a bass passage made as loud as a treble one is rougher by more than its spacing alone makes it.

By how much is the headline number. Level-free and at one level, the lowest register is 8.6 times rougher than the highest. At equal loudness it is 343 times rougher. The factor of forty between those two is the equal-loudness correction, applied twice, because pressure enters the roughness twice.

The debt’s framing expected the opposite possibility to be live. If some of the register’s roughness had been a level effect, putting level back in would have shrunk the register’s effect by that share. It grows instead, because the level that performance demands of the bass is higher, not lower, and the roughness of a pair of partials rises with it.

A bass passage is not rough because it is loud. It is rough because its notes are crowded into critical bands, and it has to be played loud to be heard, which makes the crowding cost more.

That result needs one caution attached to it immediately, and it belongs in the paragraph with the number rather than at the end. The roughness here is in the model’s own unit, a sum of pressure products. Perceived roughness grows more slowly with level than a pressure product does, so the perceptual factor between a loud bass and a quieter treble is smaller than 343. What survives any compressive rescaling is the direction: made equally loud, the bass is rougher against the treble than it is at one level, never less.

The factor of forty is sixteen decibels #

The step from 8.6 to 343 can be checked without running the loudness model a second time, and checking it shows exactly where the extra roughness comes from.

With every note of a voicing at one level, the level-aware roughness is the shape times a factor that depends only on the level. Raising every note by $\Delta L$ decibels multiplies every pair’s product of pressures by $10^{\Delta L/10}$. So the roughness of two registers at equal loudness should stand in the ratio of their shapes times $10^{\Delta L/10}$, where $\Delta L$ is how much further the lower register had to be raised.

The two outer registers were played at 82.8 and 66.8 decibels, sixteen decibels apart. Ten to the power 1.6 is 39.8, and 8.6 times 39.8 is 342 — the 343 the loudness model returned, to the rounding of its levels. The inner step is the same arithmetic: an octave down against the written register, the shapes stand at 2.12 and the levels 5.5 decibels apart, which is a factor of 3.5 and a product of 7.5, and the model’s value is 7.47.

All of the extra roughness is the equal-loudness correction, and none of it is anything else. The closed form assumes no voicing gains or loses a pair of audible partials between the levels compared, and it agrees with the loudness model to the rounding of the levels, which it could not do if any voicing had. The spacing sets the ratio at one level; the ear’s insensitivity at the bottom sets how much further apart the registers have to be played; and the squared pressure turns a level difference into a roughness factor at ten decibels a decade.

That also says what would change the number. A spectrum with more energy in its low partials is louder in the bass for the same level, needs less correction and gains less roughness; a quieter reference dynamic moves every register up the steep part of the equal-loudness contours and needs more. The factor is a property of the contours and the spectrum together, and 343 is its value for a string spectrum balanced at mezzo-forte.

The pair, with level put back in #

The previous essay’s reason for keeping surprise and roughness as two numbers was that they were independent: over sixteen arrivals they correlated 0.03. If the level-aware roughness had turned out to track the chord’s surprise, that reason would have gone.

It has not. At equal loudness the sixteen arrivals spread over a factor of 568 in roughness and the surprise of each is exactly what it was, because a surprise is a function of scale degrees and neither register nor level touches a degree. The correlation is 0.023, which is 0.029 with a rounding error’s worth taken off.

So the pair survives the third coordinate unchanged in structure, and for the reason two surprises and one event gave for adding some quantities and not others: a surprise is about what was composed, and neither the register nor the level a passage is played at is part of that. What changes is its scale: the roughness axis becomes much longer, and it becomes long in the direction the register already pointed.

The level-free version is worth setting beside it, because the comparison is the whole finding in one glance. Every horizontal line joins one arrival at four registers. In the level-free plot the lines are a factor of about eight long. At equal loudness they are a factor of several hundred long, and no line has moved up or down.

Where the candidates were, and still are #

The seven-candidate table is where the conditioning question started, and the level result has a consequence for it that is worth stating. Eighty-one chords, one number found that a roughness term inside the probability would need a coefficient that more than doubled across four octaves, because roughness fell with register and the profile term did not. At equal loudness the roughness falls not by a factor of eight across those registers but by a factor of three hundred, so the coefficient would have to vary by far more than a factor of two. Every argument that put the roughness beside the probability is stronger with level included, and the one that could have rescued putting it inside — that a register effect might be partly an artefact of holding the level fixed — is the one the arithmetic refutes.

Which computation produced the numbers #

The surprise is the identity surprise of surprise is a number conditioned as a chord given a key and a predecessor conditions it, in bits, unchanged. The voicings are the four-part nearest voicings in the choral compasses, moved by whole octaves.

The level-aware roughness is Plomp and Levelt’s pairwise formula with each partial’s amplitude a pressure computed from its level, partials under the threshold of hearing at their own frequency removed, and a string spectrum whose partials fall as the reciprocal of their number. The loudness is Zwicker’s specific loudness integrated along the Bark axis over the excitation pattern of every partial of every note. The equal-loudness level is found by bisection, register by register and arrival by arrival, against the mean loudness of the written register’s four chords at 70 decibels; each register’s figure is the mean over its four arrivals.

Where the account stops #

The roughness unit is not a perceptual one. A sum of pressure products says how much beating energy a voicing carries, and perceived roughness grows compressively with it. The ranking results are unaffected, since a monotone rescaling cannot reorder anything, and the direction of the equal-loudness result is unaffected. The size of 343 is not a perceptual size.

Every note of a chord is at one level. In performance the bass is often played louder than the inner parts and the melody louder than both, and none of that is modelled. A voicing whose bass is brought out adds roughness in exactly the pairs that involve the bass, which is where the crowding is.

The spectrum is one instrument’s, at one dynamic. A string-like spectrum at every register and every level is a simplification the comparison inherits, and a dynamic mark changes what a note is on nearly every instrument, so a louder bass is also a brighter one: a real bass line and a real treble line are rarely played by instruments with the same spectrum, and a bassoon or a double bass has a different share of its energy in the low partials that make the crowding.

The masking half of the debt is not paid. The previous essay named a chord’s loudness at a register as deciding both how rough it is and how well it masks what surrounds it. Only the first is computed here. What a loud bass chord does to the audibility of an inner voice above it is a separate computation and is left open.

What a pair of numbers cannot say about a listener #

Whether a listener hears the roughness as roughness. A dense bass chord is often described as muddy or thick rather than as rough, and whether those words name the same quantity as a beating sum is not in the arithmetic.

Whether performers balance for equal loudness. The third reading assumes a player makes each register as loud as the written one. Some do, some bring out the bass and some let it sit under the texture, and the right reading of the pair depends on which.

And whether a surprising chord is voiced differently. The pair’s independence is measured on voicings chosen by a rule about hands, and a composer or a performer may well voice an unexpected arrival differently from an expected one. That would make the correlation a property of practice rather than of arithmetic, and the corpus that would measure it is still owed.

Whose basses #

The rule of thumb every orchestration and piano-accompaniment manual gives about the register where the root an ear supplies is heard from — keep the thirds out of the bass register, spread the low voices, double at the octave rather than the third below middle C — is usually justified by the crowding of partials into critical bands, which is the spacing half of this result. What the level half adds is the reason the rule is stated so firmly rather than as a preference. A close low voicing is rough at any dynamic, and a performer who plays it loud enough to balance the texture above it is multiplying that roughness by the square of the extra pressure. The manuals’ low interval limits are written for a bass that has to be heard.

The same arithmetic says why a close voicing in the treble costs almost nothing. The treble needs less level to balance, the partials are spread across wide critical bands, and both halves of the roughness shrink together.

Still open: what a loud bass chord hides #

The level at which a bass chord has to be played to balance the texture is also the level at which it masks whatever sits just above it. The masking arithmetic the loudness figures use puts a chord’s excitation pattern along the Bark axis with a skirt that shallows as the level rises, and an inner voice a tenth or a twelfth above the bass is exactly where a loud bass’s upper skirt lands. Computing, arrival by arrival, how far an inner voice sits above the masked threshold of the chord under it — at one written dynamic and at equal loudness — would say whether balancing a low voicing for loudness also buries its tenor, and whether the progressions that are roughest in the bass are the ones whose inner lines go missing.