A bass chord low enough to balance has already hidden its tenor

A rough arrival is rough because of its spacing put level back into the pair of numbers the expectation essays report for every chord, and found that it multiplies rather than dilutes. Played at one written dynamic, the arrivals of I – vi – IV – V – I two octaves down are 8.6 times rougher than an octave up. Played so that every chord is as loud as the written register’s, they are 343 times rougher, because the bass needs about sixteen decibels more than the treble to be heard as loud, and roughness grows with the square of the pressure.

It closed on a question it had not computed. The level at which a bass chord has to be played to balance a texture is also the level at which it masks whatever sits just above it, and masking spreads upward and grows with level. So balancing a low voicing might bury its tenor, and the progressions roughest in the bass might be the ones whose inner lines go missing.

Both halves have answers, and neither is the one the question expected. The balance buries almost nothing. The register does, and it does so at the written dynamic, before any balance is applied. What the register buries takes the 343 with it.

Two sums over the same chord #

The roughness in the earlier essay is Plomp and Levelt’s sum over every pair of partials of every pair of notes, with each partial’s amplitude a pressure computed from its level. It includes every partial the score implies, and it drops only those under the threshold of hearing in silence.

The masking computation asks a different thing of each of those partials: does it stand above the threshold that every other partial of the chord raises at its frequency? Each partial of each note is a masker with its own level, spreading downward at a steep 27 decibels per Bark and upward at a slope that shallows as the masker gets louder. The intensities of every masker at a frequency are added, the threshold of hearing is the floor, and a partial is delivered if its level clears the total. That is the census a loud chord is a smaller chord takes of a triad, run on each arrival of a progression.

The heard roughness is then the same pairwise sum restricted to delivered partials. It can never exceed the full sum, because it adds a subset of the same pairs, and the figure’s generator refuses to draw it if it ever does.

The voicings are the earlier essays’ own: each chord after the tonic in the four-part voicing nearest the hands’ last position, the whole progression moved bodily by octaves. The written register already puts its bass on C3, so “two octaves down” is a bass on C1 with the tenor on A1 and the soprano on E2 — a register a pianist’s left hand reaches and an orchestra fills with contrabassoon and double bass, not a thought experiment.

The balance, restated #

The levels come from the loudness model and nowhere else. Each arrival is played at whatever level per note makes its excitation pattern integrate to the same number of sones as the written register’s mean at seventy decibels.

Two octaves down the balance asks for 82.8 decibels a note, an octave down 75.5, an octave up 66.8. These are the numbers the question was about: the extra 12.8 decibels the bass is given is the extra reach its masking acquires.

How much reach that is can be said before anything is counted. The upward slope of a masking pattern is 24 decibels per Bark, less a fifth of a decibel for every decibel of masker level, so 12.8 decibels flattens it by about two and a half decibels per Bark. A partial one Bark above a masker sees its masked threshold rise by the masker’s gain less the slope’s loss — but the partial itself was raised by the same 12.8 decibels, because the whole chord was. Balancing raises the masker and the masked together. The only thing left over is the flattening, times the distance in Bark. In the bottom octaves, where a chord’s notes lie within one Bark of each other, that product is a decibel or two.

What each voice keeps #

The first count is per voice: of each voice’s eight partials, how many clear the threshold the rest of the chord puts over them, arrival by arrival and register by register, at seventy decibels and at the balanced level.

The two panels are nearly the same picture, and that is the first result. Balancing moves no voice of any arrival by more than two partials, and in the written register it moves none at all, since the written register is its own balance. The tenor’s count over the whole progression is fourteen partials of thirty-two at seventy decibels and fourteen at the balance; an octave down it is four and four; two octaves down it is zero and zero.

The bound of two is the written seventy decibels’ own, and it is worth saying how far it travels. Written at sixty or at eighty, one voice of one arrival an octave down does move by three when the chord is balanced — an alto from six partials to three at sixty, a soprano from six to three at eighty — so the rule is not exact at every dynamic. What does travel is the tenor: across every written dynamic from fifty to ninety decibels, balancing changes the tenor’s count over the whole progression by at most two, at any register.

Read down the columns instead and the picture changes completely. In the written register the soprano delivers seven or eight partials of each arrival, the alto five to seven, the tenor three or four and the bass one. That ordering — the top voice whole, the bass reduced to its fundamental, the inner voices somewhere between — is the one the listener is given the top voice found for a single four-part chord, and here it holds at every arrival. An octave down the tenor keeps one partial of each arrival and the alto one to four. Two octaves down the tenor and the alto keep nothing at all, the soprano two or three, and the bass its fundamental.

The balance was asked about because it is the part a performer controls. The answer is that it controls very little of this. What buries the tenor is the octave the passage is written in, and a player who balances a low passage perfectly has changed its tenor’s fate by at most a partial.

The fundamental’s margin #

A count of partials says whether a voice arrives with its colour. Whether it arrives as a pitch at all is closer to a question about its fundamental, since a line whose fundamental is masked has to be followed from whatever partials are left above it, and in these registers there are none.

The tenor’s fundamental clears the chord by 7.1 decibels as written and 14.9 an octave up. An octave down the margin is 2.2, and two octaves down it is −0.9: masked. The balanced margins are 7.1, 2.0 and −1.1 — two tenths of a decibel away, in the direction the flattening slope predicts. The alto, which in close position sits a third or fourth above the tenor and inside the same critical band, is closer to the line throughout: 2.3 as written, and masked at both lower registers at either level.

The mechanism is the unit, again. The Bark scale is close to linear in hertz at the bottom, so it is close to exponential in semitones. In the vi chord the written tenor on A3 stands 0.87 Bark above the bass on C3. Two octaves down the same interval, A1 above C1, spans 0.22 Bark — a quarter of the distance, for the same written sixth. At 0.22 Bark the bass’s upward skirt has fallen by barely three decibels, so the bass fundamental alone puts a threshold within three decibels of a tenor fundamental at the same level. The bass’s second partial, C2, lies a tenth of a Bark above the tenor and masks it downward from six decibels under the bass. The alto on C2 and the soprano on E2 add their own lower skirts from a few tenths of a Bark away. Four maskers within a third of a Bark, each within a few decibels of the tenor’s level, sum to more than the tenor has.

A low chord stops being a chord found the same collapse for a triad’s count as a whole: 79 per cent of its components at E3 and 4 per cent at C1, because the critical band is thirty-seven semitones wide down there. The voice-by-voice reading adds the part the question needed: which voice goes first. It is not the bass, whose fundamental is at the bottom of everything and cannot be masked from below; it is the inner voices, whose fundamentals sit inside the bass’s harmonics.

Level against register #

If the balance barely matters, the dynamic might. Every masking slope shallows with level, and a passage played forte instead of piano moves every note by much more than a balance does.

It does. As written, the tenor’s share falls from 63 per cent of its partials at forty decibels to 16 per cent at ninety, and the alto’s from 84 to 41. Fifty decibels of dynamic is not a small change, and a fortissimo close chord in the tenor register delivers a quarter of what the same chord delivers pianissimo. That is the level effect a loud chord is a smaller chord measured on a triad, now visible in each inner voice.

But the scale of it is the point. Fifty decibels of dynamic takes the written tenor from 63 per cent to 16. One octave of register at a fixed seventy takes it from 44 per cent to 13. Two octaves take it to nothing at every level on the chart, including forty decibels, where masking is at its narrowest; down there the tenor loses to the threshold of hearing and the bass’s harmonics together. And the ring on each line, where the balance puts that register, sits on the flat part of the curve: an octave down the balance adds five and a half decibels and the tenor’s share stays at 13 per cent.

So the question “does a loud bass bury its tenor?” separates cleanly into three parts with three sizes. A register change costs the most. A change of dynamic costs a comparable amount over its whole range. The few decibels a balance adds cost almost nothing, because they are too few and because they raise the tenor with the masker.

The roughest arrivals are the emptiest #

The second half of the question was whether the arrivals that are roughest in the bass are the ones whose inner lines go missing.

They are, and so strongly that the two quantities are nearly one. Across the sixteen arrivals the full-spectrum roughness at equal loudness and the delivered share of the upper voices correlate at −0.88. The four arrivals two octaves down have roughnesses of 0.098 to 0.153 in the model’s unit and deliver 4 to 13 per cent of their upper voices; the written register’s four have roughnesses near 0.002 and deliver 67 to 75 per cent.

This is not a coincidence of two separate effects. Both numbers are functions of how many partials share a critical band. Partials inside one band beat against each other, which is roughness — the effect a third is rougher in the bass built the low-interval rule on — and mask each other, which is loss, spreading upward in the way one sound hides another first drew. The roughness the score predicts and the partials the listener loses are the same crowding counted twice, and the model can only report a large roughness for a chord by counting pairs among the partials that the same crowding has hidden.

That is what the hero figure shows once it is put back together. Counted over every partial, the balanced progression is 70 times rougher two octaves down than as written, and 343 times rougher than an octave up. Counted over the partials that arrive, it is less rough two octaves down than in any other register — the delivered sum is six billionths of the written register’s, which is to say no delivered pair is close enough to beat. The roughest register a listener receives is an octave down, at 2.8 times the written one, and an octave up the two counts nearly agree, at 0.20 and 0.19.

What the balance does do #

The balance is not inert in the delivered sum, and the one place it acts is worth stating because it moves a maximum.

At one written dynamic, with every note at seventy decibels, the delivered roughness is highest in the written register, at 9.0 ten-thousandths in the model’s unit, with an octave down just under it at 8.5 and an octave up at 3.6. Balancing raises the octave-down register by five and a half decibels and therefore multiplies its delivered roughness by three and a half, since pressure enters the roughness twice. It lowers the octave-up register by three. So the delivered peak moves one octave down when the passage is balanced, from the written bass on C3 to a bass on C2.

What the balance cannot do is reach the register below that. At C1 the delivered sum is effectively zero at seventy decibels and at the balanced 82.8 alike, because no multiplier on nothing makes something. The earlier essay’s finding that level multiplies the register’s roughness is right about the multiplication and wrong about what it multiplies in the bottom octave.

What this changes in the pair #

Eighty-one chords, one number settled that the roughness of an arrival belongs beside its surprise rather than inside the probability, and the earlier level essay found the pair’s roughness axis stretched to a factor of 568 by the balance, with the surprise unmoved. Both results survive, and the second needs its axis replaced.

The surprise is still a function of scale degrees — the final tonic costs 1.60 bits and the three chords before it 2.68, 2.68 and 2.80, in every register and at every level — so no masking computation can touch it. What changes is which roughness sits beside it. A roughness summed over every partial is a statement about the score. A roughness summed over delivered partials is a statement about what a listener receives, and it is the one the expectation essays have been describing a listener with. On the delivered axis, the pair’s long horizontal lines — one arrival across four registers — no longer run monotonically toward the bass. They rise from the treble to a peak an octave below the written register and then drop off the chart.

For a listener, a progression played in the bottom octave is not a rough progression. It is a progression that has stopped delivering its upper voices, and whatever it sounds like — the words musicians use are muddy and thick — it is not a sum of beating pairs, because the pairs are not there to beat.

Which computation produced the numbers #

The progression, voicings and surprises are the ones surprise is a number and its successors use, unchanged. Each note has eight partials whose amplitudes fall as one over their number, scaled so the note’s total level is the level stated. The loudness is Zwicker’s, integrated over the excitation pattern of every partial; the balanced level is found by bisection per arrival against the written register’s mean at seventy decibels.

The masking pattern of each partial is the codec-standard two-slope spreading function on the Bark scale, 27 decibels per Bark downward and 24 + 0.23/f − 0.2L upward, with f in kilohertz and L the masker’s level. Intensities from every other partial of the chord are summed, a voice’s own other partials included, and the threshold of hearing at the partial’s frequency is the floor. A partial is delivered if its level is above that total. The delivered roughness of an arrival is its full pressure-unit roughness times the share of its pairwise roughness carried by delivered pairs, so that both readings are in the same unit.

Where the model stops #

The spreading function is simultaneous and static. Every chord here is a sustained block. Real inner voices move while the bass holds, and a change of pitch is audible even when a held partial would be masked; a moving tenor is followed by its onsets as much as by its spectrum.

Every voice has the same spectrum. A cello tenor over a double-bass line has more energy in its upper partials than the string-like spectrum gives it, and a horn or a clarinet has a different shape again. The arch belongs to hearing, not to the series found the register arch robust to the spectrum, but the per-voice margins here are a few decibels wide and a spectrum can move them.

Delivered is a threshold, not a degree. A partial one decibel above its masked threshold counts as fully delivered and one decibel below as absent. Partial loudness, which grows gradually from the masked threshold, would round every edge here without moving any of the orderings.

The balance is per chord, not per voice. Every note of an arrival is raised together. A performer balancing a texture raises the bass more than the tenor, which the listener is given the top voice computed at one register and found changes a voice’s loudness share without changing its spectrum.

What the numbers cannot say about a listener #

Whether a masked tenor is a missing tenor. A listener who knows the progression may hear its inner line as continuing through a masked stretch, in the way a melody is heard as continuous through a brief noise. The arithmetic can say that nothing of the tenor clears the threshold; it cannot say what a listener fills in.

Whether thick is the name for crowded-and-masked. The prediction here is that a low close chord delivers little that beats and little of its upper voices, and that the adjective for that is different from the adjective for a chord that beats. That is a listening test between two registers of one progression, and it has not been run here.

Still open: whether a moving inner voice survives its own masking #

Every number above holds each chord still. The obvious next variable is time: a tenor that moves by step while the bass holds has onsets the bass does not, and a sound hides what came before it supplies the forward-masking time course that would say how long after each chord change an inner voice’s new pitch stands above the decaying excitation of the old chord. Computing the delivered share of a moving tenor over the first two hundred milliseconds of each arrival, at the registers where the static tenor has vanished, would say whether a line an arranger writes in the bass is heard as a line by its motion alone — and whether the orchestrator’s instinct to keep inner voices moving in the low register is a response to exactly this.