A tuned chord's flutter binds it only if its notes start together

A tuned close seventh chord low on the keyboard flutters at one rate found the one place in these chords where tuning adds something rather than sharpening it. A dominant seventh in close position, tuned exactly 4 : 5 : 6 : 7 over a bass below B2, repeats at a rate too slow to be a pitch, and in every auditory filter from 60 hertz to a kilohertz and a half its envelope rises and falls at that one rate. The tempered chord fluctuates too, but at a different rate in each filter.

It ended on a question with a prediction in it. A fluctuation shared by every filter at one rate is the kind of cue the ear treats as evidence of a single source, because independent sounds rarely fluctuate in step. So the tuned chord should be heard as more fused than the tempered one: one pulsing object rather than four notes. The computation proposed was the correlation between filters’ envelopes, which the tuned chord makes perfect.

It does not make it perfect, and the reason it does not is the result. The earlier essays on these chords — that a tuned seventh chord does not beat; it buzzes, and that a tempered seventh chord already has the tuned one’s low note — each asked what tuning changes and each found it changed less than expected. The flutter was the exception. This essay asks how far the exception reaches.

One rate is half of comodulation #

The auditory system’s evidence that different frequency regions belong to one source is called comodulation: their envelopes rise and fall together. It is the cue behind a well-studied effect in hearing, where a tone buried in noise becomes easier to detect when the noise in neighbouring bands fluctuates in step with the noise at the tone, because the ear can use the shared fluctuation to tell what belongs to the noise. The same shared fluctuation is one of the cues what makes two partials one note listed for grouping. In both cases what matters is not only that the regions fluctuate at the same rate but that they fluctuate together: high at the same moments, low at the same moments.

The earlier essay established the first half for the tuned chord. The second half depends on something the tuning does not fix. Inside each filter the envelope is the sum of the partials that filter passes, beating against each other, and when within each period that sum peaks depends on the partials’ starting phases. Tuning makes every difference between partials a multiple of the series fundamental, so every filter’s envelope repeats at that rate; it says nothing about where in the period each filter’s peak falls.

The earlier essays’ strings were all struck at one instant, with every partial of every note starting in phase. That is one choice among several, and the envelopes on the left of the first figure are what it gives. The right-hand envelopes are the same chord, same tuning, same levels, with each note given its own starting phase — as four voices, or four organ pipes, sounding the same chord would have. Every filter still pulses once every 61.2 milliseconds. The pulses no longer line up.

Every pair of filters #

To measure how much the filters move together, take the envelope of every filter, one equivalent rectangular bandwidth apart from 60 hertz to 1.5 kilohertz, and correlate every pair over two seconds.

Struck together, the tuned chord’s filters correlate at 0.59 on average, and neighbouring filters at 0.80. It is not the perfect correlation the prediction expected even here — filters higher up pass partials that coincide twice a period, and a pulse twice a period correlates only partly with a pulse once a period — but it is one block: every filter is linked to every other through strong correlations.

With its notes at their own phases the same tuned chord drops to 0.16, and the tempered chord struck together is at 0.16 too. On the measure the prediction proposed, the tuned chord’s special coherence belongs to the way its notes were started rather than to its tuning. The matrix keeps a trace of the tuning — its blocks are larger than the tempered chord’s — and that trace comes back below.

The rate survives, and so does most of the depth #

Whether the flutter itself — the thing the earlier essay found — survives the phases is a separate question, and the answer is mostly yes.

Averaged over the filters, the depth at the series rate is 0.31 struck together and 0.29 with the notes at their own phases. The tempered chord, which has no such rate, is at 0.03. Every filter keeps the flutter.

The one place the phases matter within a filter is the bottom. In the filters around 90 and 130 hertz, which pass the fundamentals and second partials of several notes at once — the unresolved region which harmonics carry the pitch describes at the bottom of the bass range — the envelope with everything in phase is a sharp spike once a period — depth 1.33 — and with the notes at their own phases the spikes smear into a broader swell, depth about 0.57. That is still some eighteen times the threshold for hearing a fluctuation at 16 hertz. So a listener attending to any one region of the chord would still hear it pulse, and pulse at exactly the rate the earlier essay found. What the phases take away is mostly the timing between regions.

That distinction rescues half of what the earlier essay claimed. It proposed the flutter as a cue for tuning: absent until the seventh is within about two cents of 7 : 4, then present as a single pulse at a quarter of the bass. That cue lives inside individual filters — a singer or an organist listens for the pulse to appear, not for the filters to agree — and inside individual filters the rate survives any choice of phases. So the flutter remains a tuning cue for exactly the voices and instruments that can use one. What is in doubt is only its second role, as something that holds the chord together.

One region, three, or ten #

A correlation averaged over all pairs is one number, and grouping is about structure. So the matrix can also be read the way a grouping rule would read it: link any two filters whose envelopes correlate above one half, and count how many separately fluctuating regions are left.

Struck together, the tuned chord is one region. The tempered chord struck together is ten. With each note at its own phase, the tuned chord is 3.4 on average over eight draws, and the tempered chord 10.8. With every partial at its own phase — no instrument quite does this, but it is the limiting case — the two are 9.6 and 16.4.

So the tuning is not irrelevant to grouping; it is not decisive either. At every choice of phases the tuned chord falls into fewer regions than the tempered one, because its filters share a rate and filters that share a rate are closer to lining up by chance. But the step from ten regions to one, which is what “the flutter binds the chord” would mean, needs the notes to start together. Without that, the tuned chord comes apart into three or four fluctuating regions: fewer than the tempered chord, and far from one.

The threshold of one half is a stated convention. Lower it to 0.3 and every case except the tempered chord with random partials becomes one or two regions; raise it to 0.7 and everything except the tuned chord struck together fragments into thirteen to seventeen. The tuned chord struck together is one region at every threshold from 0.3 to 0.7. It is the only case that is.

How together is together #

A real chord’s notes are never started at exactly the same instant, and the obvious question is how close they must be.

Give each note its own onset time, drawn at random within a spread, while keeping each note’s own partials together — the way a chord actually sounds when four hammers or four singers begin at slightly different moments. The tuned chord’s comodulation is 0.59 with no spread, 0.51 within half a millisecond, 0.31 within one, and 0.17 within two, where it has reached the tempered chord’s level of 0.16 and stays there however wide the spread becomes. The tempered chord does not care: its filters were never lined up, so displacing its notes changes nothing.

The reason the tolerance is so tight is the frequencies involved. The notes’ fundamentals are 65 to 114 hertz, their periods 9 to 15 milliseconds, and the partials that meet in the filters that matter are their third to tenth, with periods of one to five milliseconds. Displacing a note by a millisecond turns those partials by a large fraction of a cycle against the others, and the envelope’s peak in each filter moves to a different moment in the series period. Two milliseconds is enough to scatter them fully.

That is a very small number against what performers do. The spread of onsets within a chord played by one pianist is commonly reported in the tens of milliseconds, with the melody note leading; singers in an ensemble are further apart than that. A millisecond is not a spread a hand or a choir can hold. The one instrument that sounds a chord’s notes together to within a millisecond is an electronic one triggered by a single event.

At every bass #

The earlier essay found the flutter from C2 up to B2, where the series rate reaches the floor of pitch. Across that range the same division holds. Struck together, the tuned chord’s filters correlate at between 0.54 and 0.59. With the notes at their own phases, between 0.10 and 0.16, and the tempered chord between 0.09 and 0.16 — never more than about three hundredths apart. The division is not a property of C2; it is a property of how the notes begin.

The irony, and the connection #

Put the two results together and they make an awkward pair. The chord whose flutter would bind it into one object is a close seventh tuned exactly to 7 : 4, with every note starting within a millisecond. The instruments that can tune a seventh to 7 : 4 — voices, bowed strings, trombones, an organ tuned in just intonation — are exactly the ones whose notes start independently. The instrument that starts its notes together, if anything does, is a keyboard, and a piano’s seventh is tempered.

This connects to a rule the auditory-scene essays found from the other direction. The cue that settles it priced the cue every account calls the strongest, a shared onset, and put its threshold at about twenty milliseconds: components starting within that window are grouped as starting together, and a struck string’s partials start together to within a tenth of a millisecond. What makes two partials one note had already found that a thirty-millisecond head start takes a perfectly harmonic partial out of its note.

Set the two windows side by side. The onset cue binds a chord whose notes begin within about twenty milliseconds of each other. The flutter’s comodulation binds it only if they begin within one or two. So every chord whose notes are close enough for the flutter to line up is a chord the onset cue has already grouped, ten times over, and every chord whose notes are too far apart for the onset to group them is far outside the flutter’s window as well. There is no chord in which the flutter does grouping work the onset has not already done. The flutter is a grouping cue that works only in the circumstances where it is least needed.

Between two and twenty milliseconds — which is where most real chords begin — the onset cue says one event and the flutter says nothing in particular: three or four fluctuating regions, the same rate everywhere, no common moment. What a listener makes of that combination is a question about how cues combine that the exchange-rate essays left open, and it is the regime this essay’s numbers are most relevant to.

That does not make the earlier essay’s result any less real. A tuned close chord low on the keyboard does flutter at one rate, in every filter, at a depth far above threshold, whatever the phases. What it does not do is fluctuate in step across the ear unless its notes began in step, and so the earlier prediction — that it would be heard as one pulsing object, harder to hear into than the tempered chord — does not follow from the flutter alone.

What the model assumes #

The envelopes are those of a linear filter bank. Each filter is a rounded-exponential shape one equivalent rectangular bandwidth wide, and the envelope is the magnitude of the partials it passes summed as phasors. The auditory system’s own envelope extraction is compressive and its filters broaden with level; both would change the depths, and neither would give back the timing the phases take away.

Comodulation is read as zero-lag correlation. The auditory system is thought to compare envelopes across bands with little tolerance for delay — the effect on detection mentioned above falls away when the modulation in the flanking bands is shifted in phase — and zero-lag correlation is the plainest version of that. A listener able to align filters by delay would find more coherence in the tuned chord with its notes at their own phases, since each filter still repeats at one rate; whether any listener does is not established here.

The grouping rule is a stand-in. Linking filters above a correlation threshold and counting regions is not a model of perception; it is a way of saying how much structure the correlations contain. It was swept from 0.3 to 0.7 and reported at 0.5.

The notes are struck strings with fixed spectra. A voice’s partials are held in a fixed relation by its glottal pulse, which is why the “own phase” case moves whole notes rather than single partials. A voice also wavers in pitch, which in the earlier essay’s terms would smear the flutter’s rate itself; that is not modelled.

What the picture cannot show #

How many notes a listener hears. The prediction asked how many of the chord’s notes a listener can hear out, and nothing here answers it. In this register the notes cannot be told apart by which filters they occupy: from 60 hertz to 1.5 kilohertz no filter is dominated by a single note, because the fundamentals of a close chord over C2 all lie within about one auditory filter. Hearing out such notes depends on the ear’s reading of pitch, which the root an ear supplies and four notes cannot get below the seventh partial computed for these chords, and not on comodulation. What this essay can say is that comodulation, one of the cues that might have held the notes together, does so only when the notes start together.

And a sustained chord keeps its phases, but a chord is not only sustained. Once begun, a tuned chord’s relative phases stay where they were for as long as it sounds, since its notes share a period exactly. So a chord that happened to start together would stay comodulated. Every new chord re-draws the phases.

Still open: whether a chord that starts together is heard as pulsing #

There is one situation where the phases are fixed rather than drawn: an electronic instrument, or a sampled one, that plays a tuned chord from a single trigger with every note beginning at its waveform’s zero. There the flutter’s full coherence is available, and whether it is heard is an experiment rather than a calculation — a listening test between the same tuned chord started in phase and started with its notes a few milliseconds apart, at the basses where the earlier essay found the flutter deepest. The arithmetic predicts a difference in how many fluctuating regions the ear is offered, one against three or four. Whether listeners report one pulsing object against several is what the arithmetic cannot supply.