One fluctuation or two

The eleventh rung computed, for every member of a mistuned octave’s beat family, the share of the fluctuation inside its own auditory filter that belongs to that member. It then had to decide when two members are two fluctuations rather than one — a question the ninth rung had raised and left open, and it decided by a rule: their rates must differ by a factor of two.

The rung recorded the factor as a stand-in and said what would replace it. What decides whether two fluctuations are separable is a modulation filterbank — the auditory system analyses the envelope as well as the waveform, into channels of quality factor near one, spaced logarithmically — and two rates are two only if they land in different channels.

The separation the bank gives is 1.618, and the ninth rung’s question has an answer.

The number is the golden section and that is not a coincidence #

A resonator of quality factor Q has a response that falls to half its power where f/fc − fc/f = ±1/Q. At Q = 1 that quadratic gives f/fc = (√5 ± 1)/2, which is 1.618 and 0.618: the golden section and its reciprocal.

Take the criterion for a component being resolved to be that some channel gives it three quarters of that channel’s output, and the ratio at which two equal components each find such a channel is exactly the half-power ratio. So the separation and the channel’s own width are the same number, and at Q = 1 that number is φ.

It is worth saying plainly that this is arithmetic about a filter shape rather than a fact about hearing. What the measurements supply is the quality factor; everything else follows from choosing a resonator.

The eleventh rung’s stand-in was conservative #

Two is larger than 1.618, so the anchor has been demanding a wider separation than the bank does — which means every count it has published is a floor rather than a ceiling.

That is the opposite of what the debt anticipated. The eleventh rung’s closing section treats the factor of two as a rough approximation with the error unknown in either direction, and the direction turns out to be the safe one: any number the published estimates of the quality factor could support is below two.

The quality factor is the uncertainty and the drawing prices it. At Q = 0.5 the separation is 1.75, at Q = 1 it is 1.618, at Q = 4 it is 1.155. The whole plausible range sits between 1.1 and 1.8, and none of it reaches two.

So the correction has a known sign whatever the parameter is, which is a more useful thing to be able to say than a precise value would be.

And it changes nothing at all #

Recomputing the anchor’s grid with 1.618 in place of two produces the same numbers at every register.

The reason is visible in the rates rather than in the criterion. Where two members both survive the depth and position tests — at middle C, where the count is two — their rates are 2.51 and 10.03 a second, a ratio of four. Four clears the old criterion and clears the new one, and would clear anything between them.

The factor of two was never the binding constraint. What removes members from the count is the depth criterion and the position criterion; the separation rule was doing nothing anywhere. A stand-in that never bound is a stand-in the anchor was right to carry, and there was no way to know that without computing it.

That is worth having as a result rather than as a footnote. The eleventh rung named this as one of two things it owed, and it turns out to be a debt whose payment leaves the balance where it was — which is a useful thing to know about the other outstanding debts on this site, and is not the usual outcome.

The question the ninth rung asked #

There was a specific question underneath the general one, and it has a specific answer.

A mistuned octave at middle C puts its first two members at 1.26 and 0.89 a second. The ninth rung asked whether a listener attending to that family hears two fluctuations or one, and could not say.

The two differ by a factor of 1.42. That is inside the bank’s 1.618, so they fall in the same channel and they are one fluctuation, not two.

What a listener has is not a beat at 1.26 and a beat at 0.89; it is a single slow swelling at something between them, whose apparent rate is the intensity-weighted mean of the two — and whose regularity is worse than either, because two nearby rates beat against each other at 0.37 a second and the composite waxes and wanes over three seconds.

That last consequence is the one a tuner would recognise and it has a name in the practice: the beat that will not settle. Two members of one family too close to separate do not sound like a clean slow beat; they sound like a beat that keeps changing its mind, and this is the arithmetic of why.

A second bank in a collection that already had one #

This site has been using a filterbank on the waveform since its first phase, and putting a second one on the envelope is worth setting beside it, because the two are the same idea applied twice and the collection has never said so.

The cochlear bank sorts the components of a sound by frequency, and its channels are the critical bands — a few hundred hertz wide in the middle of the range, with the widths measured a dozen different ways over eighty years. It is what decides whether two partials are two.

The modulation bank sorts the envelope of what comes out of one cochlear channel, by fluctuation rate, into channels of quality near one. It is what decides whether two beats are two.

They are stacked: the second operates on the output of the first, so a member of a beat family is picked out by a cochlear channel and then by a modulation channel, and it survives only if both admit it. The eleventh rung computed the first stage and used a rule of thumb for the second; this one supplies the second.

The two banks have very different resolutions and that is the interesting part. A cochlear channel’s quality factor is around eight in the middle of the range — an eighth of an octave — and a modulation channel’s is one. So the ear resolves frequencies about eight times more finely than it resolves rates, in proportional terms, and a great deal of what this anchor has been counting sits in the gap between them.

What the bank does not resolve, and what that costs #

The rule cuts both ways and the anchor has only used one side of it.

Where two members are closer than 1.618 they merge, and the count falls — which is what the paragraph above describes and what the criterion was for. Where they are further apart than the bank’s own range, something else happens that no criterion here covers: the bank is drawn from a quarter of a hertz to sixty-four, and a member outside that is not analysed by it at all.

A mistuned octave’s eighth member at middle C beats at 122 a second, which is not a fluctuation anybody counts — it is a roughness, and the anchor’s own upper limit of fifteen hertz was put there for exactly that reason. So the bank’s range and the anchor’s rate window are two statements of the same boundary made for different reasons, and they agree to within a factor of four.

That agreement is not a check on anything, because both were set by hand. What it does say is that no member of any family in the middle of the keyboard falls in the awkward zone between the two — every member is either well inside the beat range or well into the roughness range, and none is on the boundary.

What the same rule says about a mistuned unison #

The octave is one interval and the anchor’s ninth rung established the general statement: any coincidence between two spectra generates a family indexed by the multiplier, with the member count equal to the partial count divided by the larger term of the ratio.

Applying the bank’s separation across those families sorts them in a way the rate rule alone did not. A mistuned unison has its members at 1, 2, 3, … times the base rate — consecutive integers — so its first two members are a factor of two apart, its second and third are 1.5 apart, and its third and fourth are 1.33. Only the first pair clears 1.618, and everything from the third member up merges into one broad fluctuation.

A mistuned fifth has members at 1, 2, 3 … of a base rate too, so the same arithmetic applies; what differs is that a fifth has fewer members, so a smaller share of them are in the merged region.

That gives the general rule the ninth rung was reaching for and could not state: a harmonic family of beats separates only at its bottom. However many members a family has, the ratio between neighbours falls as one over the member number, so members three and up are always inside the bank’s resolution and always merge. A family of eight sounds like two or three fluctuations at most, and that is a property of the integers rather than of the interval.

Which is a stronger version of the tenth rung’s headline and arrives at it from a different direction. That rung counted at most three beats and reached the number through the rate window and the position criterion; this says the merging alone would have capped it at two or three whatever those criteria did.

What the pictures cannot show #

The bank is a set of resonators and a real modulation filterbank is not measured as one. What the psychoacoustic literature reports is a masking pattern in the modulation domain — how much one modulation rate interferes with the detection of another — from which a channel shape is inferred, and different inference procedures give different shapes with the same quality factor. The three-quarters resolution floor is chosen here and it is what sets the separation: at a two-thirds floor the answer is 1.35 and at nine tenths it is 2.68, which straddles the stand-in.

So the honest version of this rung’s result is narrower than its headline. The separation is somewhere between 1.2 and 2.7 depending on two conventions neither of which is measured, the counts are the same across that whole range, and the specific answer to the ninth rung’s question — that 1.26 and 0.89 are one fluctuation — holds for every floor below about 0.83.

There is also a step from a filterbank to a count that the psychoacoustic literature does not license and this rung takes anyway. A bank is a set of channels; a count is a report of how many things a listener says there were. Between them sits a decision rule, and the three-quarters floor is that rule written as a threshold on one channel’s output. A listener who integrates across channels — comparing the pattern of outputs rather than reading one — could in principle separate two rates that no single channel isolates, exactly as a listener can hear out a partial from a chord in ways a single cochlear channel does not explain. Nothing here rules that out and nothing in this collection could.

Nothing here models attention. A listener told which beat to attend to may separate two rates a bank would merge, and the whole psychoacoustic literature on modulation masking is about detection rather than about counting. A tuner is doing neither: they are watching one fluctuation and waiting for it to stop.

One further limit belongs to the drawing rather than to the model. The bank is drawn every half octave for legibility and the separation is computed on a bank twelve times finer, because a coarse bank has gaps in it that a component can fall into and be attributed to nobody. A real bank’s spacing is not known either, and if the channels are sparse rather than dense then two components can be separated by less than a channel width and still land in different channels — which would make the separation smaller again, and in the same direction as everything else here.

And the family is computed on a piano string’s inharmonicity at one register with a spectrum falling as one over the partial number. A harpsichord, a guitar or a voice would put the same members at different rates, and where two of them land inside 1.618 of each other is a property of the spectrum rather than of the interval.

Two banks, two conventions, and one thing that is measured #

It is worth separating what is known from what is chosen here, because this rung’s whole result rests on the split.

Measured: that the envelope is analysed into channels at all, and that their quality factor is of order one. Both come from modulation-masking experiments, both are reported by several laboratories, and neither is in dispute about its order of magnitude.

Chosen here: that the channel is a simple resonator, that the resolution criterion is a three-quarters share, and that the bank is densely spaced. Each of those moves the separation, and between them they cover the range 1.2 to 2.7.

Not needed at all: any of it, for the result that matters. The recomputed counts are identical to the anchor’s published ones for every value in that whole range, because the surviving members are four times apart. So the rung’s headline — that the debt is paid and nothing moves — is robust to every convention in it, and the specific number 1.618 is the least secure thing on the page and the least load-bearing.

That ordering is worth stating because it is unusual. Most of this collection’s results get less secure as they get more specific; this one gets more secure, because what is being claimed is a null.

Whose practice this bears on #

The practice is piano tuning, where a mistuned octave’s beats are counted deliberately and where the count is the measurement. What this rung adds to it is small and specific: the anchor’s published counts are lower bounds, and the two slowest members of a mistuned octave in the middle of the keyboard are one thing rather than two.

The second of those is checkable by anybody with an instrument. Mistune an octave at middle C by a few cents and the prediction is a slow swelling with a period of about a second whose regularity wanders over three — not two independent beats that could be counted separately. A tuner who reports two clean rates there would falsify the bank at Q = 1, and would do it in a minute.

Where this ladder goes next #

Twelve rungs, and this one paid a debt without moving a number. The other debt the eleventh rung recorded is still standing and it will not be so quiet.

The auditory filter’s sharpness is not fixed: its lower skirt shallows as the level rises, by a published coefficient this collection did not carry. Every share on this anchor is quoted for a moderately loud note, and a tuner does not strike moderately — so the pedestal each member sits on grows with the strike, and the count that follows from it is not a property of the interval at all. The number of beats a mistuned octave delivers would then depend on how hard the note was hit, which is not a refinement of the eleventh rung’s result but a different kind of claim about it.