How hard the note was struck

The eleventh rung recorded two debts. The first was paid on the last page and moved nothing. The second is smaller in its statement and larger in its consequence.

Every share this anchor computes is a share of one auditory filter: how much of the fluctuation inside a cochlear channel belongs to the beat at that channel’s centre, as against the neighbouring partials that add level without adding fluctuation. The filter used is Patterson’s rounded exponential with its parameter tied to the equivalent rectangular bandwidth — and that bandwidth is a measurement made at a moderate level.

The published parameterisation is level-dependent. The filter’s lower skirt shallows as the level rises, by about 38 per cent of its 51-decibel value every ten decibels, while the upper skirt is very nearly fixed. So a loud note is analysed through a filter that admits more of what is below its centre, and the pedestal every member of a beat family sits on grows.

The size of it #

The mean share falls from 0.55 at forty decibels to 0.31 at sixty and 0.10 at seventy. The first member — the strongest and slowest — falls from 0.99 to 0.46 over the same range, and the fourth falls from 0.53 to 0.04.

That is a very large effect for one coefficient, and it is large because the shallowing is on the lower skirt specifically. A beat’s coincidence sits at a partial of the note, and the components that dilute it are the note’s other partials — which are distributed on both sides, with the dense low ones below. A filter that opens downward opens onto exactly the part of the spectrum that has the most in it.

So this is not a general blurring. It is a directional one, and the direction is the one that costs the most.

And what it does to the count #

Feeding those shares into the same criterion the last two rungs used gives a number that changes with the strike.

Two beats at forty and fifty-one decibels, one at sixty, and none at seventy or above.

That is the claim the eleventh rung said would follow, and it followed. It is also, taken at face value, false.

Why it cannot be right #

Tuners count beats on notes struck a good deal harder than seventy decibels. That is not a marginal counterexample or a matter of interpretation; it is the entire practice of laying a temperament, it has been done for three hundred years, and it is arithmetic that anybody can hear.

So the model over-predicts, and the interesting question is where.

The share criterion may be too severe. Requiring half of a filter’s fluctuation to belong to one member is a criterion this anchor chose and has swept once; a listener attending to a slow swelling in a sound that also has a faster one may not need the slow one to dominate. Lowering the floor does not rescue it — at a quarter the count is still zero by seventy decibels — so this is not the whole story.

The published coefficient may not extend this far. The 0.38-per-ten-decibels figure is fitted over a range of levels centred on fifty-one decibels, and the model here clamps the skirt at a fifth of its reference value because extrapolating the linear form past about eighty decibels makes it negative. A clamp is an admission that the parameterisation has been taken outside where it was measured, and everything past eighty on the drawing is the clamp rather than the model.

Or a tuner is not listening to the strike. This is the explanation a tuner would give, and it is the one the arithmetic supports best. A piano note’s level falls by twenty decibels within a second or two of the strike, and a tuner counts beats over several seconds — so what they are listening to is not the fortissimo transient but the decay, and the decay passes through exactly the levels at which the shares above are healthy.

So the level dependence is real and the practice already routes around it, by listening late rather than early. That is a satisfying resolution and it is one this rung would not have found without the over-prediction, which is the argument for computing a thing whose answer looks obvious.

What follows for a tuner rather than for the model #

Two consequences survive the correction, and both are checkable.

A beat is easier to count as the note decays. The share improves monotonically as the level falls, so the same mistuning is more legible a second after the strike than at the moment of it, and more legible again two seconds later — until the note gets too quiet for the beat’s depth to clear the detection threshold at all. There is therefore a window, and it is bounded at both ends by different mechanisms.

Striking harder does not help. A tuner wanting a longer look at a beat has a choice between striking harder, which extends the note’s life at the far end, and striking more gently, which puts it into the good range sooner. The arithmetic says the second is worth more than the first, because the share is a steep function of level and the note’s duration is not.

That is a testable difference between two habits, and both habits exist among tuners. It is also the kind of claim this collection can make and cannot settle: it needs somebody with an instrument, and the collection has none.

Which member loses most, and why it is not the one to worry about #

The five curves in the first drawing do not fall together, and the ordering is worth reading because it decides what the loss actually costs.

The first member holds up best — 0.99 at forty decibels and 0.46 at seventy — and the fourth collapses, from 0.53 to 0.04. The reason is where each coincidence sits. The first member’s is at the fundamental, where the neighbourhood is sparse: the next partial up is an octave away, far outside any filter at any level. The fourth member’s is at the fourth partial, where the neighbours are a quarter of an octave away on either side and a widening filter reaches them immediately.

So a widening filter takes the high members first, and the low ones survive. That is the good direction, because the low members are the slow ones and the slow ones are what a tuner counts. A count of two falling to one is the loss of the faster of two beats, which is the one that was harder to use anyway.

Read across the anchor, that says the level dependence erodes the count from the top and leaves the headline beat intact until late. A mistuned octave struck hard still has its slowest fluctuation; what it has lost is the structure above it.

What a loud chord does to everything else on this anchor #

The correction is not confined to tuning. Every share and every count this anchor has published sits at a moderate level, and music is played at many.

A fortissimo passage is analysed through wider filters than a pianissimo one, so an ensemble’s out-of-tuneness is less legible when it is loud. That runs against the intuition — a loud wrong note feels more wrong — and it runs with the practice, since intonation is rehearsed quietly and conductors ask for a passage softly when they want to hear whether it is in tune.

It also gives the previous ladder’s tuning ritual a level dependence it did not have. An orchestra matching by beats is using a criterion whose precision depends on how loudly the reference is played, and the direction is the one nobody would guess: a quietly given A is a better reference than a loud one.

The same coefficient, read as a statement about masking #

There is a second way to read the shallowing skirt and it connects this rung to a different anchor entirely.

A filter whose lower skirt shallows with level is the same object as upward spread of masking — the fact that a loud tone hides things above it far more than things below it, and hides them more the louder it is. The masking anchor states it as a property of a masking pattern and this rung states it as a property of a filter, and they are two descriptions of one measurement.

So the beat’s dilution and the chord’s self-masking are the same phenomenon, and this collection has computed them separately for two years without saying so. The connection is not a new result; what it does is put a constraint on both, because any change to the filter’s level dependence has to move both sets of numbers in step.

It also makes a prediction that neither anchor made alone. If the widening filter is what dilutes a beat, then a beat should be diluted most by partials just below its coincidence and hardly at all by partials just above it — which is a strong asymmetry, and one that could be tested by adding a tone on one side or the other and asking whether the beat becomes harder to count.

What the pictures cannot show #

The level dependence enters through one coefficient in one filter model, and the model is fitted to notched-noise masking data rather than to anything about beats. Whether a filter measured that way is the filter a listener uses to isolate a slow fluctuation is an assumption this anchor has been making since it started using filters at all; this rung only makes the assumption’s level dependence visible.

The clamp is stated in the caption and is worth stating twice. Past about eighty decibels the drawing is flat because the model refuses to make the skirt negative, not because anything about hearing stops changing. Nothing on this page should be read above eighty.

There is a fourth thing the drawing cannot separate and it matters for how the over-prediction should be read. The count that falls to zero is a count under three criteria at once — the share, the depth, and the separation — and the level moves only the first of them. So the collapse at seventy decibels is the share criterion failing, and the depth criterion, which is the one with a published threshold behind it, is untouched by any of this. A reader who wanted to keep the anchor’s counts and throw away one thing would throw away the share criterion, which is the anchor’s own invention, and would lose nothing that was measured.

And every component here is a steady tone. A struck string’s partials decay at different rates — the higher ones faster — so a decaying note’s spectrum is not the same spectrum at a lower level; it is a different spectrum, with less in the neighbourhood of every coincidence than the level alone would suggest. That would make the share improve faster than this drawing says as a note dies, and it is a second reason a tuner listens late.

What a harpsichord does, which is the control this rung wanted #

The whole argument rests on the neighbourhood of each coincidence being crowded, and there is an instrument where it is not.

A stopped organ pipe has no even partials at all, so a mistuned octave between two of them has coincidences with nothing on either side for an octave and a half. A widening filter reaches nothing, the pedestal does not grow, and the shares should be very nearly level-independent.

That is the control the model wants and cannot be given here, because the anchor’s spectrum is a power law by construction and does not carry a spectrum with holes in it. What can be said is the prediction: an organ’s beats should be as countable loud as quiet, and a piano’s should not, and the difference between the two instruments should be about the factor of five the mean share falls by between forty and seventy decibels.

There is a documentary hint in the right direction and it is only a hint. Organ tuning is described as being done at whatever registration is convenient, and piano tuning is described with careful attention to touch. That is weak evidence — an organ’s level is not the tuner’s to choose in the first place — and it is the kind of thing that would look like confirmation whether or not the mechanism were real.

Whose instrument, and the two rungs together #

The practice is piano tuning and the reading is about pianos, because that is where beats are counted deliberately and where the strike’s level is a choice the tuner makes.

Set beside the previous rung, the pair make a symmetrical point about the anchor’s two outstanding debts. One of them was paid and changed nothing, because the criterion it corrected was never binding. The other was paid and changed everything by too much, and what saved it was a fact about how the practice is actually carried out rather than a fact about hearing.

Both are the same lesson about a recorded debt, which this site has now met several times: a debt names a computation and does not name its answer, and the answer is as often a null or an over-shoot as it is the refinement the debt anticipated.

What the anchor’s numbers should be quoted as from here #

The practical consequence for this collection is a convention rather than a result, and it belongs at the end of a rung that found a parameter nobody had varied.

Every count on this anchor should carry a level. The eleventh rung’s grid is a grid at a moderate level; the tenth rung’s three beats are three beats at a moderate level; and the ninth rung’s family is a family whose shares would look different at any other. None of those pages says so, because until this one there was no reason to think it mattered.

A number that turns out to depend on a parameter nobody stated is a number that was quoted wrong, even where it is right — and this is the third time this collection has found one. A caption that names its own defaults is the fix, and it is cheap.

Where this ladder goes next #

Thirteen rungs. Beats are arithmetic anybody can hear; a tuner counts them; a cellist’s wolf is the same arithmetic coupled; a piano’s unison sits below a bifurcation; every partial beats at its own rate; a beat has a depth; the rate decides which intervals are usable; a single mistuning makes a whole family; how many arrive separately; that at most three do; that every member is the same depth; that the separation criterion was never binding; and now that the filter through which all of it is computed is a function of how hard the note was hit.

What the anchor owes now is the decay, and this rung has made it the obvious next thing rather than an afterthought. Every figure here is a steady tone at a stated level, and the resolution above is that a tuner listens to a note that is neither steady nor at one level — its partials fall at different rates, its overall level drops twenty decibels in a second or two, and the shares and depths this anchor computes are therefore a trajectory rather than a number. What would come out is the window: when after the strike a given mistuning becomes countable, and when it stops being. That is arithmetic on quantities this collection already holds — the decay rates are on the envelope anchor and the shares are here — and it would replace every number on this anchor with a pair of times.