The notes in between

The fourth rung of this ladder established that an interval is two pitch judgements and is therefore heard less finely than either of its notes — two independent errors combine as the root of the sum of squares, so an interval’s limen is about 1.41 times a single note’s.

The fifth noticed that the independence in that model is an assumption rather than a measurement. If the two notes are referred to a common anchor — a key, a drone, a remembered tonic — then their errors share a component, and the interval is heard better than the root-two model says. It computed how large that shared component would have to be to matter, and left the question of whether it exists.

Its own last paragraph named what neither rung had touched:

Every figure until now is about two notes with nothing between them, and a melody is notes with other notes between them — so the shared reference is not held across a gap but across material, and the material is itself pitched.

Two literatures that predict opposite signs #

The situation the figure is about — a melody in which two notes forming a significant interval are separated by several others — is one both of the relevant literatures are about, and they say different things.

Confirmation. A note in the key restates the tonal frame. This site has the probe-tone profiles that measure how strongly a listener holds a tonal hierarchy and how quickly a passage establishes one, and the whole of that apparatus says that intervening notes in the key strengthen the sense of where the tonic is. If the anchor for both pitch judgements is that tonic, then more material means a better anchor and a smaller interval limen.

Overwriting. A note is itself a pitch to be held. The memory that holds the first note of an interval while the second arrives is limited, it is the same memory the intervening notes are passing through, and there is a large literature on how a to-be-remembered pitch is degraded by intervening tones. If the shared component is a memory rather than a frame, then more material means a worse anchor and a larger limen.

Both accounts are about the same experimental situation and they differ in the sign of the effect. Neither is measured on this site, and both are one-parameter models that the same arithmetic prices.

What each account costs and what it buys #

The two are modelled with one parameter each and the parameters do the same job in opposite directions.

Confirming: the correlation ρ between the two notes’ errors closes a fixed fraction of its distance to a ceiling with each intervening note. Starting at 0.5 with a refresh of 0.22 and a ceiling of 0.85, eight notes take it to 0.80.

Overwriting: the correlation is multiplied by a fixed factor with each intervening note. Starting at the same 0.5 with a decay of 0.72, eight notes take it to 0.036 — essentially nothing, which is the independent case the fourth rung drew.

Put through the interval limen — where the variance of the difference is the sum of the two variances minus twice their covariance — those give limens of 4.8 and 9.3 cents on a fifth at 440 hertz for a quarter-second note.

4.5 cents apart, which is 47 per cent of the no-anchor limen. That is not a subtle discrimination between two theories; it is a factor of nearly two in a quantity this site quotes constantly.

Both accounts have a free constant in them, so the next question is whether that gap is a finding or a parameter choice. Sweeping the refresh over a fourfold range and the decay over nearly a twofold one, and reading the gap as a fraction of the independent limen:

The gap never falls below a third of the independent limen anywhere in that region, and never rises above 0.60. More usefully, the two parameters push it the same way — a faster refresh and a slower decay both widen it — so there is no combination in the plausible range at which the accounts converge. The size of the disagreement is a structural consequence of their having opposite signs, and the constants only decide whether it is a third of the limen or three fifths. (The reconstruction here gives 0.54 at the essay’s own values where the figure reports 0.47; the difference is in the single-note limen the two are scaled by, and it does not move the band.)

The sweep over length is the one worth carrying into an experiment, because it says how long the stimulus needs to be:

Two thirds of the available separation arrives by the fourth intervening note and the last eight notes are worth a twentieth of the limen. So the experiment is a short one: four notes between the two, not a phrase, and lengthening the material past eight buys nothing measurable while costing everything a long stimulus costs in attention and in the chance that the listener has re-anchored on something else.

Why the size is worth knowing even without the sign #

An honest reading of the figures is that the model does not know which account is right. What it does say is how much the answer matters, and that is a useful thing to establish before anybody runs an experiment.

Four and a half cents is:

• larger than the syntonic comma’s audibility margin in most of the contexts the comma ladder works in;
• about a fifth of a syntonic comma, so it is the difference between a tuning discrepancy being detectable in a melodic context and not;
• larger than the difference between several historical temperaments at the intervals they differ on.

Which means that any claim on this site of the form “this discrepancy is below the limen and therefore inaudible” has an unstated dependence on how much material intervenes. A comma that is inaudible between two notes played back to back may be audible when the two are separated by a phrase that establishes the key — or less audible, if the memory account is right.

Nothing on this site currently carries that dependence, and every resolution number quoted in the tuning ladders is the laboratory value for two adjacent notes.

The length sweep below bounds how bad that omission is, and the bound is reassuring in one direction only. Between two adjacent notes the two accounts differ by 0.15 of the independent limen — a cent or so — so every claim about two notes played back to back is safe under either account. What is not safe is any claim about a discrepancy heard across a phrase, and those are the claims a tuning system is actually about: nobody meets a comma between two adjacent notes, they meet it between a chord at the start of a progression and the chord it returns to.

The one place the site can check its own reasoning #

There is a self-consistency test available without any listeners, and it is worth running because it constrains the plausible range of the parameters.

The confirming account says an established key improves interval judgements. This site has an independent measurement of how long a key takes to establish: what a tonic costs in seconds put it at a few notes for an unambiguous set and rather more for an ambiguous one. So the refresh rate in the confirming model is not free — it has to be roughly the rate at which a tonal hierarchy is built, because it is the same process.

Reading the constraint back through the model gives a usable range. A refresh of 0.22 closes 63 per cent of the distance to its ceiling in four notes and 86 in eight, which is a key established over about half a phrase; a refresh of 0.40 closes 87 per cent in four, which is a key established almost at once. Both are inside what the key-finding rungs report for an unambiguous set, and both sit inside the band the sweep above tolerates. So the confirming account’s one free constant is constrained to within about a factor of two by a measurement made for another purpose, and the overwriting account’s is not constrained by anything on this site at all — which is an asymmetry in how testable the two are, and it is not in either literature’s favour.

Running the numbers the other way: a refresh of 0.22 per note takes a moderate anchor most of the way to its ceiling in about eight notes, which is the same order as the key-establishment measurement. So the confirming parameter is at least consistent with something this site measured for another purpose, which is weak evidence and is more than the overwriting parameter has.

The overwriting parameter has no such anchor. The pitch-memory literature reports interference in terms of seconds and numbers of interfering tones with results that depend heavily on whether the interfering tones are musical, and nothing on this site measures it. So the two curves in the hero figure are not equally well founded: one is pinned to an existing measurement at one end and the other is a shape with a plausible rate.

What would settle it, and why it is one experiment #

The two accounts differ in sign, which makes the experiment unusually clean: it does not need a precise measurement, it needs a direction.

Play a listener two notes a fifth apart, separated by n intervening notes, and ask whether the fifth is in tune. Do it at n = 0 and n = 8. If accuracy improves with n, the frame account wins; if it degrades, the memory account does. The predicted effect is a factor of two in the threshold, which is very large by the standards of this kind of measurement.

And the experiment has a second condition that separates the mechanisms rather than merely the signs: run it with intervening notes in the key and with intervening notes out of it. The frame account predicts that in-key material helps and out-of-key material does not; the memory account predicts that both hurt about equally, because a pitch to be held is degraded by any intervening pitch. That crossing is what would distinguish a claim about tonality from a claim about memory.

Which computation produced the numbers #

The interval limen is the fourth rung’s: the two notes’ own limens combined as √(σ₁² + σ₂² − 2ρσ₁σ₂), with each note’s limen from the duration-dependent model — the larger of the steady-state frequency limen and what a quarter-second note’s Fourier resolution allows.

The two trajectories for ρ are one-parameter and both are asserted. The refresh of 0.22 and the ceiling of 0.85 are chosen so that eight in-key notes take a moderate anchor most of the way to a strong one, which is roughly what the probe-tone literature’s establishment times suggest; the decay of 0.72 is chosen so that eight notes take a moderate anchor to nothing, which is roughly what the pitch-memory literature’s interference effects suggest.

Neither number is measured and both are stated as choices. What is robust is the shape: any monotone approach to a ceiling and any monotone decay produce two curves that separate, and the separation at eight notes is between a third and two thirds of the no-anchor limen for any parameters that reach their asymptotes over a musical phrase.

The starting correlation of 0.5 is the fifth rung’s own reference value, chosen there because it is the point at which the shared component is worth about a quarter of the interval’s variance.

Where the model stops #

Both accounts are one parameter and neither is a mechanism. A frame that strengthens and a memory that fades are descriptions, not models; what the underlying process is — how a tonal hierarchy is updated, what a pitch trace decays into — is a substantial psychological question this collection is not equipped to pose.

The intervening notes have no pitches. They are counted and nothing else. Both accounts obviously depend on which notes: a repeated tonic and a chromatic scale are different intervening material, and the frame account is entirely about that difference. Modelling it would need the intervening notes’ own contributions to the tonal hierarchy, which this site has from the probe-tone profiles, and would turn one parameter into a computation.

And time and count are conflated. Eight notes at a fast tempo and eight at a slow one are the same n here and are very different amounts of elapsed time. The memory account is presumably about time; the frame account is presumably about events. That is a third thing the experiment above would separate and a fourth condition it would need.

Whose music, and when #

The situation is a general one and the stakes are not evenly distributed across repertoires.

Where it matters most is music with a strong, continuously restated tonal centre and long melodic spans — a drone tradition, most obviously. A raga performed over a tanpura has an anchor that is literally sounding throughout, which is the confirming account taken to its limit and which would make every interval judgement in that music a two-note judgement against a present reference rather than a remembered one. The tetrachord and drone traditions are built that way and this arithmetic says the choice buys real resolution.

Where it matters least is music that changes key frequently and moves fast — where there is no stable anchor to refresh and the memory account’s decay has nothing to work against anyway.

And where the two accounts would disagree most audibly is the middle: a long classical melody in one key, with a structurally important interval spanning several bars. If the frame account is right, a listener hears that span more accurately than they hear two adjacent notes, which is a surprising claim and is exactly what a strongly established key would buy.

What the picture cannot show #

Whether a listener is judging an interval at all. The whole ladder assumes the perceptual object is the distance between two pitches. A listener following a melody may instead be judging each note against the key — a scale degree rather than an interval — in which case there is one judgement rather than two and the entire root-two apparatus is about the wrong thing. A degree is where it goes next is the essay that takes that seriously.

Nor does either account have a place for what the intervening notes are as music. A phrase that ends on the dominant and a phrase that ends on the tonic leave a listener in different states, and the frame account is entirely about that state — so the number of notes is the wrong variable and where the phrase went is the right one. Modelling that would need the boundary detector this collection already carries, applied to the material between the two notes rather than to the tune as a whole.

And it cannot show the melody. Every figure here counts intervening notes as a number. What a real melody does between two structural pitches is contour, direction, rhythm and phrase, and any of those might carry the anchor better than a count of events suggests.

Where this ladder goes next #

Six rungs. How finely two pitches can be told apart; that the octave is not where it should be; that the whole family is a function of note length; that an interval is two errors; how much a shared anchor would have to be worth; and now what happens to that anchor when there is music between the two notes.

What the ladder owes is the thing the last section names. Every rung of it treats the object being judged as an interval — a distance between two pitches — and there is a rival account in which a listener judges each note against a key and never forms the distance at all. Those two predict different things about which errors correlate: under the interval account the two notes’ errors are correlated through a shared anchor, and under the degree account there is no pair at all and the question dissolves. The whole of this ladder’s fifth and sixth rungs are about a quantity the rival account says does not exist, and nothing here has ever tested which framing a listener is using.