A golden section is a coin toss with six coins

Three essays here have established that a listener’s grip on a proportion is loose, and each has stopped short of the obvious application. The analytical literature is full of claims that a piece’s climax, or its recapitulation, or its loudest bar falls at the golden section of its length. Those claims are made about scores and defended with arithmetic to three figures, and the essay that measured a form by what a listener stores has already found that a form’s proportions are not one quantity even on the page.

This essay asks what such a claim is being tested against, and the answer needs two things neither of which is usually stated: how wide a tolerance the claim is allowed, and how many chances it had.

The tolerance is not a free parameter #

An analysis reporting a boundary at 0.618 of a piece has to allow some margin, because no boundary lands exactly. The usual margin is implicit and generous: a bar or two either way, or a rounding to two decimal places.

The honest margin is not a choice. A claim that a climax falls at the golden section rather than at the middle is a claim that a listener could tell the two apart, and how finely a listener can tell two proportions apart is the essay that priced a proportion as a ratio of two estimates. A claim finer than that is a claim about the score and not about the music.

So the tolerance is one criterion’s worth of ratio spread at the Weber fraction of the judgement — which over a stretch of minutes is thirty-seven and a half per cent, and which converts back into a band on where the boundary falls.

The band runs from 0.492 to 0.730. It is nearly a quarter of the piece wide, and it contains 0.5.

A listener cannot tell a golden section from a half #

That is the finding and it deserves stating flatly, because the whole apparatus of proportional analysis rests on its being false.

A boundary at the golden section divides a piece 0.618 to 0.382, a ratio of 1.618. A boundary at the middle divides it 0.5 to 0.5, a ratio of 1.000. Over a stretch of minutes, judged afterwards, those two ratios are inside one criterion of each other — a listener comparing them directly is at about chance.

The named proportions inside the band are 1 : 1, 4 : 3, 3 : 2 and the golden section itself. Four of the six ratios the essays here carry are one judgement.

Two things could rescue a golden-section claim from that and both are worth taking seriously. A listener counting the sections rather than timing them has three or four times the resolution, which the previous two essays priced — and at the phrase level it is enough. And a claim about the score rather than about a listener is not touched by any of this.

What the band rules out is the claim as it is usually made: about a movement, about what a listener experiences, at a resolution of three decimal places.

Why the band is the same for every piece #

One feature of the band is odd enough to be worth an explanation, because it looks like an error and is not.

The band runs from 0.492 to 0.730 for a piece of eight minutes, and it runs from 0.492 to 0.730 for a piece of one minute and for a piece of twenty. Its width does not depend on how long the piece is at all.

That follows from the scalar property this whole account rests on. A duration estimate’s spread is proportional to the duration, so the relative blur is one number whatever the absolute length, and a proportion is a ratio of two durations and therefore a ratio of two relative quantities. Doubling the piece doubles both parts and leaves the ratio’s blur exactly where it was.

The only thing that moves the band is which timing band the durations fall in, and everything over half a minute is in the same one. So a symphony movement and a song get the same tolerance, and the reason a short phrase gets a narrower one is not that it is short but that it has crossed into a finer regime of judgement.

That is convenient for the arithmetic and inconvenient for a familiar defence of proportional analysis, which is that a long piece gives a listener more to work with. It gives them more seconds and no more resolution.

How many chances the claim had #

The second half is worse and it is the half that is pure arithmetic.

A piece does not have one boundary. A movement has a recapitulation, a development’s start, a second subject, a coda, a dynamic climax, a textural climax and a final cadence — all of them defensible places to measure to, all of them reported in the literature at one time or another. An analysis that reports the one nearest to 0.618 has run a search, and the rate at which a search of $k$ candidates finds a hit within tolerance $\varepsilon$ under no hypothesis at all is

$$P = 1 - (1 - 2\varepsilon)^k$$

At a tolerance of one per cent of the piece — a bar or two in a movement, which is about as tight as anyone claims — one boundary hits 2 per cent of the time, four hit 8 per cent, and twenty hit 33. At two per cent, twenty boundaries hit 56 per cent of the time.

At the tolerance a listener’s own timing gives, the numbers stop being marginal. One boundary hits 24 per cent of the time. Three hit 56 per cent. Six hit 80. Twelve hit 96.

An analysis with six candidate boundaries, tested at the resolution a listener actually has, finds a golden section in four pieces out of five that have nothing of the kind in them.

What this does and does not say #

It does not say that no composer used a proportion. Several said in writing that they did, and a score is a score whether or not anybody hears it.

It does not say that every published golden-section claim is chance. A claim tested at one pre-specified boundary, at a tight tolerance, on a piece chosen in advance, carries real evidence — 2 per cent is a respectable rate to beat.

What it says is narrower and harder to get around. The same point in the other direction is the bar above the bar: a structure a listener genuinely induces is one a piece has to supply evidence for, and a proportion supplies none. The chance rate is a property of the procedure rather than of the piece, and a claim that does not state its $k$ and its $\varepsilon$ cannot be assigned one. Most such claims state neither, and the reason to think $k$ is large in practice is that the boundaries reported vary from analysis to analysis: a piece whose climax is at 0.618 in one reading has its recapitulation at 0.618 in another.

A listener would need a Weber fraction below 5.4 per cent to tell the golden section from 3 : 2, and below about 12 per cent to tell it from 1 : 1. Nothing anybody does over minutes is close to either.

The count is the one way out, and it is narrow #

The previous two essays supply the only route that rescues any of this, and it is worth pricing rather than gesturing at.

A listener counting the sections rather than timing them has between three and eight distinguishable proportions between 1 : 1 and 3 : 1 instead of two, depending on how many units the sections run to. At the best length — around four units each — the resolution is enough to separate the golden section from 3 : 2, which timing never is at any length.

But that route comes with three conditions and a form rarely meets all three. The metre has to hold across both sections, which rules out any proportion measured across a change of tempo. The sections have to be short enough for the count to survive, which the previous essay put at under about a hundred units. And the listener has to be counting, which is free but not automatic.

So a golden-section claim is testable exactly where a form is metrically regular and its sections are a few phrases long. That is a real category of music — a dance movement, a strophic song, a minuet and trio — and it is not the category proportional analysis usually works on, which is large through-composed movements in several tempi.

What would make a golden-section claim testable #

The arithmetic is deflationary and it is also constructive, because it says exactly what a claim would have to look like to carry weight.

Pre-specify the boundary. One kind of division, named before the piece is measured, applied to every piece in a corpus. That sets $k = 1$ and makes the chance rate 2 per cent at a tight tolerance.

State the tolerance and justify it. If the claim is about a listener, the tolerance is the listener’s blur and the test is nearly hopeless at the movement level. If it is about the score, the tolerance can be a bar — and then the claim is about composition rather than perception, which is a different and perfectly respectable thing to be about.

Use a corpus and count the misses. A proportion that holds in eleven of fifteen movements is evidence; a proportion that holds in one movement is the first term of an unreported search.

And say which measure the claim is in. A proportion in bars, a proportion in seconds and a proportion in what a listener has to store are three different numbers about one passage, and a return is shorter than its first hearing showed they can disagree by a factor of three. An analysis that measures bars and reports a perceptual claim has changed measures without saying so.

And move down the form. The essay that ran the blur down a hierarchy found the resolution climbing as the durations shorten, and a golden-section claim about a fifteen-second phrase is a claim with a much smaller tolerance and therefore a much smaller chance rate. That is where the evidence is, and it is not where the literature looks.

What replaces the claim #

Deflating a claim is only half useful if nothing takes its place, and the arithmetic here points at a replacement rather than at a gap.

The quantity a listener does have over minutes is coarse and it is not nothing: two distinguishable categories between 1 : 1 and 3 : 1 means a listener can tell a movement whose halves are about equal from one whose first part is roughly twice the second. That is a real perceptual fact about form and it has never been the thing analyses measure, because two categories look too crude to write about.

It is not crude if it is what there is. A listener leaving a concert can say whether the slow movement felt front-heavy, and the arithmetic says that judgement is available and reliable while the ratio 1.618 is not. An account of proportion built on the coarse judgement would make predictions — which movements feel balanced, which feel top-heavy, how a performance’s rubato moves a piece between the two categories — and every one of those is testable on listeners in a way that a three-decimal ratio is not.

The same is true one level down. At five to twelve distinguishable proportions the phrase level supports a real vocabulary of relations, and a composer working there is working where the resolution is. The interesting claim about proportion is not about the number but about the scale, and the account here has now computed the scale four times from four directions without once setting out to.

Which computation produced the numbers #

The tolerance is derived rather than chosen. A boundary at share $s$ divides a piece in the ratio $s/(1-s)$; the golden section’s share is $1/\varphi = 0.618$, a ratio of 1.618. One criterion’s worth of ratio spread at a Weber fraction $w$ multiplies and divides that ratio by $\exp(d' \sqrt{2}\,\sqrt{\ln(1+w^2)})$, and converting the two bounds back to shares gives the band. The half-width of that band is the $\varepsilon$ used in the chance rate.

The chance rate treats the $k$ candidate boundaries as independent and uniform over the piece, which is the null a proportional claim is implicitly tested against and is generous to the claim in one respect and harsh in another — real boundaries are neither independent nor uniform, and they cluster in the second half of a piece, which is where the golden section is.

The Weber fractions are the three bands at their midpoints, and the band a span falls in is decided by its length: everything over thirty seconds is judged as a stretch of minutes, which is why the band on the share is the same for a one-minute piece and a twenty-minute one.

Where the model stops #

Boundaries are not uniform and the direction of that error is not obvious. If defensible boundaries cluster near 0.618 for musical reasons — because recapitulations tend to fall somewhere past the middle — then the chance rate above understates the problem, since the search is being run in a region where hits are dense. If they cluster elsewhere it overstates it.

The criterion is one d′. At d′ = 1 a listener is about 76 per cent correct in a direct comparison, which is a low bar to clear and a generous definition of “can tell apart”. At d′ = 2 the band is twice as wide and the rates worse still.

And the test is one-sided. Nothing here asks how often an analysis fails to find a golden section and does not publish, which is the other half of a search and is unobservable.

What the picture cannot show #

It cannot show a performance. Everything here is measured off a score at a nominal tempo, and a performance moves every boundary — a rubato of a few per cent at a structural moment is ordinary, and a few per cent of an eight-minute movement is fifteen seconds. Whether that helps or hurts a proportional claim depends on whether performers pull toward the proportion or away from it, and one of these eight-bar phrases accelerates is the essay that shows a performance can move a structural boundary on purpose.

It cannot show a listener with the score. An analyst measuring a piece has the score, a ruler and no time pressure, and none of the blur measured here applies to them. The band is about hearing, and a great deal of analysis is not a claim about hearing.

Nor a composer’s intention. A proportion put there deliberately is there. What this essay prices is the inference in the other direction — from a measured proportion to a claim that it was put there — and that inference is the one the arithmetic damages.

And it cannot show the other proportions. Everything here is about the golden section because that is the claim the literature makes most. A claim about 3 : 2 or 2 : 1 has exactly the same chance arithmetic with the same band around a different centre, and 2 : 1 at least has the advantage of sitting outside the band the other four share.

Where this stands after five essays #

Five essays, one object, and the object has turned out to be smaller than it looked. A proportion is a ratio of two durations a listener has only as estimates; a listener has three routes to those estimates and each is best at a different scale; the finest of the three is available only where a metre holds; and at the level analyses work at, the whole apparatus of named proportions collapses into two categories.

The account has not shown that composers do not use proportions, and it cannot. What it has done is price the inference that runs the other way, and the price is high enough that the interesting question moves. It is no longer whether a piece has a golden section. It is at what scale a proportion is audible at all — which is a question about listening, has an answer in seconds, and is the one the first four essays were computing without saying so.