A melody that can accompany itself

A round is a tune that works as its own accompaniment. Sing it, and while the first voice is on its third bar have a second voice start from the beginning; the two fit.

There are not very many of them. Nursery singing has a handful, catches and canons are a specialist genre, and the overwhelming majority of tunes cannot be treated this way at all — start a second voice two bars into most melodies and the result is a mess.

That looks like a fact discoverable only by trying, and it is not. It is decidable in advance, with a model this site has been using since its first phase.

The scan #

Take a tune encoded as a list of pitches, one per quaver. For each whole-bar delay, lay a copy of the tune against itself at that displacement, take every pair of notes sounding together, and score each pair with the Plomp–Levelt roughness model — the same one behind the dissonance curve and everything the site says about consonance.

Average the scores. Repeat at the next delay. The result is a curve of roughness against entry point, and the whole of it can be computed before anybody sings anything.

Two tunes, and they do not overlap #

The hero figure runs that scan on two tunes that every English-speaking reader knows, at every delay from one bar to seven.

Frère Jacques scores between 0.035 and 0.080. Twinkle, twinkle scores between 0.081 and 0.167.

The two ranges do not overlap. The worst delay of the first is smoother than the best delay of the second — narrowly, by less than two per cent at the boundary, but completely. One of these tunes is a round at any entry point anybody chooses, and the other is a round at none.

That margin is thin enough to be worth auditing, and the section that audits it finds the whole of it comes from one thing, which is not the thing this scan was built to measure. The conclusion survives; this particular measurement of it does not.

That is the result the essay exists for, and it is worth pausing on how little went into it. No rules of counterpoint, no notion of a key, no analysis of the melodies. Two lists of numbers, one interval-roughness model, and a loop over displacements.

What separates them #

The reason is visible once the two tunes are drawn.

So the criterion is a number: how much of the tune lies inside a single triad. Eighty-one per cent works; fifty-six does not.

The mechanism is not mysterious, and it is which notes are the chord asked backwards. If nearly every note of a tune belongs to one chord, then any two notes drawn from it — at any displacement — are two notes of that chord, which is a consonance by construction. The passing notes are brief, they land against chord tones, and the roughness they contribute is averaged away.

If a tune’s notes belong to different chords at different times, then a displacement puts notes of one chord against notes of another, and the result is whatever those two chords make together. Occasionally that is fine. Usually it is not, and it is certainly not reliable across a whole tune.

A stricter test the scan also passes #

The separation reported above is complete and narrow — 0.0800 against 0.0814 at the boundary — and a result that close deserves a second, independent check rather than a confidence interval it cannot have. It gets one below, and it does not survive it; what survives is the check.

The check is the coverage number, and it is not derived from the scan at all. It counts how many of a tune’s quaver slots hold a note belonging to its best-fitting triad: 81.3 per cent for the first tune, 56.3 for the second. That is a gap of twenty-five percentage points, computed from the same encodings by a completely different route, with no model of consonance in it anywhere.

Two measurements that share their input and share nothing else are the right kind of corroboration. The narrow one says the tunes separate; the wide one says why, and the wide one is the number a person could use to guess at a new tune without running anything.

It also suggests where the threshold is, though this site has two tunes and cannot locate it — and a tune whose notes sit inside one chord is a tune a walk rather than a set describes badly. Somewhere between 56 and 81 per cent is the coverage at which a tune stops being usable as a round, and finding it would take a few dozen tunes of known status rather than two. That is a corpus problem again, and the same one that stopped the harmonic-rhythm essay short.

The separation is repetition, not harmony #

Two hundredths is a thin margin to carry a result, and the obvious thing to ask of it is which pairs are producing it. Counting them: at every delay, what share of the sounding pairs are unisons or octaves — intervals with no roughness at all, which enter the average as zeros.

Frère Jacques carries between a sixth and two thirds of its pairs at nothing; Twinkle carries none at all on four of its seven delays. Recomputing both scans over the pairs that are not unisons or octaves — which is the roughness of the material that actually sounds as two voices — gives Frère Jacques 0.0714 to 0.0956 and Twinkle 0.0814 to 0.1668, and the two ranges overlap. The round’s worst delay is now rougher than the other tune’s best.

So the scan’s headline is an artefact, and it is the artefact the essay already identified at one delay without following it through the rest. Bars one and two of Frère Jacques are identical, and so are three and four, five and six, seven and eight — the tune is built in repeated pairs. Every delay therefore lays a bar against its own twin somewhere, and the zeros that produces are what drives the average down. The scan is measuring self-similarity and reporting it as consonance.

That leaves the essay’s conclusion standing on one leg rather than two, and it happens to be the leg that was described as the corroboration. The coverage number — 81.3 per cent against 56.3 — is computed from the notes with no model of consonance in it, is unaffected by any of this, and is the measurement that actually separates the tunes. The rule below is its rule, not the scan’s.

It also sharpens what the scan is good for. A roughness average over a self-canon cannot distinguish a tune that is harmonically inert from a tune that repeats itself, and the two are correlated in exactly the repertoire rounds come from — nursery tunes are both. Separating them would need the scan run on the pairs that are genuinely two voices, which is the column computed above, and on that column this collection’s two tunes do not separate at all.

The rule the tradition never states #

Put that as a rule and it explains the scarcity of rounds.

A round must have a harmonic rhythm slower than its own canonic delay, and the practical version of that is stronger: a round is essentially a tune over one chord. That is a severe restriction on melody — no modulation, no cadential progression, no departure and return of the kind the whole of tonal melody is built from — and it is why rounds are short, simple, and concentrated in children’s and drinking repertoire rather than in art music.

It also explains the exceptions. The canons that do appear in art music are rarely rounds in this sense; they are canons at a specified interval and delay, written into a texture with a bass line supporting them, and the composer has chosen the delay to make particular simultaneities work. That is a harder and more interesting problem than the one solved here, and it is not decidable by a scan, because the composer is free to alter the tune.

Harmonic rhythm is an independent variable with a floor and a ceiling, and a round is a piece of music sitting at the very bottom of its range — one chord for the whole tune, a rate of zero.

The scan refuses the traditional entry #

Here is the part that did not come out as expected, and it is the more useful half.

Frère Jacques is sung as a four-part round with entries two bars apart. That is how it is taught, how it is printed, and how it is sung everywhere.

Two bars is the scan’s worst delay for that tune. Its best is one bar, at 0.035, and two bars is 0.080 — the roughest of the seven, and more than twice the smoothest.

The scan is not wrong. At a one-bar delay the second voice is singing bar two against bar one, and bars one and two of Frère Jacques are identical, so half the piece is in unison. Unisons have no roughness at all, and the average falls accordingly. The scan is measuring what it says it measures, and by that measure the one-bar entry really is the smoothest.

What it is not measuring is whether the result is a round worth singing. A canon in which half the notes are in unison is a canon in which the second voice is inaudible as a second voice for half its length, and the point of a round is that the parts are distinguishable. The two-bar entry is chosen so that each voice enters with different material from the one before it — bar three against bar one, bar five against bar three — which is a requirement about texture and independence, and no roughness model has any access to it.

So the honest statement of what the scan establishes is narrower than the one it was written to test, and still worth having. It settles whether a tune can work at all, at any delay. It does not settle which delay to use, and the traditional answer to that question is decided by something the model cannot see.

Against Frère Jacques’s 81.3 per cent that is a large gap, and it is the whole of why one of these tunes rounds and the other does not. A round needs a tune that is mostly one chord, and “mostly” turns out to be a number a strip can count.

The other things a canon has to satisfy #

Roughness is one constraint of several, and the ones the scan cannot see are worth listing because they are what a composer of canons actually worries about.

Parallel perfect intervals. Two voices moving in parallel fifths or octaves lose their independence — two voices that stop being two is the site’s essay on exactly that, and it is a fusion phenomenon rather than a roughness one. A canon at the unison is at particular risk, because wherever the tune moves by the same interval twice in a row the two voices are moving in parallel by construction.

A bass. A round has no bass line: whichever voice happens to be lowest at a moment is the bass, and it changes. That is workable when everything is one chord and unworkable otherwise, which is the coverage criterion arriving from a third direction.

Where the entries land metrically. A canon at a delay of a bar and a half puts the second voice’s downbeats on the third beat of the first voice’s bars, which is a different piece from one where the downbeats coincide. Every delay in the scan above is a whole number of bars, so this variable was never explored.

Whether it ends. A round does not end; the voices drop out in turn, or everybody stops. That is the same absence a cyclic form has, and for the same reason — a piece that is designed to overlap with itself cannot have a moment that is the ending for all of its parts at once.

Where the scan is weak #

Averaging is the crude part of this, and the weakness is worth naming precisely because it is fixable and has not been fixed here.

A mean over all sounding pairs treats a single very rough simultaneity on a strong beat exactly as it treats one on a passing quaver. In real counterpoint those are enormously different: a dissonance approached and left by step on a weak beat is invisible, and the same dissonance held on a downbeat is a mistake. Every counterpoint treatise ever written is largely a set of rules about that distinction, and none of it is in the model.

Weighting the pairs by metric position would help and would introduce a free parameter, which this site’s practice is to avoid where a result survives without one. The result here does survive: the separation between the two tunes is complete under the unweighted mean, and weighting could only widen it, because the tune that fails does so on held notes at strong positions.

What the model counts when it calls a simultaneity smooth is how many partials the two notes share — and what makes this tune supply smooth simultaneities at this delay is that both voices are inside the same triad at almost every slot. The harmony is not composed; it is a consequence of the tune’s own chord content and the offset.

What this cannot show #

The encoding is two tunes written out by hand at quaver resolution, printed in full in the figures above, and everything here is a statement about those encodings. A reader who disagrees with a note can see exactly which note is being disagreed with, which is the only defence a hand encoding has.

Beyond that, the model sees interval roughness and nothing else. It has no notion of a key, so it cannot tell that a simultaneity is a chord of the key rather than an arbitrary pair; no notion of voice leading, so it cannot see parallel fifths or octaves, which are the first thing a musician would check; no notion of register, since the transposition is fixed at the unison here and a real canon might enter at the fifth.

And it is entirely deaf to line. Whether the second voice is followable — whether a listener can hear two parts rather than one thicker part — is a question about auditory scene analysis rather than about roughness, and it is the question the two-bar entry is actually answering.

A round is the smallest complete form #

It is worth ending the field where it started, because this is the simplest object it contains.

The first rung of this phase asked what a piece is made of and answered: mostly itself, again. A round is the extreme case of that answer. There is one tune. Everything a listener hears is that tune, displaced. The whole texture, the whole harmony and the whole form are consequences of one line and one number.

It scores far lower, and that is the answer to whether the tonic reading is an artefact of choosing the tonic. One chord fits and the others do not, so the round’s harmony is a fact about the tune rather than about which triad the analysis happened to try first.

That is the reason a round is worth a rung at the end of a field about form. It has no sections, no cadences, no key plan, no hypermetre and no density curve, and it is unmistakably a piece of music with a shape. Whatever a form minimally is, a round is at the bottom of it — and what it is made of is a melody, a delay, and the arithmetic that says the two can be put together.

What the scan would say about a new tune #

The procedure is worth stating as a procedure, because it is short and anybody can run it.

Encode the tune at whatever resolution its fastest notes need. Compute the fraction of it lying inside each triad of the key and take the largest — that is the coverage, and it is a single number obtainable by counting. If it is above about eighty per cent the tune will very probably work at any delay; if it is near sixty it very probably will not.

Then, if the answer matters, run the scan: every whole-bar delay, mean roughness of the sounding pairs, and compare the range with the range of a tune known to work. That is a few lines of arithmetic over a list of numbers and it needs no judgement anywhere in it.

What neither step gives is the delay to use. That is decided by wanting the voices to be distinguishable, which is a question about hearing rather than about intervals, and the tradition’s answer for the tune above — two bars, the roughest of the seven — is the right one for a reason no model here can supply.

Where the ladder goes #

This rung and the phase piece are two halves of one idea: a tune against a delayed copy of itself, with the delay fixed in one case and swept in the other. Between them they cover the small family of forms in which a single line is the whole material.

They also close this phase’s form field, which opened by asking what a piece is made of and has arrived at a case where the answer is one tune and one number.