Somebody has to pay the comma

Every essay in this ladder so far has been about an instrument with a fixed set of pitches. That is not an accident of interest — it is where the problem lives, because a fixed set of pitches has to be decided in advance and the deciding is what a temperament is.

The obvious response is that singers do not have this problem. A choir can tune each chord to whatever the chord wants, adjusting continuously and by ear, and never commit to anything. That is the freedom a fretted instrument gives up in wood, and it is usually described as the reason singers are exempt. The site’s own progression essay has already shown what happens when it does: the pitch falls. What has not been shown is that the falling is the same budget as a keyboard’s, spent differently, and that the total is identical.

The progression, and where the comma enters it #

The pump is I – vi – ii – V – I, and it is one of the most ordinary progressions in tonal music. Take each root motion at its pure ratio and follow the arithmetic.

C to A is a pure minor third down, 6:5. A to D is a pure fifth down, 3:2. D to G is another. G to C is another. Four moves, four exact whole-number ratios, and the C that arrives is 80:81 of the C that left — 21.51 cents flat, which is the syntonic comma.

Nothing has gone wrong. Every interval sung was exactly right and the destination is exactly wrong, which is the shape of the problem in every rung of this ladder and is why it is arithmetic rather than a matter of care.

The budget, stated exactly #

To come home, the four moves have to add up to a return. They do not, by 21.51 cents. So the amounts by which the four intervals are deliberately detuned have to sum to 21.51 cents, or the pitch falls by whatever is left over: drift plus total detuning equals 21.51 cents.

That is the whole constraint, and it holds however the detuning is distributed and whatever the ensemble is. It is the continuous-pitch version of the invariant a keyboard obeys: a temperament cannot reduce the total error either, it can only decide which keys carry it. An ensemble cannot reduce the total either. It can only decide whether to carry it in pitch or in intervals.

The freedom that a choir has and a keyboard does not is therefore not a freedom from the comma. It is a freedom to choose again — to spend the comma one way in this progression and another way in the next one, which no instrument with fixed pitches can do.

The three policies, and what each one costs #

Sing everything pure and let the pitch go. The intervals are perfect, and the piece falls 21.51 cents per circuit. Over four circuits that is 86 cents, which is most of a semitone; over eight it is a semitone and a half. This is the policy that produces the complaint about unaccompanied choirs going flat, and the complaint is arithmetically justified even when nobody has sung a wrong note.

It is worth noticing that the complaint is always about going flat, and that the arithmetic does not require it. The census below contains a progression of the same length that pumps in the other direction by exactly the same amount, so an ensemble singing it purely goes sharp at a semitone every five circuits. That the received complaint is one-directional is therefore evidence about which progressions the repertoire it was formed on actually used, rather than about anything in the tuning.

Hold the pitch and share the comma out. Four moves, 21.51 cents, 5.38 cents each. That is a fifth narrowed by five and a third of a cent — about the size of the difference limen, so it is at the edge of being noticeable as an interval and completely invisible as a drift.

Anything in between. Half the comma absorbed and half left as drift gives 2.69 cents on each interval and 10.75 cents of fall per circuit, which is what most ensembles are actually doing when they are doing it well.

The number at the end of the line #

5.38 cents is 21.51 divided by four. It is also, exactly, the amount by which quarter-comma meantone narrows every fifth.

That is not a coincidence and it is not quite a surprise once it is written down, but it is worth stating because the two objects have nothing else in common. One is a choir adjusting by ear in the middle of a phrase; the other is a keyboard tuning laid on a harpsichord before the concert and specified in 1523. They arrive at the same number because both are dividing one syntonic comma over four steps of a chain of fifths, and because both have chosen to minimise the worst interval rather than the total.

An ensemble that spreads the comma evenly has reinvented a temperament. The difference is not the arithmetic. It is that the choir gets to un-invent it at the next progression, and the harpsichord does not.

That difference is real and it is what continuous pitch is worth. A progression that does not pump — one whose ratios return home on their own — costs a well-tuned ensemble nothing at all, while a keyboard pays its temperament’s error on every chord it ever plays, pumping or not. The choir pays only when the music asks it to.

Which raises the question of how often the music asks, and it is answerable. Running the same loop arithmetic over eleven ordinary progressions:

Five of the eleven pump and six close exactly, so the comma is a property of particular loops rather than of tonal harmony. And the two four-chord loops that dominate popular music sit on opposite sides of it: I–V–vi–IV pumps, upward, by a full comma a circuit, and I–vi–IV–V closes. Two progressions with the same four chords in a different order, one of which drives an unaccompanied ensemble sharp by a semitone over five repetitions and one of which does not.

The 5.38 cents is a property of the length as well. A five-chord pump shares the same comma over five moves at 4.30 cents each and a seven-chord one at 3.07, so the meeting with quarter-comma meantone is a meeting with four-move pumps and not a general identity. It is one comma over however many moves there are.

One caution, and it is the interesting one. Which loops pump depends on which just interval is chosen for each move, and for a whole tone the choice between 9:8 and 10:9 is itself a syntonic comma. Tuning the IV–V step of I–vi–IV–V as 10:9 rather than 9:8 turns that closing progression into a pumping one. So the table is a table under one convention, and the freedom this essay is about — the choir’s ability to choose again — is precisely the freedom to make that choice differently. A progression does not pump or not pump; an ensemble’s chain of decisions does.

How an ensemble knows where the pure interval is #

The frontier assumes an ensemble can put an interval exactly where it wants it, to within a cent or two. That is a strong assumption about a group of people with no instrument to measure against, and it is worth saying what the measuring device actually is.

It is beating. Two notes at a simple ratio share partials, and when the ratio is exact the shared partials coincide exactly and the pulsing stops. That is how every instrument in the world gets tuned, and it works for a singer for the same reason it works for a tuner: the feedback is a rate rather than a pitch, and a rate of zero is unmistakable.

The precision available this way is far finer than the quantities in this essay. A fifth above middle C mistuned by 5.4 cents beats at 2.4 times a second, which is slow, obvious and easy to remove. So an ensemble genuinely can sit anywhere on the frontier it likes; what it cannot do is get off the frontier.

And the same mechanism explains why the drift is invisible from inside. Every interval sounds right at every moment, because each one was tuned by removing a beat. There is no local evidence of the problem anywhere in the performance — the evidence exists only in a comparison between the beginning and the end, which is precisely the comparison a performer cannot make.

The beat rate also says how far an ensemble can be from the frontier without noticing, which is a different quantity from how far it can be from pure. A fifth 5.4 cents narrow beats at 2.4 a second; a fifth 1 cent narrow beats at 0.44, which is a cycle every two and a quarter seconds and is at the edge of what a held chord lasts long enough to reveal. So the practical resolution of the tuning device is about a cent on a sustained chord and much worse on a passing one, and an ensemble moving through a progression at two chords a second is tuning by a mechanism that has not had time to report. The frontier is available to a choir holding a chord and not to one singing a phrase, which is a limit on the whole argument rather than on any policy in it.

The keyboard’s version of the same decision #

Set the two families side by side and the symmetry is exact.

A fixed-pitch instrument pays the comma as a permanent, unchanging error in its intervals and never drifts. A continuous-pitch ensemble may pay it as drift, or as a temporary error in its intervals, or as any mixture; and it re-decides every few bars. Both pay it in full.

The asymmetry is entirely in when the decision is taken. A temperament is a decision made once, by somebody who is not present at the performance, about music they have not heard. An ensemble’s is a decision taken continuously by the people making the sound. That is the whole of what continuous pitch buys, and it is a great deal — but it is not exemption.

What a real ensemble does, as far as anyone can tell #

The claim that choirs drift is not folklore; it has been measured, and what the measurements show is more interesting than the folklore.

Unaccompanied ensembles do fall in pitch over long spans, and they fall by more than a comma per circuit would predict — which means drift is not only the comma. Fatigue, the tendency to sing slightly under a leading note, and the absence of a reference all contribute, and separating them requires a controlled task rather than a concert recording.

The nearer evidence for the specific mechanism in this essay is that ensembles trained to tune vertically — barbershop quartets are the standard example, with an explicit doctrine of locking chords by ear — drift more than ensembles that are not. That is the prediction the budget makes: the more exactly the chords are tuned, the more of the comma is left for the pitch to carry.

And accompanied ensembles do not drift at all, because the accompaniment is a reference and the singers absorb the whole comma into their intervals without deciding to. A choir with a piano is being held at the right-hand end of the frontier by an instrument that has no choice.

The sizes involved run from the schisma at 1.95 cents, which is below the limen everywhere, through the diaschisma and the syntonic comma at 21.5 to the Pythagorean at 23.5 — and the limen for successive tones sits near 5 cents in the middle of the range. So the quantity being distributed is four times what a listener can detect and the per-interval share, at a quarter of it, is just under.

Whose music, and when #

The pump is a claim about a repertoire, so the repertoire has to be named.

I – vi – ii – V – I is a commonplace of eighteenth- and nineteenth-century tonal harmony and of most twentieth-century popular music built on the same grammar. It is not a commonplace everywhere: music that does not move by root fifths, or that does not return to its opening chord, or that has a drone, does not pump. A raga with a fixed tonic and a sounding drone cannot drift, because the drone is the reference and the whole question is settled before any interval is sung.

So the problem this essay is about belongs to a specific practice: unaccompanied ensemble singing in a harmonic idiom that modulates by fifths and returns. Renaissance polyphony, barbershop, close-harmony ensemble singing, and a great deal of church music are inside it. Nearly everything else in the world is not.

And the arithmetic is older than the observation. The comma pump was described in the sixteenth century as a reason why pure intonation could not be maintained, at a time when the ensemble singing it applies to was the principal medium of the repertoire being written. It was an argument against just intonation before it was ever a measurement of a choir.

The third option nobody describes as one #

There is a policy the frontier does not have an axis for, and it is the one most often used: change the progression.

The pump needs four moves of a particular kind. Substitute a chord, or approach the return by a different route, and the loop closes on its own — the ratios multiply to two and there is nothing to pay. A great deal of practical advice about ensemble intonation is really advice of this kind, and so is a certain amount of harmony teaching: the recommendation to avoid a particular succession in unaccompanied writing is, underneath, a recommendation to avoid a loop whose ratios do not return.

It is worth noticing because it changes what the comma is a constraint on. It does not constrain how well an ensemble can sing; it constrains which progressions can be sung purely without moving, and a composer who knows that has a third place to put the error — in the choice of chords, before anybody sings anything.

Where the model stops #

The four-move loop is a simplification of what an ensemble does. Real voices tune to whichever note is most exposed, adjust within a chord as it sounds, and take their reference from a bass line that is itself moving. The budget is exact and its distribution over “four intervals” is a convenient fiction; what is not a fiction is that the total is one comma.

The frontier assumes the detuning is shared evenly. Nothing forces that. An ensemble might put the entire comma on one interval — the least exposed one, or the one in the least prominent voice — and keep three intervals pure. The total is the same and the worst interval is 21.5 cents rather than 5.4, which is a different and defensible policy if the exposed chords are the ones that stay pure.

And a comma is not the only reason pitch moves. Any measurement of a real performance is measuring drift from several causes at once, and this essay’s mechanism is the one that can be computed rather than the one that dominates.

What the picture cannot show #

It cannot show which note is bent. The frontier plots the size of the worst error, not its location. Whether the ensemble flattens the fifth of the ii chord or sharpens the third of the vi is invisible here and is most of what a director would actually be deciding.

And it cannot show the melody. Every number in this essay is about vertical tuning, and a singer is also making a horizontal line in which a leading note wants to be high and a descending semitone wants to be narrow. Those pull the other way from vertical purity, they are documented in the pedagogy of ensemble singing as explicit advice, and the budget in this essay says nothing about them at all.

Where the ladder goes next #

Every comma in this ladder so far has been a debt. The next rung is about the one that is not — a gap under two cents, below anything a listener can detect, which one tuning system spends deliberately because spending it costs nothing.