The progression that never comes home

The usual story about tuning goes like this. Pure ratios sound best; a keyboard has twelve keys and cannot supply enough of them; therefore keyboards are tempered, and singers and string players, who have no frets and no keys, can use the pure ratios and do.

The second half of that is false, and the reason is not a practical difficulty. It is arithmetic, and it applies to a choir standing in a field with no instrument anywhere near it.

Four chords, all of them ordinary, all of them tuned as purely as it is possible to tune them, and the tonic at the end is not the tonic at the beginning.

Following it one step at a time #

Each move is unobjectionable on its own.

I to vi. C major to A minor. The A is the sixth degree, which in just intonation sits at 5:3 above the tonic — the ratio that makes the A a pure major third above F and a pure minor third below C. Nothing controversial.

vi to ii. A minor to D minor. The D is a pure fourth above the A, 4:3. A pure fourth is the third-simplest ratio there is.

ii to V. D minor to G major. The G is a pure fourth above the D. Again 4:3.

V to I. G major to C major. The C is a pure fourth above the G. Again 4:3.

Multiply them: 5/3 × 4/3 × 4/3 × 4/3 = 320/81. Fold that into an octave by dividing by two twice and it comes to 80/81, which is a syntonic comma below unity.

The drift is not a single bad step; it accumulates. Every one of these moves is a perfectly ordinary progression and each takes a little more off, so a longer sequence in pure ratios ends further from home rather than being pulled back — which is why the problem is a property of the tuning rather than of any chord in it.

That is the syntonic comma, 21.51 cents, and it arrived without anybody making a mistake. Every interval used was pure. The problem is that “the same note” in a chord sequence means “the note with the same name”, and the same name can be reached by routes of different lengths.

Why the keyboard explanation is the wrong one #

The keyboard version of the problem is real and it is a different problem. A keyboard has twelve keys per octave, just intonation needs more than twelve distinct pitches to play in several keys, and something has to give. That is a limitation of an instrument, and it can be fixed by building a better instrument — as split-key keyboards with fourteen or nineteen notes to the octave actually were.

The comma pump cannot be fixed that way. Give a choir infinitely fine pitch control and the drift is unchanged, because the drift is not caused by rounding a pitch to the nearest available key. It is caused by the ratios multiplying to something other than one.

The shape that produces it is a move by a major third or minor third in one direction and by fourths or fifths back, and there is no way to write a functional progression that never does that.

What actually happens instead #

Real ensembles do not drift 21.5 cents per four bars, so something is being done. Four things, and they are worth separating because they are different solutions.

Temper something. Sing one of the intervals slightly out of tune, distributing the comma across the progression. This is what equal temperament does wholesale, and what an ensemble does implicitly whenever it stays with a fixed-pitch instrument.

Avoid the shape. A progression built only of fifths and fourths cannot pump, as the figure shows. This is a real constraint on what can be written in strict just intonation, and it is why the surviving repertoire of just-intonation composition is so heavily restricted in its harmonic vocabulary.

Use two different notes with the same name. The D reached as a fourth above A and the D reached as a fifth below A differ by exactly the comma, and an ensemble can use whichever fits — which is to say the ensemble is doing what a nineteen-note keyboard does, in the head. This is what the theory of adaptive just intonation formalises, and what good a cappella groups appear to do.

Drift, and reset. Let the pitch fall and pull it back at a phrase boundary. This one is not a workaround. It is documented behaviour: a cappella choirs measurably go flat over sustained passages, and the effect is large enough to be a standing complaint of choral directors.

Which note is the culprit #

It is worth locating the comma precisely, because the loose statement — “the progression drifts” — hides the fact that one specific note is being asked to be two things.

The note is D, the root of the ii chord, and it can be reached two ways from C.

Down a fifth and up a fourth: C to F to… no, that route does not reach D. The two routes that do are these. As a fifth above G, which is itself a fifth below C: two fifths up from C, giving 9:8, the Pythagorean tone at 203.9 cents. And as a fourth above A, where A is a major third above F, which is a fourth above C: that route gives 10:9, the smaller tone at 182.4 cents.

The difference between 9:8 and 10:9 is 81:80. It is the syntonic comma, it is 21.5 cents, and it is the entire content of the problem. The scale of C major in just intonation contains two different sizes of tone — a large one from C to D and a small one from D to E — and which one a given D wants depends on which chord it is standing in.

So the choice is not really about the progression at all. It is about which of two D’s is played, and every solution listed below is a different answer to that question — temper the difference away, avoid needing both, keep both available, or slide from one to the other and never come back.

The measurement, and how much of it is this #

Choral flattening is real and it is not all comma pump, which is worth being careful about because the comma pump is a satisfying explanation and satisfying explanations attract more credit than they earn.

Measured drift in unaccompanied choral singing runs to tens of cents over a piece, and the causes established in the literature include vocal fatigue, falling breath support, the tendency to approach a note from below, temperature, and the acoustics of the room. Just-intonation drift is one contributor among several, and it is distinguishable from the others only by its direction being predictable from the harmony.

The clean demonstrations are therefore laboratory ones rather than concert recordings. Given a progression engineered to pump, and singers instructed to tune each chord purely, the drift appears at the predicted rate and in the predicted direction. Given ordinary repertoire, the drift appears and its attribution is contested.

So it is not true that a progression in pure ratios always drifts. This one comes home, and the reason is visible in the numbers: it happens to traverse the lattice by a route that closes. The comma is a property of the path and not of just intonation, which is the sharpest form of this essay’s claim and the reason the pump has to be constructed rather than merely stumbled into.

The history of noticing #

The phenomenon is old, and the record of it is a good illustration of how long a clearly-stated arithmetical fact can sit in the literature without changing practice.

Giovanni Battista Benedetti wrote to Cipriano de Rore in about 1563 with what is essentially this essay’s figure in prose: a short progression, each interval taken pure, and a demonstration that the pitch descends. He used it as an argument against the pure-ratio orthodoxy of Zarlino, who was the dominant theorist of the century, and against the idea that vocal music simply realises the ratios.

Zarlino’s position had the weight of authority and it survived, largely because the alternative — that the ratios everybody agreed were the basis of consonance could not all be used at once — was unattractive and because nobody could hear the drift in a single progression. It took until the nineteenth century for the arithmetic to be generally accepted, and Helmholtz’s On the Sensations of Tone (1863) treats it as settled.

What did not follow was any change in how singers were taught, and that is the interesting part. A choir has never been instructed to sing the ii chord’s root twenty-one cents high, and the reason is that no such instruction is usable: the compromise is made continuously, unconsciously and differently in every bar. The theory was right, the practice was already handling it, and the two had almost nothing to say to each other.

The comma pump as a compositional device #

Because the drift is predictable, it can be used, and at least two traditions have.

A rising version of the pump exists — the same shape traversed the other way — and repeating it produces a sequence that climbs by a comma each time. Nobody uses it for that, because the comma is too small to be a musical event on its own. What can be used is a stack of them: repeat the loop enough times in strict just intonation and the pitch moves audibly.

The composers who have done this deliberately are those working in extended just intonation, principally Ben Johnston, whose string quartets specify exact ratios and whose notation carries accidentals for the syntonic comma precisely so that a performer knows which D is meant. Johnston’s Seventh Quartet has passages where the accumulated drift is a structural event rather than an error.

The other use is the opposite: a piece can be constructed so that the pumps cancel, which is a real compositional constraint of the same kind as avoiding parallel fifths and considerably harder to satisfy.

Where the model stops #

The account above assumes that each chord is tuned to pure ratios against its own root, and that the root is whatever the previous chord left. That is one policy among several, and the drift depends on which is chosen.

A different policy — tune every chord to pure ratios against a fixed tonic rather than against the previous chord — has no drift at all, because the reference never moves. It has a different problem instead: the ii chord in that scheme contains a fifth that is a comma narrow, which is audible as a distinctly sour chord, and the difficulty has simply been relocated from the end of the progression to the middle of it.

That is the general situation and it is worth stating plainly. The comma has to go somewhere. Distributing it over the progression gives drift; putting it in one chord gives a bad chord; spreading it over the twelve fifths gives equal temperament. There is no policy that removes it, because the arithmetic that produces it does not depend on any policy.

The three are worth pricing rather than listing, because the arithmetic says something none of the descriptions do. Take the seven diatonic triads of C major, tune the scale to a fixed tonic in pure ratios, and measure how far each triad’s third and fifth sit from pure:

Five of the six usable triads are exactly pure and the whole comma is in the sixth, in both of its intervals at once, which is why the fixed-tonic policy is described as producing one sour chord rather than a slightly worse scale. Total mistuning across the six: 43.0 cents, all of it in one place.

Now the same total in equal temperament, where nothing drifts and no chord is singled out: every major third is 13.69 cents wide, every minor third 15.64 cents narrow, every fifth 1.96 cents narrow. Total across the same six triads: 99.7 cents.

So equal temperament is not the small compromise and just intonation the large one. It mistunes the diatonic triads by more than twice as much in total — and it is the arrangement everybody uses, because the 99.7 is spread so that no chord carries more than sixteen cents and the 43.0 is concentrated so that one chord carries forty-three. A listener meets a tuning system one chord at a time, and the quantity that decides how it sounds is the worst chord rather than the sum.

What real ensembles do is not any one of these but a continuous negotiation between them, adjusting chord by chord according to which notes are exposed, which are doubled, and what has just been heard. It is not describable as a tuning system, and attempts to reduce it to one have not been successful.

What it costs to have both D’s #

The obvious engineering answer is to supply both notes and let the performer choose, and it has been tried on real instruments often enough to know what it costs.

A keyboard with separate keys for the two D’s needs, in general, one extra key for every note that can be reached by two routes — and the count grows fast. Extending the chain far enough to play in a handful of keys without a wolf takes fourteen notes to the octave; playing freely takes nineteen, thirty-one, or in the most ambitious surviving instruments fifty-three. Nicola Vicentino’s archicembalo of 1555 had thirty-six keys to the octave and two manuals, and the reports of it are consistent: the instrument worked, and almost nobody could play it.

That is the general shape of the trade. Each additional pitch removes one compromise and adds one decision the player has to make in real time, and the decisions are what the twelve-note keyboard was buying relief from. The instrument that survived is the one that made the choices in advance, badly and uniformly, and left the performer nothing to think about.

Fretted strings sit in a third position which is worth noting because it is the least discussed. A lute or a viol has frets running across all the strings, so a fret fixes one interval on every string at once and the two D’s cannot both be available even in principle — but the frets were gut, tied on, and movable, so an ensemble could and did adjust them for a piece. That is a temperament chosen per work rather than per instrument, and it disappeared with metal frets.

What the picture cannot show #

The drift chart plots the pitch of the tonic and nothing else, so it cannot show what is happening inside each chord — which is where the compromise is actually made. A chord can be pure and leave the root where it was, or impure and hold the root, and the figure sees only the second effect.

Nor does it show the listener’s tolerance. Whether a comma of drift is noticed at all depends on how wide the interval categories are and on whether the drift is compared against a fixed reference — which is why an unaccompanied choir can pump commas all evening and a choir with an organ cannot.

Nor can it show which voice is carrying the drift. In four-part writing the comma can be absorbed by any of the four, and where it goes changes what the listener hears — put it in an inner voice and it disappears, put it in the melody and the tune goes flat. Every measurement in this essay is of a single number standing for a whole texture.

Nor does it show what the drift does to a chord rather than to a note. Every intermediate harmony in the pump is still the chord it is named as, because the accumulated error stays well inside a category until the end — and that is the whole reason the progression is usable rather than merely a curiosity.

It also cannot show the time constant. Twenty-one cents distributed over four bars is imperceptible as it happens and obvious when the start and the end are compared, and that difference between the local and the global is most of why the phenomenon went unnoticed for as long as it did. The comparison worth making is against the smallest difference a listener can hear: five cents at a time is under it and the accumulated twenty-one is four times over it, so the pump is a mechanism for converting a series of inaudible errors into an audible one — and the category the notes stay inside while it happens is what keeps every intermediate chord recognisable.

The ladder from here goes to the other place where pure ratios and functional harmony come apart: what happens to two voices when the interval between them is one the partials strongly agree about.