The wolf at the end of the chain

A chain of fifths is a straight line, not a circle. It runs E-flat, B-flat, F, C, G, D, A, E, B, F-sharp, C-sharp, G-sharp, and then stops — not because there is nowhere further to go, but because a keyboard has run out of keys.

The next fifth up from G-sharp would be D-sharp, and the keyboard offers E-flat instead. Those are the same key and, in any tuning except equal temperament, not the same note.

That last interval is the wolf — from the German Wolfsquinte, and named for what it sounds like. It is not subtly out of tune. It beats at around twenty times a second in the middle of the keyboard, which is not a pitch discrepancy any more but a distinct rattling roughness, and it is impossible to mistake for anything else.

Why anybody would build this deliberately #

Concentrating the error looks perverse until the alternative is examined.

The comma has to go somewhere. Spreading it across all twelve fifths — which is equal temperament — makes every fifth two cents narrow and every major third fourteen cents sharp. Concentrating it in one fifth makes eleven fifths nearly pure, every major third in the common keys exactly pure, and one interval unusable.

The question is therefore: how often is the bad interval reached?

In quarter-comma meantone with the chain above, the wolf falls between G-sharp and E-flat. That interval appears in the keys of A-flat major, D-sharp minor and their neighbours — the far side of the map of keys — regions that sixteenth- and seventeenth-century keyboard music essentially never visits. A composer working in two flats and three sharps never encounters it, and gets pure thirds everywhere in exchange.

That is not a compromise made out of ignorance. It is an optimisation against an actual usage pattern — a decision about which keys are near and which are not, and it held for two hundred years because the usage pattern held.

What a wolf sounds like, precisely #

The reason the wolf is unbearable rather than merely wrong is worth setting out, because it explains why the same 36-cent error would be tolerable elsewhere.

Two tones sounding together beat at the difference between their frequencies. When the tones are complex — and every real instrument’s tones are — each pair of their partials beats too, and the interval’s roughness is the sum of all those interactions.

A fifth is consonant because the third partial of the lower note and the second partial of the upper note coincide exactly. Detune the fifth by 36 cents and those two partials, which are up around 800 hertz on a middle-register note, are now about seventeen hertz apart. Seventeen hertz is very close to the worst possible beating rate: fast enough to be heard as roughness rather than as pulsation, slow enough not to separate into two distinct tones.

On the roughness curve a fifth’s well is 37 per cent of the curve’s range, which is why a fifth 36 cents out is not a slightly rough fifth but a chord that has left the well entirely.

The curve makes the point sharply: consonance is a narrow well, not a broad basin. A fifth mistuned by two cents is still in the well. A fifth mistuned by thirty-six cents has fallen out of it and landed halfway up the adjacent peak.

The escapes, and why they failed #

If the problem is that one key serves two notes, the obvious fix is more keys. It was tried, repeatedly, and the instruments survive.

Split keys. Divide the black keys front-to-back into two levers, one tuned to G-sharp and one to A-flat. This solves the immediate problem completely and costs one thing: the player now has to know which of two keys to press, and the keyboard geometry becomes hostile. Split-key instruments were built through the sixteenth and seventeenth centuries and were always specialist.

The archicembalo. Nicola Vicentino built one in 1555 with thirty-one notes to the octave across six manuals, which is enough to make meantone close properly — thirty-one-tone equal temperament is very nearly quarter-comma meantone, and the chain wraps round with no wolf at all. It is a beautiful solution and essentially unplayable.

Simply retuning. A harpsichord can be retuned in an hour and a moderately skilled player did it as a matter of course, moving the wolf to wherever the next piece would not go. This was the ordinary answer, and it is why the wolf’s position is a variable rather than a constant.

None of these scaled. What scaled was giving up on pure thirds, and the eighteenth century did.

That figure’s failure is instructive and worth naming: a chart of where the twelve notes sit says nothing about the wolf, because the wolf is not a note out of place. Every note in meantone is exactly where it should be. The wolf is an interval between two notes that were never meant to be adjacent, and only a picture of the chain shows it.

Counting the beats #

The wolf’s unpleasantness has a number attached to it, and computing that number is the clearest way to see why the interval fails while a merely imperfect one does not.

Take a fifth in the middle of the keyboard: A at 220 hertz below E at 330. The third partial of the A is at 660 hertz and the second partial of the E is at 660 hertz. They coincide exactly, there is no beating, and the interval is still.

Now widen the fifth by 36 cents. The E moves to 336.9 hertz, its second partial to 673.8, and the two partials are 13.8 hertz apart. That is the beat rate, and 13.8 hertz sits almost exactly at the top of the roughness curve — fast enough that the ear cannot follow individual pulses, slow enough that it does not resolve into two separate tones.

Move the same interval up an octave and that pair’s beat rate doubles to 27.6 hertz; move it down an octave and it halves to 6.9. Reading a worst register off those three numbers is the obvious next step and it is the wrong one, because a fifth between two complex tones has more than one pair of partials in it and the roughness is the sum over all of them. The section after the figure does that sum.

And a pure fifth is 3:2, whose sum repeats after two cycles of the upper note and three of the lower — the coincidence that makes the interval easy to set by ear, and the coincidence a wolf destroys.

Which register the wolf is worst in #

Summing the site’s own Plomp–Levelt roughness over every pair of partials of both notes, at every register a keyboard has, gives an answer with no interior maximum in it at all — and it is worth putting the wolf beside a pure fifth at the same pitch, because the wolf is a defect and a defect is a comparison.

Absolute roughness falls from the bottom of the keyboard to the top and never turns round. The peak of that curve is at a fundamental of about sixteen hertz, which is below the lowest note on a piano and below the bottom of hearing. The ratio rises and never turns round either, and so does the plain difference between the two columns, whose largest value is up near three kilohertz. Neither quantity has a maximum anywhere a keyboard plays.

What the model is saying is that a pure fifth in the bass is already rough — 0.498 against the wolf’s 0.504, a difference of one per cent — because down there the third partial of the lower note and the second of the upper sit inside a single critical band whether the interval is tuned or not. The bass has no room left to be spoiled. Two octaves above middle C the pure fifth has fallen to a twentieth of its bass value and the wolf has barely moved, so the bad interval stands at nine times its good neighbour and is unmissable.

So the claim to keep is the one the closing caveats of this essay already make — that a wolf two octaves up is intolerable and a wolf in the bass is merely bad — and the claim to drop is the middle-register peak, which came from watching one pair of partials and treating its beat rate as the whole of the roughness. Both statements were in this essay and they disagreed with each other; the sum decides between them.

What the wolf did to composition #

Instrument constraints leave marks on repertoire, and this one is legible.

Seventeenth-century keyboard music does not modulate far. That is usually explained as a stylistic preference, and it is at least as much a physical fact: the far keys were not available, and a composer who wrote them was writing something that could not be played on the instrument in the room. Frescobaldi’s toccatas roam harmonically within a narrow orbit of keys, and the orbit’s edge is where the wolf is.

When composers did reach for the far keys, they did so deliberately and for effect. The sudden appearance of a remote chord in seventeenth-century music is not a modulation in the later sense; it is a lurch into a region that sounds physically different, and it was used the way an orchestrator uses a shrill instrument.

The disappearance of the wolf therefore removed a compositional resource at the same time as it removed a constraint. Nineteenth-century music can go anywhere, and everywhere it goes sounds the same.

The wolf is a general phenomenon #

Wolf intervals are not confined to keyboards or to Europe, and the general principle is worth stating: any attempt to close a chain of pure intervals into a finite cycle produces a discrepancy, and the discrepancy can be spread or concentrated but not removed.

Fretted instruments have the same problem in a different form. A fret is a straight bar across all the strings, so it fixes the same pitch ratio on every string — which is fine in equal temperament and impossible in meantone. Sixteenth-century lutes and viols were therefore fretted in something close to equal temperament, two hundred years before keyboards adopted it, and the resulting mismatch between a lute and a meantone harpsichord in the same room was a known nuisance. The same problem appears wherever two systems of intonation have to share a piece.

Wind instruments carry it in the bore. A player’s embouchure can bend a note by twenty or thirty cents, which is why an ensemble’s intonation is negotiated rather than fixed, so the wolf is negotiable — but the instrument still has notes that are naturally sharp or flat, and orchestral players spend their careers compensating for them.

A keyboard with twelve keys to the octave is the constraint the whole argument rests on: it is what forces the chain to close, and it is the only reason a leftover has to be put anywhere.

What the wolf reveals #

The most useful thing about the wolf is what its existence proves: that the chain of fifths is genuinely a chain and not a circle, and that the circular diagram everybody learns is a picture of equal temperament rather than a picture of music.

In equal temperament, G-sharp and A-flat are the same note, and that identity is so completely built into modern musical thinking that it takes an effort to see it as a convention. It is one. In meantone they are 41 cents apart — a fifth of a semitone, which is a large interval — and a musician who wrote G-sharp meant something different from one who wrote A-flat.

The notation still carries the distinction. Written music distinguishes G-sharp from A-flat everywhere, at some cost in complexity, for an instrument on which they are identical. That is not redundancy; it is a fossil of a period when the difference was audible, preserved because it still tracks harmonic function even when it no longer tracks pitch — one of several things notation records for reasons that have outlived their cause.

Naming the animal #

The word is worth a paragraph, because it is one of the very few pieces of technical vocabulary in music that is straightforwardly onomatopoeic.

Wolfsquinte — wolf-fifth — appears in German organ-building literature by the seventeenth century, and the earlier Latin sources describe the interval as ululans, howling. Organ builders had the strongest reason to name it: an organ cannot be retuned between pieces, its tone does not decay, and a held wolf fifth on a full principal chorus is a sound that empties a building.

The name attached to the interval and then generalised. A wolf note on a cello is a different phenomenon entirely — a body resonance that coincides with a played pitch and makes it stutter — and it inherited the name because it produces a comparable pulsing. String players still speak of a wolf on a particular instrument at a particular note, and it is a property of that instrument’s construction rather than of any tuning system.

Two unrelated defects, one name, because both of them beat at a few times a second in a way that sounds animal.

Living with it #

The practical accommodations are more interesting than the theoretical escapes, because they are what actually happened.

Choose the key. A harpsichordist selecting a programme was, among other things, selecting a set of keys that avoided the wolf, and that consideration sat alongside the musical ones rather than beneath them.

Move it before the concert. Retuning a harpsichord takes under an hour and shifts the wolf anywhere on the chain. It was routine, and it means that the wolf’s canonical position between G-sharp and E-flat is a convention of modern writing about historical tuning rather than a fixed historical fact.

Voice around it. A wolf fifth is only intolerable when both notes sound together and sustain. A player who arpeggiates the chord, or who leaves out the fifth, or who lets the upper note pass rather than settle, gets away with it. Continuo players did all three by reflex.

Let the winds cover it. In ensemble music the keyboard’s wolf can be buried under an instrument that can bend its pitch, and the resulting compromise is what everybody actually heard. The pure-meantone keyboard sound that survives on recordings today is in that sense unrepresentative: it is what the instrument does alone, which is rarely how it was used.

And of the four triad qualities the diminished is the one a wolf produces where a fifth should be — which is why a chord containing the wolf is not described as out of tune but as a different chord.

The family, and what a smaller wolf costs #

The meantone tunings are not a list but a one-parameter line: a fifth narrowed by some fraction of a syntonic comma, with everything else following from that one choice. Two of the things that follow are straight lines in the fraction, and dividing one by the other gives the exchange rate the whole historical argument turns on.

Every cent of major-third impurity accepted buys exactly 2.75 cents off the wolf, on every member of the family, and the ratio is eleven over four for a structural reason rather than an empirical one: a third is four fifths and the wolf carries whatever eleven of them did not spend. It does not depend on which comma, which member, or which end of the line the reckoning starts from.

That is a favourable rate, and it is most of why sixth-comma won the eighteenth century. Silbermann’s tuning gives up seven cents of third — audible, and about half of what equal temperament gives up — and takes twenty cents off the wolf, from thirty-six down to sixteen. Sixteen cents wide is not pleasant, but it is inside the range an ordinary badly tuned fifth occupies rather than outside it, so the far keys become merely poor rather than forbidden. That is the difference between a constraint and a defect.

Running the line the other way finishes the argument. The wolf reaches zero — meaning the twelfth fifth comes out exactly pure while the other eleven are two cents narrow — at a tenth of a comma, and the major third is then thirteen cents sharp, within a cent of equal temperament’s fourteen. A meantone with no wolf in it is equal temperament with the error relabelled, and the family says so without anyone having to argue the point.

Where the model stops #

The register sweep is a table and not a figure. The dependence is computed above and drawn nowhere, and it is the one result here that would benefit most from a picture, since two monotone curves crossing scale is hard to hold in prose.

The roughness model is a model. The dissonance curve computes sensory roughness by the Plomp–Levelt method, which accounts for beating between partials and for nothing else. It says nothing about harmonic expectation, cultural familiarity or context, all of which affect whether a listener calls an interval unpleasant.

The figures draw one member of the family. The table above prices the rest of it, but every picture on this page is quarter-comma, so the wolf a reader sees is always the widest one that was actually used.

The chain’s start point is a choice. Beginning the chain at E-flat puts the wolf between G-sharp and E-flat. Beginning it elsewhere moves the wolf elsewhere, and that was the tuner’s actual decision. The figures pick one convention and do not show that it was a decision.

The ladder from here #

Later rungs: beat rates and how a tuner uses them. The full meantone family. Split keys and enharmonic instruments in detail. Fretting, and why lutes reached equal temperament first. Well temperaments as the negotiated settlement. Key character, and whether it survived the transition. Adaptive tuning by computer, which finally makes the archicembalo’s solution practical. And the wolf in other traditions, where a chain of pure intervals meets a finite instrument and the same arithmetic bites.

Bach’s Well-Tempered Clavier is named for a temperament that is not equal, and the title is a claim: that every key is usable, which in 1722 was news. It is not a claim that every key is the same, and the pieces do not sound as though it were.