Acoustics & Music Theory
why it sounds like that
Pitch and tuning

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.
16 min read 9 figures The comma that will not closeSmall whole numbers

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.

The chain of fifths in quarter-comma meantoneThe fifths laid end to end as the chain they are. The bar under each shows how far that fifth departs from a pure three-to-two, and the last one — where the chain is forced to close — is the wolf.E♭B♭FCGDAEBF♯C♯G♯D♯-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4-5.4+35.7each fifth's departure from a pure 3:2, in centsone fifth carries the whole error
Fig. 1 The chain of fifths drawn as the line it is, in quarter-comma meantone. Each bar under a fifth shows how far it departs from a pure three-to-two. Eleven of them are about five cents narrow. The last one, forced to close a gap that eleven narrowings did not close, is thirty-six cents wide.

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.

220 Hz against 223 HzTwo tones 3 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 3 beats a second and used by every tuner who has ever worked by ear.00.511.52seconds3 beats per second — the difference, exactlythe carrier is drawn slower than it sounds, or it would be a solid band
Fig. 2 Two tones a few hertz apart, and their sum. The slow swelling is the difference frequency. A pure major third has no swelling at all; a meantone third achieves that, an equal-tempered third does not, and a wolf fifth swells so fast that the swelling stops sounding like swelling.

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.

Roughness across an octaveSensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The deep wells land on the fourth, the fifth and the octave; the thirds sit on shoulders rather than in wells, which is a real feature of this model and not a defect of the drawing.semitones above the lower tone6/55/44/33/25/32/1CC♯DE♭EFF♯GA♭AB♭BCroughest at about a semitonesmooth at the simple ratios
Fig. 3 Sensory dissonance across an octave, computed by summing the roughness between every pair of partials of two complex tones. The deep wells at the fourth, the fifth and the octave are where the partials line up. A wolf fifth sits on the steep wall just to the right of the fifth’s well, which is the worst neighbourhood in the picture.

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.

How far each system sits from equal temperamentDeviation from equal temperament, in cents, for each degree of the chromatic scale. Zero is equal temperament by definition; the just major third is about fourteen cents flat of it and the Pythagorean major third about eight cents sharp.024681012-20-1001020semitones above the tonicCC♯DE♭EFF♯GA♭AB♭BCjust intonationPythagoreanquarter-comma meantonesharp of equalflat of equal
Fig. 4 Three systems against equal temperament. The meantone line’s virtue is what it does to the thirds, and its cost is invisible here — the wolf is one interval between two adjacent chain positions, and a plot of absolute pitch cannot show it at all.

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 the beat rate doubles to 27.6 hertz, which is past the roughness peak and into a region that starts to sound like a low buzz. Move it down an octave and it halves to 6.9, which is a distinct wobble rather than a howl. The wolf therefore has a worst register, and it is the one keyboard music mostly occupies.

Two tones in the ratio 3 to 2Two sine tones whose frequencies are in the ratio 3 to 2, and their sum. The combined pattern repeats every time both waves return to the start together, which for a simple ratio is soon.one frame = one repeat of the combined wavelower — 2 per frameupper — 3 per frametheir sumthe pattern repeats after 2 cycles of the lower tone
Fig. 5 Two tones in the ratio three to two, and their sum. The pattern repeats after two cycles of the lower tone, which is what a pure fifth means. Detune it and the repeat never quite happens; what was a short cycle becomes an endless slow slide, which is the beating.

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 circle of fifthsThe twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering.CaGeDbAf♯Ec♯Ba♭F♯e♭C♯b♭A♭fE♭cB♭gFdone step= one fifth= one sharpouter ring: major keys · inner ring: their relative minors
Fig. 6 The circle of fifths, which is true of equal temperament and of nothing else. On a meantone instrument this diagram is a chain with a break in it, and the break is a place composers knew about and avoided.

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.

Notes on the keyboardA piano keyboard with the notes under discussion marked. The keyboard is used throughout this site because it shows distance rather than name, and distance is what the theory is about.CEG3 notes sounding
Fig. 7 The keyboard, which is where the whole problem lives. Twelve keys to the octave is not a fact about music; it is a fact about hands, and every difficulty on this page follows from a decision about ergonomics made before anybody understood the arithmetic.

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.

Twelve fifths do not make seven octavesPitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round just over seven times and finish 23.46 cents past seven octaves — the Pythagorean comma, and the reason no keyboard can be tuned in pure fifths.CC♯DE♭EFF♯GA♭AB♭Bstartseven octavestwelve fifthsthey miss by 23.46 centsone turn is one octave · angle is pitch class · radius is how far it has climbed
Fig. 8 Twelve fifths against seven octaves, drawn without the correction that closes them. Every wolf that ever howled is the same gap, moved to whichever interval its owner used least.

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.

Four triads as stacked intervalsEach triad drawn as the semitone distances above its root, with the nearest simple frequency ratio beside each note. The chords differ only in the size of the two stacked thirds, and that difference is the whole of their character.C1/1E5/4G3/243major0–4–7C1/1E♭6/5G3/234minor0–3–7C1/1E♭6/5F♯33diminished0–3–6C1/1E5/4A♭8/544augmented0–4–8semitones above the root
Fig. 9 Four triads as stacked intervals with their nearest simple ratios. In meantone the major triad’s third is exactly 5/4 — genuinely beatless, in a way no modern keyboard can produce — which is what the whole arrangement was for, and what the wolf was the price of.

Where the model stops

Beat rates are register-dependent. The wolf beats at twenty hertz in the middle of the keyboard and at five in the bass, because beating goes with absolute frequency difference and not with the interval. A wolf in the bass is bad; a wolf two octaves up is intolerable. No figure here shows that dependence.

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.

Meantone is a family. “Quarter-comma” is one member; sixth-comma, fifth-comma and two-sevenths-comma meantones were all used, each with a differently sized wolf and differently pure thirds. The figures here draw one of them.

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.

Where this goes next

024681012-20-1001020semitones above the tonicCC♯DE♭EFF♯GA♭AB♭BCjust intonationPythagoreanquarter-comma meantonesharp of equalflat of equal Where to hide the comma, which is the only real question Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written. CC♯DE♭EFF♯GA♭AB♭Bstartseven octavestwelve fifthsthey miss by 23.46 centsone turn is one octave · angle is pitch class · radius is how far it has climbed Twelve fifths and seven octaves, which are not the same thing Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that. 00.511.52seconds3 beats per second — the difference, exactlythe carrier is drawn slower than it sounds, or it would be a solid band Beats are arithmetic that anybody can hear Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.