Keys that had characters, and could be measured

Johann Mattheson, writing in 1713, gives E minor as suited to “something thoughtful, profound, grieved and sad, yet in such a way that hope still remains”. Christian Schubart, ninety years later, has D flat major as “a leering key, degenerating into grief and rapture” and A flat major as “the key of the grave”. The tradition of assigning characters to keys runs from the seventeenth century to Berlioz and beyond, and it is embarrassing to modern eyes, because on a modern piano it cannot possibly be true.

On an equal-tempered instrument every key is an exact transposition of every other. The intervals in F sharp major are the same sizes as the intervals in C major, to the last decimal place. Anything a listener reports about the difference has to come from register, from the feel of the keys under the hand, or from expectation — the same kind of confound that makes claims about consonance hard to test — and none of those is what Mattheson was describing.

He was not writing about an equal-tempered instrument.

What an irregular temperament is #

The constraint on any keyboard tuning is severe and has one form. Twelve fifths have to span exactly seven octaves, or the instrument does not close and some key is unplayable. Twelve pure fifths overshoot seven octaves by a Pythagorean comma, 23.46 cents. So a temperament is a decision about how to distribute one comma over twelve links, and nothing else.

Equal temperament makes the obvious choice: a twelfth of a comma from each fifth, which is 1.955 cents apiece. Every fifth is the same size, every third is the same size, and the instrument is perfectly uniform.

A well temperament — the word is a translation of wohltemperirt — makes an unequal choice. It takes more of the comma out of some fifths than others, while still spending exactly one comma in total. The chain still closes, every key is still playable, and the keys are no longer the same as each other.

The difference between those two figures is the whole subject. On one instrument there are twelve keys; on the other there is one key available in twelve positions.

Which computation produced the number #

Nothing in these figures is tabulated. A temperament is entered as twelve numbers — how many cents to remove from each fifth in the chain C–G–D–A–E–B–F♯–C♯–G♯–E♭–B♭–F–C — and everything else is derived.

Werckmeister III, published in 1691, is entered as a quarter of a comma removed from each of four fifths: C–G, G–D, D–A and B–F♯. The other eight are pure. Four quarters make one comma, so the chain closes.

The pitch of each note then follows by accumulating along the chain:

$$c_{k} = \sum_{i<k}\left(701.955 - t_i\right) \bmod 1200,$$

where $t_i$ is the narrowing of the $i$-th fifth. The major third above any root is the difference between two of those numbers, and its departure from a pure 386.314 cents is the length of the bar.

The closure is asserted rather than trusted. The routine that builds a temperament adds up the twelve narrowings and refuses to return anything if they do not come to a Pythagorean comma to within a billionth of a cent, and it separately checks that the accumulated chain arrives at exactly 8400 cents — seven octaves — after the twelfth fifth. A narrowing list that does not close describes an instrument nobody can play in every key, and the arithmetic catches that here rather than in a figure that looks convincing.

For Werckmeister III the thirds come out between +3.9 and +21.5 cents sharp of pure. The spread is 17.6 cents, which is most of a fifth of a semitone, and it is not distributed randomly: the keys with few sharps and flats have the smooth thirds and the remote keys have the harsh ones.

Why the smooth keys are the near ones #

The pattern is not a coincidence and not a choice. It falls out of where the tempered fifths sit in the chain.

A major third is four fifths’ worth of chain — the same fact that makes the syntonic comma appear at all. If all four of those fifths have been narrowed, the third comes out close to pure, because narrowing four fifths by a quarter-comma each removes a whole comma from the stack — which is exactly the amount by which a chain of pure fifths overshoots a pure third. If none of them has been narrowed, the third comes out a full syntonic comma sharp, which is the Pythagorean third and is unpleasant.

Werckmeister put his four narrowed fifths where C, G, D, A and E live. So the thirds in those keys are good. F sharp major’s third spans four untouched fifths and is therefore as bad as a third gets.

The consequence, stated as a composer would experience it, is that modulating outward from C is a directional journey. It does not merely change which notes are played; it changes the size of the third, monotonically, until the far side of the circle. That is a musical resource, and it is a resource an equal-tempered instrument does not have. Whether it also changes how rough the chords are is a separate question with a different answer, and it is the subject of a later section.

Kirnberger’s scheme is worth pausing on because it shows the constraint being gamed. Four quarter-syntonic-comma fifths make C’s third exactly 5:4. But four syntonic comma-quarters do not add up to a Pythagorean comma; they fall short by a schisma, 1.95 cents. Kirnberger dumps that leftover on the F♯–C♯ fifth, which is far enough away that nothing much notices. The result is one perfect key, several good ones, and no fifth anywhere that is more than two cents from pure.

Compared on one axis #

Laid over each other, the temperaments are strikingly close together — which is the second thing worth noticing about them.

Compared on one axis, the three temperaments and just intonation differ by a few cents everywhere and the differences all sit on the thirds and sixths. That is the whole disagreement, and it is small: every one of the systems keeps every interval inside its own category, so what separates them is shade rather than identity.

No note of Werckmeister III is more than 8 cents from its equal-tempered position. A listener asked to identify the pitch of a single note would not reliably tell the two apart. The audible difference is entirely in the chords, and specifically in how much the partials of the notes beat against each other — which is a property of the intervals, not of the pitches.

That is why the effect survives being described as “colour” rather than as “tuning”. Nothing sounds out of tune. Some chords are simply calmer than others.

The triad, not the third #

Every bar in every figure above measures one interval. A key is a chord, and this site has a measure of what a chord costs — the summed roughness over every pair of partials — so the obvious control is to rank the twelve keys by that instead and see whether the ordering survives.

It does not. It very nearly reverses.

C major, whose third is nearly pure and which the hero figure puts at the smooth end, is the fourth roughest of the twelve. F♯ major, the harsh key, is smoother than it.

The reason is arithmetic and it is worth having in full. Differentiate the roughness against each impurity at middle C and a cent of fifth is worth 3.8 cents of third: the slope is 2.3 × 10⁻³ per cent on the fifth and 6.0 × 10⁻⁴ on the third. A fifth’s partials coincide at the third and second, which are loud; a third’s coincide at the fifth and fourth, which are quieter, and there are fewer such coincidences inside eight partials.

And a well temperament arranges for exactly the trade that cancels. The keys with good thirds are the keys whose fifths were narrowed to buy them, and the keys with Pythagorean thirds have pure fifths. Werckmeister’s 5.87-cent fifth costs 0.0135 in roughness and his 21.5-cent third costs 0.0130. The two are within four per cent of each other, and the whole twelve-key spread in total roughness is seven per cent.

So the strong reading of the third-size figure is wrong: a well temperament does not make some keys calm and others rough. It moves the roughness out of one interval and into another, at nearly constant total.

What that leaves standing is the quality, and it is the better claim anyway. The two impurities beat at completely different rates. A fifth 5.87 cents narrow puts the root’s third partial against the fifth’s second at 785 hertz, 2.7 hertz apart — a slow countable throb. A third 21.5 cents sharp puts the root’s fifth partial against the third’s fourth at 1308 hertz, 16.4 hertz apart, which is not a throb at all but the beginning of roughness. Equal amounts of beating, at rates a factor of six apart, in different registers of the same chord.

That is what a key character is, on this arithmetic: not more or less, but a different kind of impurity, moved between the two intervals of a triad by where the tuner put the comma.

The contrast that makes the point #

The alternative to spending the comma unevenly is spending nearly all of it in one place, which is what meantone does.

Drawn as a chain rather than as a wheel the same tuning reads as eleven equal narrowings and one leftover, and the leftover is the wolf fifth. The two pictures are the same twelve numbers: a wheel shows which keys pay and a chain shows which fifths were altered, and the whole family of well temperaments is a decision about how to distribute one fixed quantity between those two readings.

Meantone buys pure major thirds in eight or nine keys and pays with one interval so bad it cannot be sounded. It is not a circulating temperament: some keys are simply off limits, and two hundred years of European keyboard music stays inside the ones that work.

The well temperaments of the late seventeenth century onward exist precisely to remove that restriction. They are worse than meantone in the good keys and enormously better in the bad ones, and the historical shift from one to the other tracks the point at which composers started wanting to modulate anywhere. The characteristic sound of a remote key in Werckmeister is not a wolf; it is a bright, hard, faintly strained major third. Which is roughly what Schubart says about F sharp major, though he puts it more colourfully.

Setting one by ear #

A temperament is a list of cent values on paper and a sequence of counted pulses in practice, and the gap between the two descriptions is worth closing.

A tuner does not measure frequencies. A tuner listens for the beating between two nearly-coincident partials and adjusts until that beating happens at a prescribed rate. A pure fifth beats not at all, because the third partial of the lower note and the second partial of the upper are the same frequency. A fifth narrowed by a quarter of a comma beats at a rate that follows directly from how far it has been narrowed and from how high it is.

This is what makes an irregular temperament practical to set and easy to describe. Werckmeister’s own instructions do not give cents, which did not exist as a unit until 1885; they say which fifths to tune pure and which to narrow, and by how much in terms of an audible pulse. Eight pure fifths are the easiest thing in the world to tune, because pure means silent. Four narrowed ones are set by counting.

The consequence is that an unequal temperament is in one respect easier than an equal one. Equal temperament requires twelve fifths each narrowed by the same small amount, none of which is beatless, and each of which beats at a different rate because the beat rate scales with absolute frequency. It has to be checked against thirds and sixths as it goes. Werckmeister requires eight silences and four counts.

That fact is a useful corrective to the assumption that equal temperament is the simple default and everything else is an elaboration. Historically the arrow points the other way: equal temperament is the hardest of these to set by ear and the last to be reliably achieved, and it became standard only once the instrument-building trade could produce it consistently.

Where the model stops #

The roughness comparison depends on the timbre. Summing Plomp–Levelt over eight string partials with amplitudes falling as 1/n is what decides that a cent of fifth outweighs a cent of third by 3.8; a spectrum with a weak third partial or a strong fifth one would move that ratio and could move the ranking back. The direction of the effect is robust — a fifth’s coincidences are always at lower, louder partials than a third’s — and the factor is not.

Third size is not the only variable. These figures measure one thing, and the character of a key on a real instrument depends on several. The fifths vary too, and the section above is what happens when they are let in. So does the register the key sits in on a given keyboard, the way a harpsichord’s plucking point interacts with different string lengths, and — for anything with open strings — whether the tonic has one.

The writers disagree with each other. Mattheson’s E minor and Schubart’s E minor are not the same character, and Schubart was writing in 1806, when several of the instruments he knew may already have been close to equal. If key character were a straightforward consequence of temperament, and if everybody used similar temperaments, the descriptions should broadly agree. They do not, and that is real evidence against the strong version of the claim.

Nobody knows which temperament any particular piece assumed. Well temperaments were a family, not a standard, and tuners worked by ear from recipes that varied by region and by decade. Any statement of the form “this piece was written for Werckmeister III” is a hypothesis about an undocumented practice.

A spread of seventeen cents is a real difference and a small one. It is easily audible in a sustained major triad and much harder to hear in fast passagework, in a texture with a lot of doubling, or on an instrument whose decay is quick. The claim that supports itself is that different keys sounded measurably different; the claim that a key had an emotional character is a further step, and the arithmetic is silent about it.

Vallotti and Young are nearly the same temperament. They differ by a rotation of one position round the chain, and their spreads of third size are identical. Any theory of key character fine enough to distinguish them is finer than the evidence.

Whose music, and when #

The literature on key character is overwhelmingly German and overwhelmingly eighteenth century, and it belongs to a period when keyboard instruments were tuned unequally and everybody knew it. The three temperaments drawn above were published in 1691, 1754 and 1779 by, respectively, an organist writing for organ builders, an Italian friar, and a pupil of Bach’s writing for keyboard players.

The most argued-about instance is the Well-Tempered Clavier, which Bach compiled in 1722 and again in 1742, with a prelude and fugue in every major and minor key. It has been used as evidence for equal temperament for two centuries, and the argument does not hold: wohltemperirt is the word for exactly the family of irregular temperaments described here, and if Bach had meant gleichschwebend — equal-beating, the contemporary German term for equal temperament — the title would have said so. What the collection demonstrates is that all twenty-four keys are usable, which is the definition of a circulating temperament and not the definition of an equal one.

Beyond that the evidence thins rapidly. A decorative squiggle on the 1722 title page has been read since 2005 as a tuning instruction, which would settle the matter if it were true; it remains one interpretation of an ornament. The honest summary is that Bach expected a circulating temperament, that the twenty-four keys would not have sounded alike on it, and that which specific temperament he preferred is not recoverable.

Outside Europe the question mostly does not arise, because it is a question about fixed-pitch instruments with twelve notes that are expected to play in every key. A tradition that does not transpose has no reason to close the circle and therefore no comma to distribute.

The ladder from here #

Later rungs: meantone’s bargain, in the detail of what two hundred years of playing in eight keys was actually like. Equal temperament as a decision rather than a discovery — who wanted it, when, and what the arguments for it were, none of which was that it sounded better. The comma pump in real repertoire. Singers and string players, who have no temperament at all. And nineteen, thirty-one and fifty-three, where the number of notes changes and the whole problem is restated.

The eighteenth-century writers were describing something real and describing it badly. The thing itself is seventeen cents wide and can be drawn.