A split key buys one of everything, and two buy C and E

From the sixteenth century to the eighteenth, an organ or harpsichord builder who wanted more from a meantone keyboard could cut a black key in two. The back half sounded one note and the front half another: a G♯ and an A♭, a D♯ and an E♭, on the same physical accidental. It was expensive, it was awkward under the hand, and as earlier essays recorded, it was built anyway. What nobody here has asked is how a builder chose. With two extra keys to place there are many possibilities, and what surviving instruments and contracts mostly describe is one of them: G♯/A♭ and D♯/E♭, with the others rarer.

There are two obvious stories about that. In one, those two simply repair the worst damage: the wolf lies between G♯ and E♭, so a split at each end of it attacks the wolf directly. In the other, the arithmetic has an answer about how to buy the most triads per key, and those two accidentals are the optimum. The second turns out to be simply false, in an instructive way, and the first is true but says less than it seems to. What a meantone keyboard’s arithmetic actually says is stronger than either and points somewhere else.

The picture needed is a line and not a circle.

A meantone keyboard is not a circle. Twelve keys hold twelve neighbouring positions on the chain of fifths, from E♭ at three fifths below C to G♯ at eight above, and the fifths that would close it never meet. A split key adds one position at one end of that stretch. Nothing else about the instrument changes, and the dial above reads off what each new length contains. All of it is exact arithmetic on positions, and none of it depends on the temperament, only on the chain being long enough that no two positions fall on one pitch.

Every key buys one of everything #

A major triad on a position needs that position, the next one up, and the one four up: the fifth is one step along the chain and the third is four. A minor triad needs the position, the next one up and the one three down. So in a stretch of n neighbouring positions there are exactly n − 4 major triads and exactly n − 4 minor ones, and the count takes no notice of which end the stretch was extended at. Twelve keys hold eight of each. Fourteen hold ten. A key added at the top and a key added at the bottom each add one major and one minor triad.

The same reasoning holds for whole keys. A major key’s tonic, subdominant and dominant occupy seven neighbouring positions: one below the tonic to five above. So n − 6 major keys. A minor key’s tonic and subdominant triads and a major dominant with its raised leading note occupy ten: four below the tonic to five above. So n − 9. That raised leading note, five fifths above the tonic, is where a minor key’s expense lies, as the same distance under two names makes clear from the other direction: G♯ and A♭ are not one thing on such a keyboard, and only one of them leads to A. Twelve keys hold six major keys and only three minor ones, on G, D and A. Each further key adds exactly one of each.

So the first story’s hope fails at once. There is no best place to buy triads. The arithmetic of counting is perfectly indifferent to where a split goes, and a builder looking for the most chords per lever would have no reason to prefer any accidental to any other, so long as it extends the chain. A split that does not extend the chain, a second copy of a position already present, buys nothing at all, but nobody builds those.

There is one exception, and it is magnificent and was also built. In quarter-comma meantone, 31 fifths very nearly close, to within about six cents. So the last key of a chain of thirty-one is a different object from the first thirty. It buys five major triads at once, because each of the four positions that lacked a neighbour now has one. Vicentino’s thirty-one-note archicembalo of 1555 is therefore not simply a very long version of a two-split organ. It stands on the far side of a discontinuity. The argument for building to thirty-one keys is a real one and the argument for building to twenty-nine is not, and the essay on the equal divisions reaches 31 from the other direction.

What the two splits were for #

If the count cannot choose, the choice has to be about which keys. Lay each minor key’s reach along the chain, and the answer can simply be read.

The two accidentals that were split most often are exactly what completes the minor keys on C and E. C minor needs A♭ for its subdominant and it has only G♯ on a twelve-key keyboard. E minor needs D♯ for its leading note and it has only E♭. Add those two and the minor keys run C, G, D, A, E: five keys in a row around the middle of the chain, where the instrument’s modes already lie. The major keys become E♭, B♭, F, C, G, D, A and E. Neither pair of splits on one side does that.

So placing the two splits is a choice among three things with exactly equal counts. Both on the sharp side gets B minor and loses C minor. Both on the flat side gets F minor and loses E. One on each side keeps a repertoire centred, with the same reach on either side of the modes it mostly uses. That is the sense in which the wolf story is true: those are the positions at the ends of the chain, so they are the positions on either side of the wolf. But the wolf is not why they are the right ones. A split on the flat side at A♭ moves the wolf along by one, and the wolf is still there at thirteen keys, and at fourteen, and at every number up to thirty. A split key does not repair the wolf at all. It moves it to somewhere it is less often needed, as the earlier essay on the fractions already said, and it is the keys on either side of it, not the distance from it, that decide which move is worth paying for.

The surprising connection is to the leading note. The objection to these keyboards is usually put in terms of a chord that sounds wrong. But the expensive thing in a minor key is the single note five fifths above its tonic, and the history of split keys can be read as a history of what that one note cost. The rule that a dominant takes a raised leading note is a piece of voice-leading, not of tuning. The notations invented for the overflow showed how much later spelling is shaped by the same note. On a meantone keyboard that habit has a price in levers, and the fact that minor keys were bought three and then five at a time, at n − 9, while major keys were being bought at n − 6, is a fact about that one habit.

What a missing key costs, and where it does not show #

It would be natural to expect that the eight triads missing from a twelve-key keyboard are obviously rough, so that the split is buying a large reduction of an easily measured quantity. Measured by the Plomp–Levelt sum, that is not what happens.

By the roughness sum, a wolf triad is seven per cent rougher than a good one. Every one of the eight is rougher than every one of the sixteen, so the separation is clean, but the size is unimpressive. The sum is dominated by all the many pairs of partials in a triad that are not affected by moving one note by forty cents, and the wrong note is itself about as rough against other partials as the right one is. What actually distinguishes the two is the one thing the sum weighs lightly: the pair of partials that should coincide. A pure major third on middle C has its lower note’s fifth partial and its upper note’s fourth partial exactly together, so they do not beat. In the wrong third the same two partials are thirty-one hertz apart, which is close to where a fluctuation is at its roughest, and they do it in the very region that a listener is attending to. That difference is about what a tuner counts, and not what a sum over all pairs reports. It is a fair warning about how far the sum can be trusted in any argument about tuning. Forty-one cents is an enormous distance to a singer, and on this sum it is nearly nothing.

Nor does this argue that the wolf is imaginary. It argues that the penalty is carried in one pair, and that any argument for a split key resting on a roughness total will understate it. The figure’s buttons are the honest test. And the pitch name matters too: a triad with the wrong note also has the wrong function, since a player reaching for a D♯ to lead up to E is handed an E♭, a fifth of a tone sharp of the note wanted, and the semitone that should have been small is now too small to be a leading note’s step at all.

This also changes what a split was worth to the person paying for it. A patron who heard a divided key demonstrated did not hear a chord made seven per cent smoother. They heard a chord in which one pair of partials stopped fluctuating, in the register where the ear follows a third most closely, and a leading note that now did what a leading note is expected to. Both of those are things a roughness total underprices, and it should not be surprising if the value a sixteenth-century musician put on them was much greater than any figure here would suggest. The figure above can bound the acoustic difference from below, and no more than that.

The other way to get the same chords #

A keyboard does not have to be split to play more triads. It can be tempered less, so that the wolf shrinks. That is the line an earlier essay drew, and it can be read here for what it costs against the alternative.

Tempering less shrinks the wolf only by spoiling every other third. At a sixth of a comma the good major thirds are 7.2 cents wide and the wolf is still 26.7. At a tenth the wolf has gone, at 15.3, and so has anything that could be called a good third: the major ones are 12.9 cents wide. There is no point on that line that gets the eight missing triads at less than the cost of all the others, and this is exactly what a temperament cannot do proved for the sum of the twelve thirds. Each lever on a split key, by contrast, buys a real third at the full quarter-comma purity, and does so for exactly one extra major and minor triad. Tempering spends a little of every chord. Splitting spends a lever per chord and leaves the others alone.

That sets out the choice a builder had, in a form that a builder could have recognised. Pure thirds on a fixed set of chords, with more levers for more chords; or a tempered range across many chords, with all thirds suffering together. The well temperaments of the following century found a third way, with thirds of different sizes in different keys, and the character of those keys is a separate story, which trades in a different unit. So split keys belong to a repertoire that wanted its central keys pure and could live with a limited reach, and they lost out to one that wanted to modulate. Nothing here shows they should have lost.

What this chain leaves out #

A key is a later word. Sixteenth- and seventeenth-century theorists did not think in major and minor keys, and a builder ordering a keyboard was thinking of modes, finals and transpositions, of the cadences a choir might sing, of playing from a part at a pitch the singers could manage. The modern phrase “minor key on E” is a shorthand here for the triads that would be used at a cadence on that final with a raised leading note and a subdominant. On E, in particular, the traditional Phrygian cadence moves through F and does not use D♯ at all, so the demand for that accidental came from music treating E as something closer to a later minor. The figures say what is needed for a given set of triads. Whether that set is what the music of a given year and place wanted has not been counted, and it is exactly what would have to be counted.

Which accidentals were split is recorded here only as what the secondary literature reports as most common. The G♯/A♭ and D♯/E♭ pair is well attested on Italian and English instruments, including the Temple Church organ of the 1680s, but this essay has no census, and some keyboards split others or more. A table of surviving contracts by date and place would be the check, and the ordering of a builder’s priorities could then be read directly from it.

The temperament is one point on a line and the third is one spectrum. The chain arithmetic holds at any fraction of a comma. The roughness comparison uses one register, one spectrum and one spacing, and it treats a split as giving a note in tune with its neighbours. Real instruments of this period were tuned by ear, counted by beats, in temperaments only approximately regular, with reed and flue stops that would not stay together for long.

And the hand has been left out. A divided accidental is more than a lever. The keys are narrow, an organist plays by habit, and a piece written for fourteen notes must be played by someone who has learned where they are. The cost of one key is not one key, and it is the reason the thirty-one-note instrument had so few players. The arithmetic can state the benefit, and it cannot say what that benefit would have to be to pay for a change of technique.

Whose keyboards #

The instruments in question are keyboard instruments in regular meantone from about 1500 to about 1750, in Italy, Spain, the Low Countries and England particularly. Voices and viols do not face the problem at all, as the progression that never comes home reports: their notes are not fixed in advance. The whole calculation is a statement about what fixed pitches on a limited number of levers can do, and it says nothing about what the music is.

Still open: what the third key should have been #

Two splits are one on each side because the middle of the repertoire is symmetrical around D. A third split has to break that symmetry, and the arithmetic says it buys a new minor key on either F or B and one new major key on either A♭ or B. Which of those a builder should choose is a question about a repertoire, and it is countable: take a body of keyboard music of a given period and place, tally which triads and which cadences its pieces actually use at which transposition, and read off, for each number of levers from twelve to nineteen, the placing that plays the most of it as written. The computation would say whether the historical order of splits follows the music’s demands, and whether there is a date at which the music started demanding something no such placement could give.