The clef is an integer

The first rung of this ladder found the thing about the staff that every reader knows and nobody says: its vertical axis is not a pitch axis. It counts letters. Seven positions carry twelve pitches, so the same vertical distance is two different intervals before an accidental is allowed and six after, and an accidental is not an extra symbol on a complete scale but the repair for a scale with five values missing.

It named the clef and left it owing, and the sixth rung named it again as one of two things the ladder does not have.

The debt turns out to be small, and paying it makes the staff look like a worse instrument than it did.

A clef says one number #

A clef does exactly one thing: it names the letter that sits on one of the five lines. Everything else follows, because the axis counts letters and the positions are therefore a ladder of consecutive letters with no gaps in it.

So a clef is an integer. There are eight in European practice and their content is entirely in where they put their line:

Read the right-hand column and the pattern is not subtle. Each clef sits exactly two positions — one third — below the one above it. Eight clefs, a third apart, covering fifteen positions of offset from the lowest to the highest.

That is a lattice rather than a list, and what its spacing is worth turns out to be very little.

Take the eight ranges this rung uses — four voices at thirteen positions and four instruments at twenty-three to twenty-five — slide each across every offset, and let each pick its best clef from a lattice at a stated spacing:

The total overflow does not depend on the spacing at all across the first three rows, and the reason is worth having: every one of these ranges is wider than the staff, so it overflows at both ends whatever clef is chosen, and shifting the window by one position adds a position at one end and removes one at the other. The ledger lines a reader actually draws are fixed by the range and the staff, and the clef cannot change how many there are.

What the clef changes is how they are distributed, and that is the last column. A lattice of seconds balances the two ends to within 3.88 positions on the worse side, a lattice of thirds to 4.25, a lattice of fourths to 4.46. Doubling the number of clefs from eight to sixteen buys 0.37 of a staff position — a third of a ledger line — and halving them from eight to six costs 0.21.

So the choice of a third is not a fine optimum; it is the middle of a flat region, and the whole range of defensible spacings differs by less than one position of ledger line. That is a good reason for a convention to settle on something round and a poor reason to admire it, and it means the eight clefs are eight because sixteen would be a memory burden rather than because eight is the number the arithmetic wants.

The rest of this rung is about what the lattice can and cannot do with that spacing.

Eleven positions, and nothing is eleven wide #

The staff has five lines and four spaces between them, which is nine positions, plus the space immediately below the bottom line and above the top, which makes eleven.

Eleven diatonic positions is an eleventh — a compound fourth. A note eleven letters above another is a fourth plus an octave.

No voice is that narrow and no instrument is remotely that narrow. Four voices is the number the exercise is in and each of those four overflows the staff on its own. This site’s own SATB compass figures give the four voices spans of twelve to thirteen positions, so every one of them overflows the staff. The instruments are far worse: a viola’s range is twenty-four positions, a cello’s twenty-four, a bassoon’s twenty-five.

So the question “which clef fits this instrument” has no answer, because none of them does. The question a clef answers is which end sticks out.

Each voice’s own clef is the right one #

The overflow can be minimised, and the answer is close enough to the practice to be worth stating as a result.

For the bass voice the best clef is the bass clef, leaving one position off the staff. For the tenor voice it is the tenor clef, one position off. For the soprano voice it is the treble clef, one position off.

Three of the four clefs named after voices are, by this arithmetic, the best available clef for the voice they are named after. That is not a tautology — the naming is historical and the arithmetic is not — and the alternative clefs are much worse: the French violin clef leaves fifteen positions of a bass part off the staff.

The alto voice is the exception and the exception is informative. For the range this collection uses — F3 to D5 — the best clef is the mezzo-soprano, at one position off, and the alto clef leaves three. The alto clef is used for the alto voice in practice, and the arithmetic prefers its neighbour.

The reason is the lattice’s own granularity. The clefs are a third apart, so a range that sits a third higher than the clef was designed for changes the answer, and an alto range topping out at D5 rather than at B4 is exactly a third higher. The sensitivity of the answer and the spacing of the clefs are the same quantity, which means the arithmetic cannot distinguish between “the alto clef is wrong for altos” and “this collection’s alto range is a third too high”. It is almost certainly the second.

The instruments cannot be done with one clef at all #

For a twenty-four-position range the best single clef leaves seven positions off the staff, and seven positions is three ledger lines.

That is not a workable notation, and the practice is not to attempt it. A viola part changes between the alto clef and the treble; a cello part between bass, tenor and treble; a bassoon part between bass and tenor; a trombone part between bass, tenor and alto. Every instrument whose range is wide enough to make one clef untenable is an instrument notated in more than one clef, and the instruments notated in a single clef throughout — the violin, the flute — are the ones whose usual range, as opposed to their extreme one, is narrow enough at one end for ledger lines to do the rest.

So a clef change is not a convenience. It is the mechanism by which a five-line staff is used for a range it cannot hold, and the reason the C clefs survived into modern practice at all is that they are the middle of the lattice, where the instruments with the widest ranges live.

That is what it costs to make the vertical axis an integer. A distance on the page is a count of letters, and the number of semitones it stands for is not determined by it — so the axis is not a ruler, and the accidental is what repairs the gap between the two.

What the third between clefs is worth #

The lattice’s spacing is the design decision, and it can be priced.

Take four hypothetical clef sets: clefs spaced one position apart (fifteen of them), two apart (eight, which is the real set), three apart (five) and four apart (four). For each, ask what the worst overflow is after choosing the best available clef, over every position a range of a given width could sit at.

Eight clefs a third apart get everything fifteen clefs would get, on any range wide enough to matter. The reason is arithmetic: once a range is wider than the staff by more than the clef spacing, the overflow is set by the range’s width rather than by how finely the clef can be positioned, and a finer lattice has nothing left to improve.

And going coarser costs immediately. Five clefs a fourth apart lose two positions on the wide ranges and two on the narrow ones.

So the third is not a convention. It is the coarsest spacing that gives up nothing, which is the same shape of argument this collection has made about the eleventh partial’s tolerance and about where a piano’s hammers strike: a design decision that looks like taste and turns out to be the corner of a trade-off.

The transposing instruments are the same arithmetic #

There is one more device in the practice that the integer model handles without extension, and it is worth pointing at because it is usually taught as a separate mystery.

A transposing instrument’s part is written in a clef and moved by a fixed interval. A B♭ clarinet’s part is written a major second above what it sounds; an E♭ alto saxophone’s a major sixth plus an octave. The reason given is fingering — the same written note means the same fingering across a family — and that reason is true.

The arithmetic adds a second one. A transposition is another integer on the same axis, so a transposing part is a clef choice with more resolution than the lattice offers: the eight clefs step in thirds, and a written transposition can step by any interval at all. An instrument whose range sits awkwardly between two clefs can be moved instead of reclefed.

Which is exactly the distribution in the practice. A clarinet covering nineteen semitones before it can overblow is the reason its family needs the device at all. The instruments notated in a single clef with a transposition — clarinets, saxophones, horns, trumpets — have ranges of twenty to twenty-two positions and could not be fitted by any clef; the instruments notated at sounding pitch in two or three clefs — cello, bassoon, trombone, viola — have ranges of twenty-three to twenty-five and are fitted by changing clef. The two devices are alternatives for the same problem and the practice uses one or the other rather than both, which is what a reader would predict from the arithmetic and is not what the pedagogy says.

Which computation produced the numbers #

One conversion and three comparisons, none of them approximate.

A pitch’s diatonic index is how many letters it sits above C0: the octave number times seven, plus the index of the letter at or below its pitch class. That is the coordinate the staff actually uses, and converting to it is the only step in this essay that touches semitones at all.

A clef’s eleven positions follow from its named pitch and its line: line n is two positions above line one, so the bottom of the staff is the named pitch less twice the line number less one, and the eleven positions run from one below the bottom line to one above the top.

The overflow of a range under a clef is the larger of how far the range’s bottom falls below the staff and how far its top rises above it. The larger rather than the sum, because a reader’s difficulty is set by the end that runs furthest off rather than by the total — three ledger lines at the top and none at the bottom is harder than one at each.

The lattice comparison sweeps a range of a given width across eleven consecutive starting positions and reports the worst case, so no set is being flattered by a range that happens to sit where one of its clefs is centred.

The ranges are the one place a choice was made. The four voices are this site’s own SATB_RANGES, used unchanged by every part-writing figure here. The four instruments are stated full compasses rather than practical ones, which makes their overflows the largest an honest figure can report — a viola part that stays inside its usual working range needs fewer ledger lines than the number above.

Whose music, and when #

The eight clefs are European and the set was larger before it was smaller. Fifteenth- and sixteenth-century vocal partbooks used all of them and more, choosing per part so that each singer’s line sat inside the staff — which is exactly the optimisation above, carried out by scribes, one part at a time. The set contracted through the eighteenth and nineteenth centuries to the four in use now, and the three that survive as more than curiosities are treble, bass and alto with tenor for cello and trombone.

The contraction has a cost and it is the one the arithmetic predicts. A modern soprano part in treble clef and a modern bass part in bass clef both sit comfortably, because those two clefs are the ends of the lattice; a modern alto part in treble clef sits three positions high, and the practice absorbs that with ledger lines and with the octave-transposing treble clef, which is a ninth clef that says its number is seven lower than it looks.

What a tablature keeps is the other half of this: a notation that fixes the action rather than the pitch has no clef problem at all, because it never had a pitch axis to run out of.

That last device is worth naming because it breaks the model. An octave-transposing clef is not an offset on the letter axis; it is an offset plus a lie about which octave, and it exists because the lattice ran out at the bottom for tenor voices. The lattice has eight positions and the music needed nine.

What the picture cannot show #

Ledger lines are counted as positions, not as difficulty. Two positions off the staff is one ledger line and reads easily; seven is three ledger lines and does not, and the relation between the count and the reading effort is not linear and is not measured here.

A range is not a distribution. An instrument’s compass includes notes it rarely plays, and the right optimisation weights positions by how often they occur — which would move every answer toward the middle of the tessitura and would need a corpus.

The alto result is a range disagreement and is reported as one. With an alto compass a third lower the alto clef wins, and this collection’s compass figures are not authoritative about voices.

And nothing here is about reading. Whether a reader finds one clef easier than another for reasons of habit rather than of geometry is the whole of why the C clefs are unpopular, and it is not a quantity any of this touches.

Where this ladder goes next #

Seven rungs, one in each of six fields and now a second in one of them, and what all seven have found is the same shape: a notation fixes some coordinates of a musical object and leaves the rest to be supplied, and which ones it fixes is a claim about what the object is. The staff fixes the letter and leaves the pitch; the signature fixes the bar and leaves the grouping; the dynamic mark fixes an order and leaves the spectrum; the tablature fixes the action and leaves the sound; three notations fix three different things about one chord; and the clef fixes where the eleven positions sit and leaves everything outside them to ledger lines.

What the ladder still owes is the rung the sixth named beside the clef: the notations invented for music this system cannot hold. Graphic scores, chromatic staves, the tablatures written for instruments that did not exist when the staff was designed. The arithmetic above is the way in, because it says precisely what the staff’s capacity is — eleven letters — and every one of those inventions is a response to a specific overflow. A chromatic staff is an axis with twelve positions per octave instead of seven; the trade it makes is computable from the same integers, and it is the trade every proposal since the seventeenth century has failed to win. Two names for one key is why: the seven letters are not a lossy compression of the twelve but a different object, a window on a chain of fifths, and a chromatic staff throws that away to buy the eleven positions this essay has been counting.