What a tablature keeps

A stave note is a pitch and a time. It is not an instruction: it says what should come out, and leaves the player to work out how.

A tablature is the other thing. It says which string and which fret, or which hole to cover, or which valve to press, and it says nothing at all about what the result sounds like — a lute tablature read on an instrument in a different tuning produces different music, correctly, because the tablature was never about the pitches.

The usual account of this is that tablature is the cruder of the two: a beginner’s notation, tied to one instrument, superseded. That account is wrong in an interesting way, and the way to see it is to take one written note and ask what the stave failed to say about it.

Four places for one note #

Four positions, and the choice between them is not arbitrary — every guitarist has opinions about it, and the opinions are about sound rather than convenience. What follows is where the sound difference comes from.

The string gets shorter and the hand does not move #

The speaking length of a fretted string is the scale length divided by two to the fret over twelve. At the first fret that is 61.2 centimetres; at the fifteenth it is 27.2. The string has lost more than half its length.

The player’s right hand has not moved. It is where it always is — somewhere over the soundhole or between it and the bridge, a fixed distance from the bridge because the bridge is fixed.

So the fraction of the string being plucked changes, and it changes a great deal: from 0.196 of the way along at the first fret to 0.440 at the fifteenth, which is very nearly the middle.

That fraction is the only thing that decides which partials a pluck excites. A string plucked at a point cannot excite any mode that has a node there — the same argument that puts a piano’s hammer at a seventh or an eighth of the string — so the pluck imposes a comb, and the comb’s teeth are at multiples of the reciprocal of the fraction.

Losing the second partial is not a subtlety. It is the octave, it is normally the strongest thing in the sound after the fundamental, and taking it out is what produces the particular hollow quality of a note played high on a low string. The second partial is 7.8 decibels below the fundamental at the first fret and 20.6 below it at the fifteenth: a difference of nearly thirteen decibels in one partial, between two ways of playing the same written note.

Seventeen per cent in a centroid is a smaller number than thirteen decibels in a partial, and both are describing the same thing. The centroid is a summary and the comb is the structure; a listener hears the structure.

Which is why players choose #

The practical consequence is a body of technique that has no representation on a stave at all.

This is the same decision a keyboard maker made once, permanently, when they fixed the striking point at a seventh or an eighth: the piano’s hammer position is chosen and then unchangeable, so its spectrum is a property of the instrument, while a fingerboard hands the same parameter to the player and hands it to them implicitly. One instrument decided the question at the factory and the other decides it every note.

A phrase played across the strings — each note on a new string, low on the neck — is bright and even, and every note has the same relationship between its length and the plucking point. The same phrase played along one string is progressively duller as it climbs, and the timbre is a continuous gradient rather than a set of similar sounds. Both are correct readings of the same notation, and a player choosing between them is choosing something the notation cannot express.

There is a second consequence, smaller but exactly measurable, and it is a pitch difference rather than a timbre one.

Fretting a string stretches it slightly, because pressing it to the fret makes it longer, and the effect is larger where the string is shorter. So the four positions do not even produce the same pitch: the model here puts them within one cent of each other on an uncompensated instrument, and once the saddle is set back by the millimetre and a half a real guitar uses, the spread opens to four and a third cents — because compensation flattens a short sounding length more than a long one, and the four positions have very different lengths.

What fretting alone does to the pitch, before any compensation, is stretch every stopped note sharp — pressing a string to a fret raises its tension — and the error grows up the neck, which is one of the reasons a guitar cannot be in tune in the first place. That is a fact about the instrument and not about either notation, and every guitar’s saddle is set back to absorb it.

What the tablature cannot say #

The symmetry is the point, so it has to be shown in the other direction too.

A tablature specifies the action exactly and the sound not at all. Read it on an instrument tuned a tone lower and it produces the same fingering and different music. Read a lute tablature — where the tuning is a renaissance tuning, and the courses may be doubled at the octave — on a modern guitar, and what comes out is not the piece.

That is not a defect either. It is what the notation is for: a tablature is written for an instrument, and the instrument’s tuning is part of the reading. Scordatura, where a composer retunes a string for one piece, is trivial in tablature and requires a written explanation on a stave.

So the two notations partition the same information differently, and neither contains the other:

The stave keeps the pitch and discards the action. It survives transposition, it survives being played on another instrument, and it cannot say which of four sounds was meant.

The tablature keeps the action and discards the pitch. It survives a retuning, it says exactly which of the four sounds was meant, and it cannot be read by anybody holding a different instrument.

And what each one keeps is what its tradition thought the music was. Lute music of the sixteenth century is written in tablature not because its composers were unable to write on a staff — many of them wrote vocal music on one — but because the piece is a sequence of hand positions on an instrument whose sound depends on them. Vocal polyphony is written on staves because the piece is a set of lines, and a line is a sequence of pitches that any voice in the right range can carry. The stave’s abstraction is what lets a melody be recognised across every transformation that leaves its shape alone; the tablature’s concreteness is what lets a lutenist reproduce a sound. Neither abstraction is free and each is paid for in the other’s currency.

The instruments that have both problems #

The clearest test of the argument is an instrument where the two notations are both used and disagree.

A woodwind’s fingering chart is a tablature by the same definition — and one hole does a dozen jobs on a modern instrument, so the chart is longer than the fingers are: it records which holes are closed rather than which pitch results, and where two fingerings give one note it records that there are two. The stave records the note and cannot say which fingering — the identical division of labour, one instrument family along.

And on instruments whose partials are not a harmonic series at all — a drum, or a bell, which has no fundamental to be in tune with — a tablature has nothing to choose between, because there is no second place to make the same pitch.

Brass is the extreme case. A note on a natural trumpet has one fingering because it has no valves, and a note on a valved trumpet often has three or four; the alternatives differ in intonation by measurable amounts and in timbre by rather less. And the clarino register, where a natural instrument can play a melody at all, is defined entirely by which partials are close enough together — a fact about the tube that no notation of either kind contains.

A keyboard is the instrument on which none of this happens: one key, one pitch, one timbre, no choice — which is why keyboard music has never needed a tablature and why the stave, which was built for it, is silent about a decision that instrument does not have.

Put the other way round: the size of a tablature’s advantage is the number of ways an instrument can make one note. It is zero on a keyboard, four on a guitar’s middle C, and unbounded on a family of instruments where the tuning itself is part of the piece.

How typical the opening example is #

That formulation names a quantity, and a quantity named is a quantity that can be evaluated over the whole instrument rather than at the one note the essay opened with. Running the same comb arithmetic over every pitch a guitar can reach, and taking for each the spread in the second partial between its best and worst position, gives the distribution the middle C example was drawn from.

It was drawn from the top of it. Within fifteen frets, thirty of the forty reachable pitches have more than one position, their second-partial spreads run from under a decibel to thirteen, the median is 6.0 decibels — and middle C, at 12.8, ranks second of the thirty. Exactly one pitch on the instrument, the G below it, offers a larger timbral choice. The essay opened on very nearly the best case the fingerboard has.

That is worth saying plainly because it changes what the thirteen decibels is evidence for. It is a real measurement of a real note and it is not a typical note; the typical note with a choice offers about half as much, and the quietest quarter of them offer two decibels or less, which is at or under what a listener would reliably hear as a difference in balance at all. A tablature’s advantage is real and it is unevenly distributed, and averaged over the instrument it is a good deal smaller than one example suggests.

The reach matters more than the pitch does. Restricting to the first twelve frets — which is most playing on most instruments — no pitch has four positions at all, the maximum is three, and the median spread falls from 6.0 decibels to 3.5. Middle C within twelve frets is rank twelve of twenty-seven: unremarkable. What produces the large numbers is specifically the availability of a position high on a low string, where β approaches 0.44 and the second partial is nearly extinguished, and that position is the one a player is least likely to use for reasons — intonation, left-hand comfort, the dullness itself — that the notation does not record either.

The hand against the fret #

There is a control implied in the caveats below and it is worth running rather than asserting, because the answer turns out to depend on where along the neck the note is stopped.

The claim that needs testing is that moving the plucking hand changes β by more than changing the fret does — which, if it held everywhere, would make the whole effect above a second-order one hiding under a first-order one the player controls directly.

Take the four positions of middle C and, at each, sweep the hand over a realistic range of eight to twenty centimetres from the bridge. Against the fret choice’s 17 per cent of centroid movement with the hand held at twelve:

The two effects trade places along the neck, and the mechanism is the essay’s own. β is a distance divided by a length, so a fixed range of hand positions in centimetres covers a wide range of β on a long string and a narrow one on a short string. Low on the neck the hand has more authority than the fret does and the caveat is right. High on the neck the string is too short for the hand to reach the fractions that matter — at the fifteenth fret, moving from eight centimetres to twenty takes β from 0.29 to 0.44, all of it in the region where the second partial is already dying — and the fret choice wins.

So the honest version of the caveat is that the hand dominates exactly where the fret effect is weakest, and gives way exactly where it is strongest. The player who most needs the option of choosing a position is the one playing high on a low string, and that is the player whose right hand can do least about it.

Which computation produced the numbers #

The speaking length is the scale length times two to the minus fret over twelve, which is the equal-tempered fret rule and is what the frets on the instrument were positioned by. The plucking fraction is a fixed twelve centimetres from the bridge divided by that length; twelve centimetres is a stated typical hand position and every number scales with it.

The comb is the standard plucked-string spectrum: amplitude proportional to the sine of n·π·β over n squared, where β is the fraction. Its zeros are at every n that is a multiple of one over β, which is where the “loses partials 2, 5, 7” figures come from — 1/0.44 is 2.27, so the nulls fall nearest partials 2, 5, 7 and 9.

The spectral centroid is the power-weighted mean partial over the first twenty partials. Truncating at twenty rather than at forty changes the absolute numbers and not the ordering, because the high partials carry little power in a 1/n² spectrum.

The fret-stretching figures come from the same model the tuning ladder uses: a stated action height, the string’s extension under it, and a stiffness that turns a fractional stretch into a hundred times as much tension. The four-and-a-third-cent spread is the same four positions evaluated with a saddle setback of 1.5 millimetres — a typical value, stated, not fitted.

The count of thirty-seven pitches and twenty-seven with alternatives is an enumeration over the standard tuning and the first twelve frets, not an estimate.

The distribution over the whole instrument is that same enumeration carried one step further: for each reachable pitch, every position it has is scored with the same comb, and the figure kept is the difference in the second partial’s level between the brightest and dullest of them. It is reported at both twelve and fifteen frets because the two answer different questions — twelve is roughly where most playing happens and fifteen is the reach the opening figure uses — and the two medians differ by a factor of nearly two, which is the reason for stating the reach every time a number like this is quoted.

The hand-against-fret comparison holds the pitch and the position fixed and sweeps the plucking distance from eight to twenty centimetres, which is about as far as a right hand travels on a classical guitar without changing technique. The range is stated rather than fitted, and widening or narrowing it moves every row in the table in the same direction, so the crossing point between the two effects moves while the ordering along the neck does not.

What the picture cannot show #

It has no body in it. Everything here is the string’s own spectrum, and what a listener hears has been through the instrument’s resonances, which are strongly frequency-dependent and change the balance again. The body is the filter, and it is applied after the comb, so a partial the comb removed cannot be restored by it — but one it left in may be attenuated to nothing.

It has one plucking distance. A player moves their hand. Playing near the bridge is a deliberate effect with its own name on every plucked instrument, and it changes β by more than the fret does. The claim here is that the fret changes β even when the hand does not move, which is the part that is invisible.

It ignores the finger. A nail, a plectrum and a fingertip release the string differently and impose their own low-pass on top of the comb, and the difference between them is larger than the difference between two positions.

It says nothing about the left hand. Vibrato, the pressure of the stopping finger, and whether the note is stopped or open all change the sound at least as much as the plucking fraction, and the open string is a categorically different object — it has no finger damping it and rings far longer, which is why guitarists work so hard to hide the difference.

And it is a model of a pluck, not a recording of one. No measurement here comes from an instrument. The comb is exact for an ideal flexible string released from a triangular displacement, and every real correction — stiffness, the finger’s width, the string’s damping — softens the nulls rather than moving them.

The ladder from here #

This rung took a notation that names actions and one that names sounds and found neither contains the other. The last rung of the ladder stays with the same shape of question in harmony, where there are three notations in daily professional use for the same object — a figured bass, a Roman numeral and a chord symbol — and the number of four-part realisations each of them admits can be counted exactly. The three counts differ by four orders of magnitude, and each notation’s number says what its tradition thought a chord was.