The notehead that is not a note

The fifteenth rung prices a rest as an attribute of the note before it: a binary decision plus a length, 0.67 bits where a tenth of the notes carry one. It ends by naming the object that does not fit that shape.

A tie is not an attribute. It is a second notehead, drawn on the staff, with a pitch and a position of its own, joined to the first by a slur — and a reader who plays it has made an error. Every quantity so far is charged per notehead, and a tied continuation is the one notehead in the notation that carries no event at all.

Charged twice, and the second charge is the larger one #

The decision is the obvious cost and it is the smaller one. Is this notehead an event or a continuation? — a binary question per notehead, worth its own entropy, 0.52 bits a sounded note where a tenth of the noteheads are continuations.

The other cost is what the notation has already spent by the time the question is answered. A tied continuation is drawn like any other notehead: it sits on a line or a space, at a place in the bar, and a reader takes both in before the slur tells them the notehead is not a note. On the fifteenth rung’s own accounting that is 4.79 bits of reading per notehead, and a tenth of the noteheads being continuations means one extra notehead for every nine sounded ones.

Multiplied out, the extra reading is 0.53 bits a sounded note, and the total is 1.05 — twice the decision alone, and a fifth of what a note of music costs to read.

That is a large number for a mark nobody thinks of as difficult. It is larger than the whole duration term the previous rung computed, larger than the difference between reading a scale and reading a leaping line, and it is charged on every page of orchestral music ever written.

Why the cost cannot be avoided by reading better #

A reasonable objection is that a trained reader sees the slur first and never reads the second notehead properly at all. The objection is probably right about what happens and it does not remove the cost.

The slur is drawn between two noteheads, so it cannot be seen before both of them are located: what the eye needs in order to know that the second notehead is tied is the second notehead’s position. So the position is read whatever happens, and only the pitch could be skipped — and the pitch of a tied continuation is the same as the pitch of the note before it, which means a reader who reads it and a reader who infers it arrive at the same place by different routes.

Charging the whole 4.79 is therefore an upper bound and charging the position alone — 2.23 — is a lower one. The tie’s cost lies between 0.77 and 1.05 bits a note at a tenth, and the conclusion that it exceeds the decision that identifies it holds at either end.

Three notations for one lengthened note #

The staff has three ways to make a note longer than the value it is written at, and they can be priced against each other.

At a tenth the three are 0.52, 0.10 and 0.26 bits, and they do not change places anywhere on the axis. The dot is the cheapest of the three at every density, by a factor of five over the tie.

Which raises a question the ranking itself cannot answer. If a dot is five times cheaper, why does a staff have ties at all?

The general mechanism is the expensive one #

Because a dot cannot do what a tie does, and the ways it cannot are exact.

A dot multiplies a value by three halves, and nothing else. A note lasting five quavers cannot be dotted; a note lasting seven can be double-dotted and one lasting nine cannot. The dot reaches the durations that are a value plus half a value and no others, which is a sparse subset of the durations music writes.

A longer note value reaches the powers of two and their dotted forms, which is a larger subset and still a subset.

A tie reaches everything, including — and this is the one that decides it — durations that cross a barline. A bar line is not a mark a reader can ignore: it is an assertion about where the accents are, and every note value on the staff is defined relative to a bar that the value must fit inside. A note lasting from the fourth beat of one bar into the second of the next has no symbol, and there is no symbol that could be invented for it without abandoning the bar.

So the notation’s hierarchy is a cost ordering with a coverage ordering running against it, and the staff uses each device exactly where the cheaper one will not reach. The general mechanism is the expensive one, which is the shape of nearly every device a staff has: the ledger line, the octave sign, the accidental are all the expensive general answer to a case the cheap specific one cannot express. The clef is the same object one level up — a cheap way of moving the whole axis, used because moving it note by note is the expensive general answer.

What the extra noteheads do to a page rather than to a note #

There is a second reading of the same arithmetic that a copyist would reach for first, and it is about the page rather than the reader.

A repertoire tying a tenth of its noteheads writes 1.111 noteheads for every note it sounds. That is eleven per cent more marks on the paper, eleven per cent more horizontal space at any given spacing rule, and — on the tenth rung’s own accounting of what a page holds — eleven per cent fewer bars to a system.

So a tie costs twice on the reading axis and once more on the page, and the third cost is the one that has an economic history behind it. Paper was expensive, engraving was expensive, and the notational conventions that survived are the ones that fitted more music into less space — which is an argument against ties and in favour of dots, running in the same direction as the reading cost.

That the convention has drifted toward more ties since paper stopped being expensive is consistent with that and is not evidence for it. It is the kind of correlation a historian would want three more centuries of to believe.

The barline is what makes the tie necessary and expensive at once #

There is a circularity here worth drawing out, because it says something about the notation as a design rather than about any one mark.

The tie exists because the barline exists. A notation without bars — a chant notation, a proportional notation, a tablature with no metre in it — has no need for a mark that joins a duration across a boundary, because there is no boundary. So the barline creates the tie, and the tie is the most expensive mark on the staff per occurrence.

And the barline is what makes the position axis cheap. Without it every point in time is equally likely and a note’s position costs the full three bits of a quaver grid; with it the metrical hierarchy makes the downbeat far more likely than the offbeat, and the measured cost of a note’s position drops accordingly.

So the barline buys a saving on every note and pays for it with a tie on a few, which is the trade a notation makes rather than a defect in it. The saving is 2.23 bits a note against the three a flat grid would cost; the cost is 1.05 bits a note at a tenth tie density. On these numbers the barline is worth having and the margin is not enormous.

That is a strange sentence to be able to write about a device nobody has questioned since 1600, and the honest reading of it is that the numbers are within their own uncertainty of each other rather than that the barline is marginal.

The one place the second charge is not paid #

There is an exception to the whole argument and it is common enough to matter.

A tie within a bar is very often written where a longer value or a dot would have done — a crotchet tied to a quaver rather than a dotted crotchet — and engraving conventions differ about when. Those ties are optional and are the ones the arithmetic says are expensive for nothing.

A tie across a barline is not optional and never was. There is no symbol for it and no symbol could be added without abandoning the bar, so its cost is the price of the notation’s central device rather than an engraver’s choice.

Counted separately, the two would divide the tie’s total cost into a part that is a design necessity and a part that is a habit — and the second is removable by an engraver with no change to the notation at all. How much of a page’s tie cost is optional is a question about engraving practice, and it is the one number in this rung that somebody could act on.

The direction of the answer is guessable and its size is not. Conventions since the nineteenth century have moved toward writing more ties and fewer dots, on the argument that a tie shows the beat structure and a dot obscures it — which is a claim that a tie buys something on the position axis to pay for what it costs on the others, and is exactly the trade this accounting is set up to price and has not.

What the pictures cannot show #

The tie share is a parameter and not a measurement, exactly as the rest share was on the previous rung and for the same reason: this collection’s corpus has no ties in it, because its format cannot express one. A tenth is chosen as a plausible orchestral figure and nothing here measured it.

The three notations are priced with three different models and the models are not commensurable in the way the drawing implies. A tie’s cost is derived — an entropy plus a notehead’s reading load. A dot’s is asserted: one binary symbol, one bit. A longer value’s is asserted too: a choice among six shapes, log₂ 6 bits. Two of the three numbers are stipulations, and while their ordering is not sensitive to the stipulations, their ratios are entirely.

The barline comparison at the end compares a saving measured on three tunes against a cost computed from an assumed density, which is the weakest arithmetic on the page and is reported because leaving it out would be worse.

The comparison between the three notations also assumes the reader is decoding each mark from scratch, and a trained reader is not: a dotted crotchet is a single learned shape, and so is a crotchet tied to a quaver. The twelfth rung’s whole result is that a reader reads figures rather than notes, so a durational pattern that recurs is chunked exactly as a scale fragment is, and its cost falls to whatever it costs to recognise one chunk. That applies to all three notations and it applies most to the one a reader meets most often, which is a self-reinforcing advantage the arithmetic has no term for at all.

And nothing here is about performance. A tie is one of the marks sight-readers most often miss, which is a fact about errors and not about information, and this ladder has no error model anywhere in it.

What a tie is for, which the accounting cannot see #

Everything above prices a tie as a cost, and a notation does not add marks that only cost. It is worth stating what a tie supplies that the arithmetic has no term for.

A tie makes the beat structure visible. A dotted crotchet spanning the second and third beats of a bar of four writes one symbol across a strong beat; the same duration written as two tied crotchets puts a notehead exactly on the third beat, where the metrical hierarchy says a reader is looking. So a tie tells the eye where the beats are, and a dot hides one.

That is a benefit on the position axis and it is invisible to this accounting for a specific reason: the position axis is measured over where onsets fall, and a tied continuation is not an onset. A measure that counted noteheads rather than events would find that a tie reduces the position term, because it puts a notehead in the most predictable place a bar has.

So the tie’s two charges have a credit against them that this ladder’s own measure is constructed not to see. Whether the credit is larger than the charge is a question about which of two accountings is the right one, and the honest answer is that it depends on what a reader is doing — a sight-reader tracking beats wants the notehead, and a reader decoding durations does not.

That ambiguity is the reason engravers argue about ties and do not argue about clefs.

Whose notation this is about #

The staff as it has been since about 1600: bars, note values as powers of two, dots, and ties. The claim is narrow and it is a claim about that design rather than about music.

What the arithmetic supports is the ordering — dot, value, tie — and the reason for it, which is that coverage and cost run opposite ways. What it does not support is any of the individual numbers, and the number an engraver would want is the one nobody has: how many ties a page actually has.

That is a corpus measurement of the plainest kind and it needs no listener, no analyser and no model. It is the same measurement the fifteenth rung asked for about rests, and both would come from the same source: a few hundred pages of engraved music, counted. Two numbers, one afternoon, and they would turn the last two rungs from parametric statements into measurements — which is the cheapest outstanding debt on this anchor and has been since the ladder started charging things per note.

Where this ladder goes next #

Sixteen rungs. The stave is not a ruler; two names for one key; the time signature is a claim; a mark that is not a level; what a tablature keeps; three notations for one progression; the clef is an integer; the notations invented for the overflow; the axis that is not a time axis; the two axes multiplied; the eye that has to cross them; what is being crossed is not uniform; what is above and below it is not independent; when is as much to read as what; how long it lasts, which the corpus could not hold; and now the mark that puts a notehead on the page carrying nothing.

Everything above adds. The pitch term plus the position term plus the duration term plus the tie term is a page’s cost, and the addition assumes the four are independent — which the fourteenth rung named as the interaction it could not reach.

It can be reached, and on the corpus this ladder already has. A tune does not put its leaps and its offbeats at random with respect to each other, so a reader who has seen where a note falls already knows something about how far it moved. The quantity is the mutual information between the two axes, it is measurable on the same hundred and one notes every rung here uses, and what it says is how much every reading load on this anchor has been over-charging.