Four parts are easier to read than two

A reader does not read notes turned a note into a number of bits — minus the log of the probability of the interval that reached it — and found that eight notes of a leaping line are four times as much to read as eight notes of a scale, at the same width on the page. It ended by naming what it had not done:

Everything here is one line of one voice, and a reader of a score is taking in several at once — which multiplies the load and does not multiply it by the number of staves, because the parts are related.

Everybody who plays a keyboard knows the direction of the answer. A four-part chorale is easy to read and a two-part invention is not, and the chorale has twice as many notes on the page.

Two parts cannot state a chord #

The measurement turns on a fact established elsewhere and it is the reason the answer is a discontinuity rather than a curve.

Why the exercise is in four parts varied the voice count over one choir’s compass and found that the family starts at three: a triad has three pitch classes, and a realisation that drops one is not a cheaper realisation of the chord but a realisation of a different chord. Under that rule a duet has no complete voicing of any triad at all. Not few. None.

That rung drew the consequence for the rules of part-writing — two-part treatises talk about intervals and four-part ones talk about chords. The consequence for a reader is larger and it is the whole of this argument.

A reader of a texture that states its chords is not reading independent lines. Given the harmony and the previous sonority, the next one is nearly determined: the parts must lie in their compasses, the chord must be complete, the two parallel prohibitions must be avoided, and no voice may leap far. What is left to read off the page is the choice among whatever survives all that, and it is a small number.

A reader of a texture too thin to state its chords has no such help. The second line is a melody, and a melody costs what the twelfth rung says a melody costs.

So the two textures are priced in different currencies, and that is the finding rather than a defect of the method.

The comparison would be unfair if a two-part texture could be given the harmonic reading and were being denied it. It cannot. Allow two voices to take any pitch classes from the chord without stating all of them, and the count of reachable continuations does fall to about two — but what has been counted is no longer the reader’s problem, because a reader given two notes cannot infer which chord they belong to and therefore cannot use the count. The information is only available to somebody who already knows the harmony, and two parts are exactly the case in which nobody does. The thin texture is charged as two melodies because two melodies is what is on the page.

What is left to read, chord by chord #

The gap between the two bars is the measurement. A chord of this progression admits about a hundred and sixty complete voicings inside the ranges, which is what the notation leaves open. Once the previous sonority is on the page and the voices are not allowed to leap, between four and eight of them remain.

That is 2.39 bits a chord, for four noteheads. Add the cost of knowing which chord it is — charged here at the worst case, one of seven diatonic degrees chosen with no preference at all, which is 2.81 bits — and the whole event costs 5.20 bits for four notes, or 1.30 bits a notehead.

The same progression read as independent melodies is 1.89 bits a note under the twelfth rung’s own measure, whatever the part count. Two of them is 3.79 bits for two notes.

So a four-part chorale delivers twice as many parts for 37 per cent more information, and each of its noteheads is worth 69 per cent of a notehead of an invention. The direction every keyboard player reports comes out of the arithmetic, and the size of it is a third.

The cost per notehead keeps falling as parts are added: 1.30 bits at four, 1.21 at five, 0.90 at six. Two things push it down together. The chord has to be named once however many parts sound it, so that 2.81 bits is divided among more noteheads each time; and a thicker texture is more constrained, because every extra voice is another compass to fit inside and another pair of parts that must not move in parallel. A six-part texture is nearly free to read a notehead at a time and is close to unwritable, which is the same trade the voice-leading ladder priced from the writer’s end.

The four-part figure is an upper bound and the two-part figure is not. The 2.81 bits for naming the chord assumes a reader with no expectations about which chord follows which, and this collection has a whole ladder saying otherwise. A reader who knows that a dominant is usually followed by a tonic pays less than 2.81, and the melodic figure has no such slack in it: 1.89 bits is already the measured distribution of the intervals.

It is the voice leading, and it can be spent #

The obvious objection is that the comparison has been rigged by choosing a well-behaved chorale, and the objection is exactly right. It is also the second finding.

A four-part texture allowed to move each voice by up to twelve semitones costs 2.41 bits a notehead, which is worse than reading four independent melodies would be. The advantage is not a property of having four parts. It is a property of having four parts that move by small amounts, and it is spent as soon as they do not.

The crossing is at about four and a half semitones a voice a chord. The cheapest realisation of this progression that obeys both prohibitions moves its four voices twenty-nine semitones over seven changes, which is a little over one semitone a voice — comfortably inside, by a factor of four.

So a chorale is not easier to read because it is a chorale. It is easier to read because its parts are voice-led, and this says how much voice leading it takes. That is a claim about the writing rather than about the notation, and it is the first thing on this anchor that is.

It also makes a prediction that eye-tracking equipment could settle, on material that takes an afternoon to write. Two four-part textures, the same chords in the same key at the same tempo, one realised by the solver and one realised with the voices allowed to leap: the second should cost a reader nearly three times what the first does, and the reader’s eye–hand span should shrink in it by the same ratio the twelfth rung predicted for a leaping single line. Both pages carry the same chords, the same number of notes and the same rhythm, and they differ only in a quantity the page does not name.

Why the invention is the hard case, and it is not the counterpoint #

There is a temptation to read all this as counterpoint is difficult, and it would be the wrong lesson.

A two-part invention is hard to read for a reason that has nothing to do with its being contrapuntal. It is hard because two parts cannot assert a harmony, so nothing about the second line is predictable from the first. Add a third part playing the same kind of line, and the texture crosses the boundary: it can state a chord, and the reader gets the whole apparatus of prediction that a chord brings. A three-part invention should be easier to read than a two-part one, which is the reverse of what a count of notes says and is worth putting to any keyboard player as a test.

The measurement says something sharper about three parts still. At two semitones a voice, three parts leave an average of 1.1 reachable voicings — the continuation is very nearly forced — where four leave 5.4. Three-part writing is the cheapest of all to read and the most constrained to write, and the two are the same fact seen from opposite sides: the reader has nothing to choose because the writer had nothing to choose. That is the reading-side version of what the voice-leading ladder found when it asked why the exercise is in four parts, where three parts made the two famous prohibitions cost exactly nothing because the cheapest writing obeyed them by accident.

Four parts is where a writer first has choices to make, and it is still nearly three times cheaper a notehead than an independent line. The exercise is in four parts for the writer’s sake and it happens to be the last texture that is easy for the reader too.

What the notation leaves open, and what closes it #

There is a number from the sixth rung of this ladder that puts the whole result in proportion, and it is very large.

Three notations, one progression counted the four-part realisations each notation admits. For the eight chords used here, a set of chord symbols admits about $1.6 \times 10^{20}$ realisations under the two parallel prohibitions — sixty-seven bits over seven changes, or 9.58 bits a chord.

That is what the notation leaves open. The measurement above says the reader has 2.39 bits to resolve. The difference — a factor of a hundred and fifty — is not supplied by the page and is not supplied by the chord symbol. It is supplied by the requirement that the voices move as little as possible, which is a stylistic convention with no notation at all.

A reader of a chorale is running the voice-leading solver. That is a strong claim and it is what the arithmetic amounts to: the page carries far more than the reader takes off it, and the surplus is closed by an expectation about how parts move, learnt from the repertoire and written down nowhere on the staff.

Which computation produced the numbers #

The progression is I–vi–ii–V–I–IV–V–I in C, realised by the voice-leading ladder’s own layered solver: the cheapest complete voicing path under a ban on parallel fifths and octaves, over one choir’s compass cut into the stated number of parts.

For each transition, the reachable set is every complete voicing of the next chord that avoids both prohibitions against the previous sonority and whose total motion is within the stated allowance times the number of parts. The bits a chord asks are the logarithm to base two of the size of that set, averaged over the transitions.

Naming the chord is charged at the logarithm of seven, which is a diatonic degree chosen with no preference. It is an upper bound and it is quoted as one.

A texture that cannot state its chords is priced instead at the twelfth rung’s readingLoad for a tune, once per part, which is the cost of reading each line as a melody.

Where the model stops #

Reading is not decoding. Everything here is a count of alternatives, and a reader is not enumerating them. What an information measure is good for is ratios between two things measured the same way, which is how every number above is used.

The motion bound is a proxy for the style. Real part-writing prefers small motion, resolves tendency tones, avoids exposed leaps and doubles by rule; only the first of those is in the model, and only two prohibitions.

The harmony is charged at a flat rate. Naming one of seven degrees uniformly is the worst case, and a reader with any sense of progression pays less — which makes the four-part figure looser than it should be, in the direction that weakens the finding rather than strengthens it.

The compasses are one choir’s. The voice-leading ladder’s family of ensembles cuts one fixed outer range into $n$ parts, so a texture of six is six narrow voices rather than six ordinary ones.

And a chorale is not a keyboard piece. Four staves and two staves are different objects on a page even at the same note count, and nothing here has a term for how many systems the eye has to cross.

What the picture cannot show #

It cannot show rhythm. Every part here changes on every chord. A texture in which one voice holds while another moves is a different reading problem and probably an easier one.

Nor can it show a second-order melody model. The twelfth rung’s own defect is still here: the 1.89 bits for a line counts each interval independently, so a scale and a chromatic run come out dearer than a reader would price them.

It cannot show a familiar chord. A diminished seventh is one object to a musician and four noteheads to this model, and the recognisable sonorities are exactly the ones a reader stops decoding.

Nor can it show the key signature. A line in five sharps is the same set of intervals as the same line in C and is not the same thing to read.

It cannot show a real invention. The two-part figure is two independent melodies, and a real invention’s second voice is the first voice delayed — maximally related, and invisible to a model whose only relation is between simultaneous notes. That is the clearest thing this measure gets wrong, and it makes the invention look harder than it is.

And it cannot show a page. How much music a page holds is a capacity in notes, and four parts on four staves fit far less music on a page than one part on one — so a chorale that is cheaper a notehead may still turn the page three times as often.

Whose notation, and when #

The staff, the compasses and the two prohibitions are the European part-writing tradition’s, and the interval distribution behind the melodic figure is from simple Western tunes.

The historical observation available is about pedagogy rather than about repertoire. The four-part exercise has been the standard training object since the middle of the eighteenth century, and the standard explanation is that four parts are what a choir has. The voice-leading ladder gave a second reason: four is the smallest number of parts at which the prohibitions have a price and the largest at which the price is small. This adds a third, from the other side of the page — four parts are the thickest texture that is still cheap to read, and a student who can read the exercise can check their own writing.

None of the three explanations excludes the others, and it is worth noticing that they agree. A convention that survives two centuries usually has more than one thing holding it up.

Where this ladder goes next #

Thirteen 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; the fact that what is being crossed is not uniform; and now the fact that what is above and below it is not independent.

What the ladder owes now is the rhythm. Every line measured here and on the rung before it is a run of equal notes, and every teacher’s account of sight-reading difficulty puts duration at least level with pitch — a syncopation across a beat is the thing a student stumbles on, not an interval. The ingredient is already here: the metrical ladder computes a weight for every position in a bar, and a weight is a probability waiting to be turned into bits by exactly the substitution the twelfth rung made for intervals. What would come out is the second axis of a reader’s load, on the same scale as the first, so that a page’s difficulty could be quoted as one number instead of two — and, more usefully, so that the trade a composer makes between a hard rhythm and a hard interval could be priced. It needs no corpus that this collection does not already hold.