Where the chord actually lands

Expectation is a curve, not a list drew the hazard across a whole bar and reported that it runs from 0.043 on the weakest quaver to 0.290 on the downbeat, a ratio of six and a half. Every figure that has used it since has put the chord change on the downbeat.

That is a parameter the ladder carries and has never varied. The curve is computed at eight positions and read at one of them, and the one is the extreme.

It matters more than a factor of six and a half, because an arrival is not the only thing a listener pays for. By the time a chord arrives on the seventh quaver, six beats have gone by on which it did not, and each of those was charged — that is the surprise of nothing happening, which found the charge to be about a third of a bar’s timing bill. What an arrival at a given beat costs is therefore the waiting plus the arriving, and the ladder has been quoting only the second half.

The costs are a distribution #

Write the full cost of an arrival at beat j as the sum of the hold costs at every beat before it plus the arrival cost at j, and something falls out that is more than a bookkeeping tidy-up.

Two to the power of minus that cost is exactly the probability that the first change in the bar comes at beat j. The nine numbers — eight beats and the case where no change comes — sum to one.

That is a real check rather than a decoration. A pile of surprises that does not sum to a probability is not a model of anything, and every figure here asserts the sum before it draws. A hazard is a probability per beat; the costs it induces over a bar must be a probability over the bar, and they are.

The waiting turns a partial order into a total one #

The table has a property the metrical weights do not, and it is the most useful thing on this page.

Longuet-Higgins and Lee’s weights give a bar of eight quavers four distinct values: the downbeat, the third beat, the second and fourth beats, and the four off-beat quavers. Four of the eight positions share the lowest weight and are, to the metre, interchangeable. Two more share the middle weight.

The full costs are all different. The four weak quavers cost 5.02, 5.31, 5.78 and 6.07 bits — a spread of a whole bit between positions the metrical hierarchy calls identical — and the two middle beats cost 3.35 and 4.11.

The reason is the waiting. Two positions of equal weight are equally unlikely to carry the change, but the later one has more beats of not-carrying behind it, and each of those was charged. So a metrical hierarchy is a partial order and an arrival cost is a total one, and the thing that breaks the ties is time rather than weight.

That is a claim a listener could be tested on and this collection cannot test: it says the last quaver of a bar is a more marked place for a chord to arrive than the second quaver, by about a bit, even though both are the weakest positions the metre knows about. Every account of syncopation the collection draws on prices those two the same, syncopation is a number about the metre included, because all of them are functions of weight alone.

The spread is three times what the ladder reports #

The fifth rung’s figure quotes the ratio between the hazard on the downbeat and the hazard on the weakest quaver: 6.7, which in bits is the 2.74 that separates an arrival cost of 1.79 from one of 4.52.

The full cost separates them by 4.28 bits, which is a factor of 19.5.

The difference is the waiting. A change on the last quaver is expensive twice: it lands on the least likely position, and it lands after seven beats of not landing. Those two costs compound rather than trading off, so pricing only the arrival understates the metrical penalty by a factor of about three.

That is a correction to a number this ladder has published, not a new quantity. Everywhere the collection says a change on a strong beat is cheaper to expect than one on a weak beat, the size of the difference has been the arrival term alone.

Move them off the beat and the sign changes #

The curve takes a schedule of change positions and it has never been given one that is not a downbeat. Give it the fourth quaver of each bar and the same arithmetic runs the other way.

Two bars with the changes on the downbeats cost 3.6 bits of timing. The same two bars with the changes on the fourth quaver cost 9.0. The metreless control — a listener for whom every position is equally likely — costs 6.0 either way, and the two real schedules sit on opposite sides of it.

So the metre is worth 1.7 times in one direction and 1.5 times in the other, and a model with a metre in it is better than one without only when the music agrees with the metre. That is obvious once stated and it has never been drawn here, because the schedule was fixed at the value that makes the metre look good.

Five schedules, none of which is the default #

The comparison generalises. Take four bars at one change a bar and put the changes in five different places.

Changes on every downbeat cost 11.6 bits over the four bars. Anticipated by a quaver — arriving on the last eighth of the previous bar, which is one of the commonest gestures in the repertoire the ladder draws its harmonic rhythm from — they cost 24.3 for the same four changes. Per change that is 1.79 bits against 4.52, a factor of two and a half.

Two more sit between. Changes on the third beat cost 2.02 bits each, barely worse than the downbeat, because the third beat carries the second-strongest weight in a bar of four. Changes on the backbeat cost 2.79 each and there are twice as many of them, so the total is the largest on the figure.

The reading to take from that is not that syncopation is expensive. It is that the expectation bill of a passage is set by the schedule at least as much as by the harmony, and this ladder has been holding the schedule fixed while sweeping everything else. Syncopation is a number about the metre prices the same phenomenon on the rhythm ladder in a different currency, and the two are now comparable in a way they were not.

The case nobody had priced #

The last column of the distribution is a bar in which the chord does not change, and it holds a third of the probability.

That is a strange thing to find in a model whose harmonic rate is one chord a bar. It is not a contradiction: the expected number of changes in a bar is exactly one, and the hazards are set so that it is. But the changes are Bernoulli per beat, so the count in a bar has a distribution: sometimes none, sometimes two, and the mean comes out at one.

The ladder’s harmonic rate is an expectation and has always been read as a description. A passage generated by this model would have a third of its bars with no chord change in them and a fair share with two or three, which is a much more variable texture than “one chord a bar” suggests to a reader.

It also puts a number on how loose. Of a hundred bars generated by the model at one chord a bar, about thirty-three carry no change, about thirty-seven carry one and about thirty carry two or more. How often the chord changes reports harmonic rate as a single figure with limits set by perception at either end, and every figure on this ladder and that one has drawn the middle of that range as though it were a metronome.

Whether that is a defect depends on what harmonic rhythm is. If it is a rate — a mean over a passage — the model is right and the reading was loose. If it is a period — a chord every bar, on the bar — then the model is wrong in a way that matters, because a listener who has learned that the harmony changes at the barline has an expectation this hazard cannot express. A metre has to be able to change its mind is the collection’s account of the analogous problem one level down.

What the harmonic rate does to the spread #

The factor of 19.5 is not a constant. It grows with the harmonic rhythm, and it grows fast.

At two changes a bar the spread is 6.35 bits, a factor of 81. At one it is 4.28, a factor of 19.5. The hazard ratio underneath it is 6.7 at every rate, because it is a property of the metrical weights alone.

So the arrival term is rate-independent and the waiting term is not, which means the whole of the growth is in the waiting. Faster harmony makes each individual silence more informative, and a syncopated arrival in fast harmony has to be paid for through a great many of them.

That is a testable prediction about where syncopation is expensive, and it has the right sign: a displaced chord in a slow chorale is a mild event and a displaced chord in a fast harmonic texture is a strong one. The collection’s own account of groove — the milliseconds that are the groove — is about displacements far smaller than a quaver, and this arithmetic says nothing about those, but the direction agrees.

What a real schedule would look like #

None of the five schedules on the figure is a repertoire. They are constructions chosen to span the space, and the honest question is where actual music sits in it.

The collection cannot answer that, and the reason is worth being precise about because it is not the corpus this anchor keeps recording as owed. What is missing is not a count of root motions or a coefficient in a model. It is a count of where in the bar chord changes fall, which is a much easier thing to obtain and which nothing here needs a theory of harmony to produce: it is a histogram over eight positions.

There is a good reason to expect it to be strongly peaked on the downbeat and the third beat, because that is what a barline is for. There is an equally good reason to expect a long tail on the last quaver, because anticipation is a standard gesture in most of the repertoire the ladder’s other numbers come from. Between those two the model’s own estimate of the expectation bill could be out by a factor of two, and the sweep above is the honest form of the statement until somebody counts.

The distinction between the two missing counts matters for scheduling the work. The correlation between the identity and timing surprises is a joint distribution over two continuous quantities and needs analysed harmony. This one is a marginal over eight positions and needs a score with barlines in it.

When the segmentation decides where the change is #

There is a circularity here that the collection has met before and should say out loud.

The position of a chord change is not an observation. It is the output of a segmentation, and which notes are the chord showed that a segmentation is decided by the metre — the same metre whose weights supply the hazard. So the model is scoring the arrival positions against a curve derived from the machinery that produced the arrival positions.

That does not make the numbers wrong; the two uses of the metre are different. The segmentation uses the weights to decide which notes are chord tones and the hazard uses them to decide how likely a change is at each position. But it does mean a passage segmented under a different barline would produce a different schedule and a different curve, and the expectation bill of the two readings is not directly comparable.

Three decisions that constrain each other is the collection’s account of that entanglement at the level below, where key, metre and segmentation are resolved together rather than in sequence. The expectation bill is a fourth decision that depends on all three and has never been included in the joint search.

Which computation produced the numbers #

The hazard is the fifth rung’s, unchanged: metrical weights scaled so that the expected number of changes in a bar is the stated harmonic rate, capped below a probability of one.

The cost of an arrival at a beat is the sum of the hold costs at every earlier beat of the bar plus that beat’s arrival cost, all in bits. The no-change cost is the sum of the hold costs across the whole bar. Two to the power of minus each cost is that outcome’s probability and the figures assert that the whole set sums to one before drawing.

The schedules are lists of beat positions over several bars, priced by the same curve. Each figure asserts that every position it was given lies inside the passage, because a bar of three has six positions where a bar of four has eight and an index past the end would be dropped in silence — the figure would draw a schedule with fewer changes than its own name claims.

Nothing about the harmony enters any of this. The identity of the chord that arrives is the fourth rung’s quantity and is independent of where it lands, which is the assumption two surprises and one event swept and could not settle.

Where the account stops #

It cannot say a listener resets at the barline. The distribution above is over one bar with the survival probability restarting at its downbeat, which assumes a listener who knows where the bar is and treats each one afresh. The right period at the wrong phase is the collection’s account of how hard finding that barline is.

It has no anticipation in it. A chord arriving on the last quaver of a bar is very often heard as the next bar’s harmony arriving early, and under that reading it is not a late arrival in this bar but an early one in the next. The model has no way to express that, and it is the single most common syncopation in the repertoire the harmonic rates come from.

It cannot price a change that is prepared. A suspension, an anticipation and a passing chord all put notes of the arriving harmony before the arrival, so a listener’s hazard is not what it was at the start of the bar. The model’s hazard depends only on position.

And it says nothing about duration. A change on the last quaver of a bar is followed by a chord that lasts into the next bar, and a change on the downbeat by one that lasts a whole bar. Those are different musical objects and the arithmetic here treats them as the same event at different addresses.

Where this ladder goes next #

Eight rungs: a measured hierarchy of pitch stability, a chord that fails to arrive, a surprise given a number, that number conditioned on a key and a predecessor, expectation drawn as a curve, two surprises added under a swept correlation, the third term the curve had all along, and now the curve read at the beats a change actually lands on rather than at its peak.

What is owed after this is the register. Every quantity on this ladder is a set of pitch classes at a position in a bar, and a chord is not a set of pitch classes: it is voiced, in a range, with a bass note, and both surprises should depend on that. A dominant seventh in close position in the treble and the same chord spread across two octaves with the seventh in the bass are the same event to every model here and are not the same event to a listener. The collection has the ingredients — a chord is a register has the voicing, and the roughness of a voicing is computed from its own spectrum rather than from its name — so a version of the identity surprise that is conditioned on voicing needs no corpus. What it needs is a decision about whether the conditioning belongs in the probability or beside it, and that decision is the rung.