The cadence as evidence

Three decisions that constrain each other ended by naming the rung the zero demanded. Every model so far infers a key from a bag of notes; the thing that most obviously declares a key is a cadence; and the closure ladder’s five-component cadence vector is exactly an ordered-pair measurement that has never been used as evidence for a key.

The two passages are one note apart and the cadence evidence is decisive about them in opposite directions. That is the case the previous rung said it could not construct, and it is worth noticing that the whole difference between them is an accidental — one sharp, on one note, in the last three bars of eleven.

A histogram counts that accidental once among some thirty-three note events. An ordered-pair reading counts it as the thing that makes a leading tone a leading tone, which is a component of a cadence, which is the evidence.

What the score is #

The closure ladder’s five components come from closureVector, and they are the same five what makes an ending an ending established: root motion by a descending fifth, arrival on the tonic, a leading tone resolving, metrical placement and duration.

Two of the five are properties of where a pair sits in a bar rather than of the notes, and they are the same for every key hypothesis, so they cannot discriminate between keys and are dropped here. What is left is three harmonic components, and the score for a key is how many of them the passage’s ordered pairs satisfy when read in that key.

That is a deliberately weak scoring rule. The closure ladder’s own finding was that the five components have no derivable weighting — it is a vector with no total, and it says so — so counting them is the weakest thing that uses all of the ones available.

What an ordered pair contains that a bag does not #

It is worth being explicit about the information, because the argument is not that one model is better tuned than the other.

A bag of pitch classes over eleven bars is twelve numbers. Every ordering of those bars produces the same twelve numbers, so a bag cannot represent anything about which chord followed which — and the whole of tonal function is a statement about that. A dominant is a dominant because of where it goes; a degree is where it goes next makes the same point about a scale system that defines its degrees behaviourally and has no other way to.

A cadence is the smallest object that carries the information a bag discards. Two chords, in order, is the least a model can hold and still be able to tell a dominant from a subdominant with the same notes in it.

The passage, and why it is built that way #

C major and G major share six of their seven pitch classes. A passage drawn only from those six is exactly ambiguous between them, and it is the ambiguity every profile key-finder resolves by weighting — by which of the six notes it hears most of — rather than by evidence.

The body here is eight bars of that kind. The ending is three bars, and the two endings differ in one note.

The profile model’s ranking is identical for the two passages and its top correlation differs by 0.008. The cadence evidence flips.

Why the profile model is not simply worse #

The temptation is to read this as the ordered model winning, and that is not what the figures show. A key-finder that keeps the order makes the same point about a different ordered model in the same phase, and comes to the same conclusion: what is bought is a distinction, not accuracy.

The profile model’s reading of both passages is E minor, which is not absurd — the body is heavy in E, G and B — and it is a reading about the content, which is what it was built to measure. Counting produced the hierarchy is the essay about where those profiles come from, and what they were fitted to was listeners’ judgements of how well a note fits a context, which is a question about content.

The two models are measuring different quantities and the disagreement is not an error in either. A passage can be made of the notes of one key and go to another, and the two readings are both true statements about it.

What the rung adds is that only one of the two quantities is available to a listener without waiting. A histogram needs a window; a cadence is complete the moment its second chord arrives. So the cadence evidence is the one that can be fast, and the fourth rung of the key-relations ladder measured how long the slow one takes: a handful of chords.

There is a third quantity worth naming, because it separates the two models more sharply than either figure does. The profile model’s answer is a ranking with small gaps: its top two correlations here differ by 0.08, which is not a confident reading of anything. The cadence evidence’s top two differ by a factor of three on one passage and by half on the other. A count of satisfied components is a coarse measure with big steps, and a coarse measure with big steps is exactly what a decision procedure wants.

There is one signal in the list that this essay has been treating as a duration and which is better understood as a level.

A rest is a diminuendo, and a long one. A pause after a cadence is therefore not merely a gap where evidence stops arriving; it is itself evidence of the same kind as the dynamic signals, arriving at a rate the score never writes down. That is the one component of the closure vector whose size a composer sets without notating anything at all.

Which computation produced the numbers #

The passage is a list of roman numerals with the key each is written in, which makes each chord a set of absolute pitch classes.

Reading a passage in a candidate key means renaming every chord: for each absolute pitch-class set, find the numeral that produces it in the candidate key, and refuse the chord if none does. The refusal is the interesting half. A chord that is not diatonic to a candidate key contributes nothing to that key’s score, which is a much sharper instrument than a profile model’s — where every note contributes something to every key and only the weighting differs.

Each surviving consecutive pair is then run through closureVector and its three harmonic components are counted. Nothing is fitted; the components are the closure ladder’s own and the count is the weakest aggregation available.

The profile reading is keyCorrelations on the passage’s pitch-class histogram, unchanged from the key-relations ladder’s first rung: the Krumhansl–Kessler profiles, rotated to all twenty-four keys, correlated with the count.

Where the model stops #

Counting components is a weighting, and the section on the sweep prices it. Equal weights are a choice, the finding survives most other choices and not all of them, and the component that fails is not the one anybody would have guessed.

The weighting, swept #

Counting three components equally is a weighting like any other, and the honest thing is to turn it rather than to say it was turned. Sweeping every whole-number weighting from one to eight on each of the three, which is 728 of them:

The C-cadencing passage is unanimous and the G-cadencing one fails on thirteen per cent of the weightings. The asymmetry is the passage’s own: reading the G passage in C costs one chromatic chord and reading the C passage in G costs one, but the eight ambiguous bars are written in C, so a weighting that leans on the body rather than on the ending has somewhere to lean.

Which component does the leaning is the part worth having, because it is the opposite of what the closing caveats predicted. The leading-tone component, taken alone, names C for the G passage — the component singled out as the one that identifies a key rather than merely a closure is the one that gets this passage wrong when it is trusted on its own. Holding the other two at one and raising it, the G reading survives to a weight of just under three and flips there:

So weighting the leading tone twice does not sharpen the peak, as the caveat that used to stand here said; it halves the margin, and at three it loses the reading altogether. Root motion and arrival on the tonic each name G alone, with a margin of two and of one, and it is the pair of them holding the reading up against the component that was supposed to be carrying it.

None of that overturns the rung — the finding holds at equal weights and at every weighting anybody would defend on grounds other than this measurement — but it does relocate where it is fragile, and it is a useful reminder that a component’s name is not evidence about what it does. The leading tone identifies a key in the abstract; in this passage it is also present in the body, several times, in the wrong key.

And two of the five components are thrown away. Metre and duration are the components the closure ladder found were doing real work, and they are dropped here because they cannot discriminate between keys. That means the score used for key-finding is not the closure vector but its harmonic half, which is a smaller object than the rung’s premise implies.

A cadence is being detected in a stream where it is known to be. The passages are constructed, the endings are unambiguous, and the pairs are read consecutively with no segmentation problem. A real passage does not arrive with its ordered pairs marked, and which notes are the chord is the decision that would have to come first.

And a passage with no cadence in it gets no evidence at all. The score is zero for every key on a passage that never closes, which is not a failure — it is the model saying it has nothing to go on — but it means the cadence evidence cannot stand alone. A working key-finder would need both, and combining a count of components with a correlation is a problem in units, of the same shape as the one the closure ladder recorded when it refused to total its own vector.

One combination needs no units at all and is worth reporting because of what it does rather than because it works. Rank the twelve keys by each model and add the two ranks. On the C-cadencing passage that gives C at 2, G at 4 and E at 6 — a clean answer, because both models agree. On the G-cadencing passage it gives C and G tied at 3. A rank sum cannot resolve a disagreement in which each model is confidently first and last; it records it, which is the correct behaviour and is not a key-finder.

Whose music, and where the ambiguity is a device #

The scoring rule is about a repertoire and it is worth saying which. Descending-fifth root motion, arrival on a tonic and a leading tone resolving are the closure of European tonal music from roughly 1650 to 1900, and the profile model’s own numbers were measured on listeners raised in it. Neither model claims anything about music that does not cadence.

Inside that repertoire, the construction this rung uses is not artificial in kind even though these passages are. A body of material that could be in either of two keys, resolved at the last moment by a cadence, is a standard device — it is what a galant transition does, what a sequence does before it lands, and what every modulation to the dominant does on its way. The eight ambiguous bars here are a caricature of it; the arithmetic is the same.

The place the reading breaks is the repertoire built on ambiguity that is never resolved. A piece that draws on the six shared notes and cadences on neither key is not badly analysed by either model — it is a piece about which both models are right to be uncertain, and that is a much later kind of music than the profiles were measured on.

What a chord that fits nothing is worth #

One property of the scoring rule is worth drawing out, because it is where the ordered model gets most of its sharpness and it is not obvious.

Reading a passage in a candidate key means renaming every chord, and a chord that is not diatonic to that key gets no name and contributes nothing. So a single chromatic chord — the F sharp triad in the G-cadencing ending — removes several pairs from consideration in every key that does not contain it, and the removal is silent. A key that can name every chord accumulates evidence; one that cannot is not penalised so much as starved.

That is a very different instrument from a correlation, which cannot be starved: every note contributes to every key’s score in proportion to that key’s profile, so no key ever runs out of evidence and no key is ever refused. The profile model’s flatness in the figures above is that property showing.

Whether refusal is the right behaviour is a real question. A key-finder that refuses every chromatic chord will do badly on chromatic music, which is most of the nineteenth century; the profile model will do mediocrely on all of it and fail on none of it. That is a genuine trade and this rung has only run the easy half.

What the picture cannot show #

Whether a listener uses cadences this way. The claim is that the information is there, not that anybody extracts it. The profiles have an experiment behind them and this scoring rule has none.

Nor how it behaves on real music. Both passages here were built to separate the models, and a construction that separates two models has demonstrated a difference rather than an advantage. The joint reading of the previous rung at least ran on constructed material with a stated ground truth; this runs on material built to have one.

And the two measures are not on one axis. Every figure here normalises each to its own maximum in order to draw them together, which makes their shapes comparable and their heights meaningless. A model that combined them would have to solve the problem the normalisation hides.

The one place the two agree #

On an ordinary tonal passage — one whose notes and whose cadences belong to the same key — the two models agree, and it is worth saying why that is not a triviality.

Tonal music makes them agree on purpose. A key is established by both means at once: the notes of the scale are used, and the cadences close in it, and a passage that did one without the other would be a passage somebody wrote to be strange. The agreement is a property of the repertoire rather than of the models, which is why it takes a constructed passage to separate them.

That is the strongest version of the rung’s finding. The two models are not redundant; they are made redundant by the music they were built for, and the redundancy is what the music is doing. Pull the two apart and each still says something, and what each says is different.

Where this ladder goes next #

Six rungs. A progression is a path across a space with a geometry; it never comes home in just intonation; its rate has limits set outside music; the objects it connects are produced by a segmentation that depends on the metre; the three decisions taken together change two readings in five; and now, the ordered pairs the ladder had been discarding turn out to carry key evidence that the bag of notes cannot, on passages one note apart.

The rung after it is the one the combination problem names. Cadence evidence is a count and profile evidence is a correlation, and a key-finder that used both would have to put them in the same units — which is not a technical difficulty but the same question the closure ladder refused when it built the vector, of how to weigh components that were measured on different scales. The honest form of it is that the evidence is a vector, and choosing among keys on a vector means either finding a defensible total or reporting the whole thing and letting the reader see the disagreement, which is what these figures do.