Modulation — where it appears
Named by 40 essays across 8 fields — each of them below, with the objects they name alongside it.
Keys are neighbours, and the map is computed
Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.
The boundary is where the neighbourhood changes
A section boundary can be found by an operator that never sees a section. It walks the diagonal of a similarity matrix asking one local question — do the bars behind me resemble each other, do the bars ahead resemble each other, and do the two groups resemble each other — and where the answer is yes, yes, no, there is an edge. What it cannot find turns out to say more than what it can.
The key plan is the form
The large shape of a classical movement is not a shape at all, it is a journey — out to one key and back. Which key is not a matter of taste. Of the two keys that share six of their seven notes with home, only one introduces a note the home key does not use in any of its chords, and that note is the arriving key's own leading note. The departure is audible because of one accidental.
The same thing somewhere else
A measure built on which notes are sounding calls a passage that comes back a fifth higher a stranger. There is a dial that fixes this, and turning it is supposed to be a trade — more sensitivity to a transposed return, less specificity against a coincidental one. It is not that trade. Two different statistics answer opposite ways, and the setting that would compromise between them is the worst one available.
Three ways to measure how far a key is
Notes in common and steps round the circle of fifths are the same measurement — among the twelve major keys the first is seven minus the second until it bottoms out. Voice-leading distance between the tonic triads is a different one, and it disagrees in exactly one place: E and A♭ are four steps from C, share three of its seven notes, and are two semitones away, which is nearer than the dominant.
How much evidence a modulation needs
Run a key-finder bar by bar over a progression that moves to the dominant at bar six. With four bars of history the answer becomes the new key at bar seven and holds. With three bars it never gets there at all, and reports E minor and B minor on the way. The window decides the lag as much as the music does.
The circle is a circle, and the map is not
Among the twelve major keys, notes in common is a strict function of distance round the circle of fifths — one value for each step count, no exceptions — so there is nothing else to measure and the map really is one-dimensional. A second axis appears only when the minor keys are added, and it is a different kind of move: the relative shares all seven notes and the parallel is one semitone away.
What a choir does that a soloist cannot
Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift measured for tuned singing describes only ensembles that hold their pitch still.
Two keys at once
Every key figure so far assumes one key is sounding, and the standard key-finder has no value it can return that means two. Play one progression in C and the same progression a major third away at the same time, and the model does not report uncertainty: it reports E minor, at a correlation of 0.886, against 0.959 for the same progression in one key. It is as confident as it ever is, and neither key it names is being played. Give it the missing hypothesis — pairs of key profiles rather than single ones — and it recovers both keys at every separation, all seven of seven.
The alternation a key-finder cannot follow
Two keys sounding together are not in the key-finder's vocabulary, and an earlier essay ended by pointing at the other case and saying what was missing: a passage whose alternation rate can be varied while everything else is held still. Built, it gives a rule with three numbers in it — the second key is never named below a block of three bars, the tracking clears ninety per cent above twice the window, and the reading is late by half the window throughout.
A key-finder that keeps the order
Every key-finding model until now begins by throwing order away — a histogram over a window, correlated against twenty-four profiles — and the thing that most obviously declares a key is an ordered pair of chords. A model that keeps the order costs 84 states and 7,056 transitions against 24 hypotheses and none, and what it buys is the one test a histogram was said not to pass: two keys in turn and two keys at once have histograms 98 per cent alike, and are two different sequences.
A modulation and a borrowing are one number apart
The key-finder that keeps the order has one tuned parameter, and said so. Sweep it and the parameter turns out to be the whole verdict: below a threshold the model hears two keys alternating, above it one key with borrowed chords. The threshold rises with how slowly the keys alternate — and the value it chose sits above every threshold in range, so its finding about fast alternation was a consequence of the tuning.
The passage built to make them disagree
A cadence count and a key-profile correlation have been put on one scale and run on a passage where the two agree, which tells nobody anything. Building one where they disagree — the notes of one key and the cadences of another — measures the exchange rate at about six bars per cadence, and finds something the agreement case concealed: the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.
The margin the dynamic program already had
Nine earlier essays produce a single best reading, and the passages worth arguing about are the ones where two readings are nearly equally good. What is needed for that has been inside the model from early on: a dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed, and printing it turns every analysis here into a measurement of ambiguity.
The beat a tuner can actually use
There is a modulation index for every interval on every timbre, and the published threshold for detecting a fluctuation is a function of exactly that and its rate. Running one against the other turns a table of decibels into a window in cents — and reverses the ordering, because the interval with the shallowest dip has the widest window.
The mean survives the window
A roughness that moves has a mean, a depth and a rate — all three of which a listener could only have through a temporal window. Applying the window already to hand settles which of the three survives, and the answer refutes the guess: a running average cannot change an average, so the octave's factor of nineteen stands and the movement is what goes.
How much of the reading arrives late
Every margin reported earlier is two-sided: the best path through a key at a bar is the score into it plus the score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line. On a thirty-two-bar song it removes more than half the certainty, and on a passage built to be ambiguous it changes the key named at nine bars out of eleven.
The listener who forgets
Setting an analyst's reading of a key against a listener's measures what arrives late. Both passes assume perfect recall of their own half — which is as wrong going forward as knowing the future is going back. Put a decay on the forward pass and a listener with a memory of a few bars keeps a quarter of the confidence and nine tenths of the answers.
The bars a key is made of
A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.
The key-finder with no tonic
Thirteen essays of key-finding have run over twelve major collections and not twenty-four keys, so a passage in A minor is read as C. Adding the missing twelve costs almost nothing and fixes almost nothing — because the model has no tonic in it at all, and below three raised sevenths in a passage the extra states are worth exactly zero.
The partial that gets louder as it goes sharp
Eleven earlier essays sweep a set of partials and hold their amplitudes still, and a real tract does not move with the fundamental. Put the formant account and the sweeping account together and every partial acquires an amplitude modulation at the vibrato rate: 0.07 decibels on the fundamental and 8.6 on the twelfth partial of the same note. They are in phase below a formant and anti-phase above one — not ninety degrees apart — and the whole note swings 0.82 decibels, because they cancel.
The rate that does not rise with the partial
Twelve earlier essays give every vibrato the same six hertz, and the measured spread is 5.5 to 7.5. Putting the two fluctuations a choir contains on one axis shows why the rate matters: the beating between mistuned voices rises with the partial and leaves the range a listener follows as fluctuation at 1,217 hertz, while the vibrato's own modulation is six hertz at every partial. Above that frequency a section fluctuates by vibrato alone — and if every singer had the same rate, it would barely fluctuate at all.
Every member of a beat family is the same depth
A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.
One fluctuation or two
Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.
How hard the note was struck
The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.
Given the bar in octaves, the degree comes back
A bass note is a bare pitch class in the usual figure, and a register in any realisation anybody plays. Voice each bar in octaves and a listener knows which three pitch classes are sounding and which is at the bottom, which names the chord outright — and the rule that uses it reads the right scale degree in 92 per cent of bars on a real bass line, against 68 for the published cue on the same line and 89 for that cue on a line made entirely of roots. The question was whether the register recovers the 89. It recovers it and passes it.
A sharper cue is worth nothing to a reading that moves
The rule that names the chord outright reads 92 per cent of scale degrees right where the published bass cue reads 68 — at the key cost these readings have always been run at. Sweep that cost and the advantage is not a property of the cue. Where a change of key is cheap the sharp rule reads 46 per cent and the vaguest rule 47, because a reading that will move key on one bar's evidence follows a sharp cue wherever it points. The cue's whole value is borrowed from the model's reluctance to be moved.
The two parameters turn out to have a ceiling between them
The essay before this one asked whether the bass cue's weight and the cost of changing key are one quantity with two names, and said the test was a contour: if they multiply, the curves of equal degree share are hyperbolae. They are not. Along the ninety per cent contour the product of the two runs from 2.7 to 15.2, because past a bass weight of about one and a half the reading saturates and more cue buys nothing. And the sweep finds something no single-parameter sweep here could: the key cost has a best value, and a reading that will never change key is worse than one that will.
The ceiling is thirteen bars with names
The key-finder's two cues stop buying anything at about 93 per cent of scheme bars read right, and the obvious suspect was the chord segmentation feeding it. The reading was never given a segmentation: every bar arrives with its true chord. What the ceiling is made of can be listed instead, and it is thirteen bars of 188 — seven in a bridge of secondary dominants, four in a minor episode whose chords C major also owns, and two at the edges of a modulation. No weight of any cue moves one of them.
The bridge is not a labelling problem
Seven of the key-finder's thirteen remaining misses are the thirty-two-bar song's bridge, and the essay before this one proposed that they might be an argument about labels rather than a failure: relabel the eight bars with the keys their dominants point at, and the reading's E and D majors might turn out to be right. They do not. The reading gets one bar of eight under either label, because four of the eight are read as the chord's own root taken for a tonic — a dominant seventh heard as a chord in its own right. And one of the bars the relabelling could have bought is unwinnable in principle: an applied dominant is the same four notes in the major and the minor of the key it points at.
Looking ahead rescues the relative minor, not the bridge
A key-finding reading that scores each bar by its own notes cannot hear an applied dominant, whose meaning is in the bar after it. Given one cue about a pair of bars — a dominant earns credit when the next bar is rooted on its key's tonic — the thirty-two-bar song's bridge is read as the chain of dominants it is, each chord both the arrival of the last and the departure to the next. But the same cue hears a blues's opening tonic seventh as the dominant of its own IV, and at the setting that read the listing best it costs five bars for three. The cue earns its place only in a reading reluctant to change key, and there what it rescues is the rondo's episode in the relative minor.
A second slot hears every modulation as a tonicization
A key-finder that names one key per bar reads a chain of applied dominants by moving the key with every chord. Give its state a second slot — the key a bar is in, and the key its dominant points at — and the thirty-two-bar song's bridge is read exactly as the scheme labels it, every bar, for the first time in these essays, and the blues's opening stays in its tonic. At every setting that does that, the rondo's two eight-bar episodes are heard as tonicizations inside C, and no setting of the three costs reads both. A modulation to a closely related key and a long tonicization are the same chords; what separates them is how long they last, and a cost for entering a state cannot count.
A modulation is a dominant that keeps coming back
A key-finder with a slot for an applied dominant read the thirty-two-bar song's bridge as its scheme labels it and heard the rondo's two episodes as tonicizations inside C, and no setting read both. The obvious repair — charge a tonicization by the bar, so a long one costs more than a short one — makes it worse: the reading gives up the bridge before it gives back a single episode, because the bridge spends six bars in applied chords and the episodes three and two. By the bar, the chain is the longest tonicization in the listing. What the episodes have and the bridge does not is a dominant that resolves to the same tonic again and again.
A return is cheaper to avoid than to pay for
The essay before this one found what separates a modulation from a tonicization: an episode's dominant keeps resolving to the same target, and the bridge's dominants never do. So the key-finder was given a memory of where its last dominant went, and charged for a return — the prediction being that one setting would at last hear the episodes as keys and the bridge as a chain. None does. Every return the charge is aimed at is avoided for less than a nat, by hearing a D7 as a minor chord or by moving the passage into another key, while the reading the charge was meant to force costs twelve.
The leading note is already inside the chord that hides the key
A key-finder kept hearing a rondo's episodes as passages inside C, and the escape it used was to hear a D7 as a D minor chord, throwing away its F♯. Charge a chromatic note by how strongly it points at another key, and that escape closes. Nothing else moves, because every F♯ and G♯ in the episodes was already inside the chord the reading heard — as the third of V of V. The charge takes a chromatic note seriously everywhere except where it was needed, and in the blues it mistakes a flattened seventh for a leading note.
A vibrato hides a choir's tuning above four hundred hertz
Thirteen essays drew a vibrato that had always been running; a real one grows out of a straight tone over the first quarter-second of a note. Counting the beats a listener can follow between two voices of a section shows what that front edge is worth. A vibrato takes most of a section's countable beats away above about 400 hertz — seven in ten of them at the singer's formant — and adds some below. The straight start gives them back only while the vibrato is still growing, and in the band where a voice carries that is less than one beat.
A choir's vibratos must lock to two milliseconds
A vibrato hides a section's mistuning from a listener counting beats, and the obvious way it could stop hiding it is for the singers' vibratos to move together. They would have to move very nearly together. Half-entrained vibratos leave a section as hidden at the singer's formant as independent ones; locked to nine-tenths they expose three-quarters of what a straight section does; only past nineteen parts in twenty does the section's tuning show as if they sang straight. At six cycles a second that is two vibratos peaking within two milliseconds of each other — and the better a section is tuned, the tighter the lock has to be.
Singers who hear each other late lock their vibratos in opposition
A choir's vibratos give back the section's tuning at the singers' formant only if they are locked to within about two milliseconds. Singers who lock do it by hearing each other, and they hear each other late. Two vibratos pulled towards what each hears of the other settle near in step while the delay is under about forty milliseconds, and even then come within two milliseconds only if each singer corrects a phase error in about thirty. From forty to a hundred and twenty milliseconds late — a third to two-thirds of a vibrato cycle — they lock in opposition, peaks seventy to ninety milliseconds apart, and hide the section's tuning more completely than vibratos that never locked at all.
Sixteen singers cannot all oppose each other
Two singers who hear each other a third of a vibrato cycle late lock their vibratos in opposition and hide their tuning better than if they had not listened at all. Sixteen cannot do it, because no arrangement puts every one of them opposite every other. At the latencies where a pair opposes, a section of sixteen does not lock at all: each singer keeps a vibrato rate of its own, and a listener hears the section's tuning exactly as through independent vibratos. Outside that band the section locks in step and gives most of its tuning back. A section that listens to itself can expose its tuning or leave it as it was, but never hide it.
A row that hears its neighbours moves its tuning, it does not hide it
Sixteen singers who all hear each other alike cannot lock in opposition, because opposition between every pair is impossible. A row that hears mainly its neighbours has no such obstacle — opposition along a chain is consistent — and at the latencies that make a pair oppose, it half-alternates: neighbours opposite each other, every second singer nearly in step. That hides the tuning between neighbours as well as a duet hides it, and exposes it between every second singer more than independent voices would. Over the whole row the two cancel to the hundredth. With no latency, the same local hearing stops the row locking at all, and hides nearly half of what a whole-row lock would show.
Named alongside it
The objects these essays reach for when they reach for this one.
Key-findingBeatingVibratoInferenceFormantScale degreeUnisonCadenceRelative minorSegmentationTonal functionCritical bandwidth