Acoustics & Music Theory
why it sounds like that
Concept

Tonic — where it appears

The note a passage is heard as centred on and as returning to. It is not in the pitch set: the same seven notes support seven tonics, and which one is home has to be established by what the music does with them.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.

The same seven, started later

A mode is not a new scale. It is the same seven notes with a different one treated as home, and the entire change in character comes from reassigning which degree the semitones fall next to.

scales · Modes The seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense.

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

harmony · Progression The seven modes, brightest first. The same seven pitch classes started on each of its degrees in turn, ordered by how many of their notes are raised. Each row differs from the one below it by exactly one note, and that note moves down one semitone each time.

Nothing in the census knows which note is home

All four properties six earlier essays are about are invariant under rotation and transposition — one property tuple over all eighty-four rotations and transpositions of the diatonic set. So the census that separates 349 shapes cannot separate a major scale from its own Aeolian mode, and everything that makes one note a tonic is outside it.

scales · The diatonic set The price of a tonic. Every note of Dorian is given the same duration except its tonic, which is lengthened; the horizontal axis is the share of the total that goes to it. The key-finder answers with the parent key until 25.0 per cent of the time is spent on the modal tonic, and with D minor above it. At the left-hand edge every note has equal weight, which is the pitch-class set itself — and with every weight identical the correlation is not merely low but undefined, because a flat histogram has no variance to correlate with anything.

What a tonic costs in seconds

The standard key-finding algorithm cannot be run on a pitch-class set at all — a flat histogram has no variance and the correlation is undefined. Give it durations and it answers with the parent key for all seven modes identically, and it takes between 15.8 and 30.0 per cent of the total time spent on one note before it names that note instead.

perception · Modes The mode is its tritone. Every set of degrees that lies inside at least one of the seven modes — 510 of them — scored by how many modes it leaves standing, with the 15 minimal sets that decide printed in full. All 15 have two members and every one of them either is a tritone or contains the tritone above the tonic. The only degree every mode has is the tonic itself; there is no degree belonging to just one mode. Averaged over every order the degrees could arrive in, 5.67 of the seven have to have been heard before the mode is settled — most of the scale, because the deciding pair arrives when it arrives.

The one note that decides the mode

A key signature names the seven and not the rotation, so the mode has to be heard. Enumerate every set of degrees that lies inside at least one of the seven modes — 510 of them — and count how many modes each leaves standing. Fifteen minimal sets decide, every one has two members, and every one is the mode's own tritone. The only degree all seven share is the tonic; no degree belongs to a single mode; and averaged over the orders the degrees could arrive in, 5.67 of the seven have to have been heard first.

scales · Modes A model that cannot say two keys does not say it is unsure. I – IV – V – I played in C major and in a second major key at the same time, with the second key moved round the circle of fifths. For each separation: the correlation the standard key-finder gives its single best answer, and the correlation reached by the best PAIR of key profiles — a hypothesis the finder does not have. The pair recovers both keys that are sounding at every separation, 7 of 7. The single answer names neither of them at 4 of the 7, and its confidence does not fall when it is wrong: at four steps apart it reports E minor at r = 0.886, against 0.959 for the same progression in one key.

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.

harmony · Key-relations How fast two keys can alternate before the finder stops following. The share of bars a moving key-finder names correctly, once its reading is shifted back by its own lag, against how many bars each key holds for. One line per window. Below a block of three bars the second key is never named at all — 2 of the sweep's readings report a single key for the whole passage — and above about twice the window the tracking is over ninety per cent. The lag itself is about half the window: 0 bars at a window of 3, 0 bars at a window of 4, 3 bars at a window of 8.

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.

harmony · Key-relations The twelve keys a key-finder has never had. Every scheme the key-finder reads, read twice: over the twelve major collections the model has always used, and over twenty-four with the harmonic minor added. The pale bar is the first and the dark one the second. On 5 of the 6 the extra twelve states change nothing a reader would see — the largest loss of certainty is 3.2 per cent, on the rondo — and no bar of any of them is renamed. The exception is the ostinato, every bar of which the twelve-collection model calls E♭ and the twenty-four-collection model calls C minor — the same seven notes, the right name. So the missing states were not costing the key-finder its answers. What they were costing is the ability to say which of a collection's seven degrees is home, and that is a different repair.

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.

harmony · Key-relations The mixture at which the chain acquires a tonic. Eight turns of i–iv–v–i in A minor, natural throughout, read at every mixture of the two emissions: nought is the original set overlap against the triad on each degree, one is the measured probe-tone profile rotated to each candidate key. The blocks along the top are the name the model gives, the line below is how far ahead of its best rival that name is. Below a mixture of 0.55 every bar is called C major, which is the collection and not the key; at and above it every bar is called A minor. The margin collapses to 0.33 bits at the crossing and recovers to 4.56 — higher than the 3.38 it started at, because a profile has an opinion about this passage and an overlap does not.

A tonic bought with the function

Putting the measured probe-tone profile inside the ordered key-finder is one term, and it does what was predicted: the natural-minor passage is named A minor at every bar instead of C major. It also does two things nobody predicted. It renames a scheme that has been read in the wrong key at every bar since the day it was written, and it destroys the chord's function while it is buying the key.

harmony · Key-relations Every question is answered in a corner, not on a ridge. The degree share times the minor share, over the plane of two cues: how much of the emission is the probe-tone profile, across, and how much a bass note is worth, down, with every chord's root in the bass and a bass rule that rewards the triad rooted on the bass. It runs from 0% at a profile share of 0 and a bass weight of 0 to 87% at 0.6 and 1.5. Bass 0: 0%, 0%, 0%, 0%, 41%, 32%, 46%. Bass 0.25: 0%, 0%, 0%, 0%, 52%, 55%, 44%. Bass 0.5: 0%, 0%, 0%, 0%, 68%, 62%, 51%. Bass 1: 0%, 0%, 0%, 0%, 78%, 77%, 77%. Bass 1.5: 0%, 0%, 0%, 0%, 87%, 80%, 86%. Bass 2: 0%, 0%, 0%, 0%, 87%, 87%, 87%.

Two cues meet in a corner

The profile finds a key's tonic and the bass finds a chord's degree, and until now each was swept with the other held at nothing. Swept together across 42 settings, the plane they make is not the ridge that was predicted. The tonic is a step in one direction, at a profile share of 0.55, and the bass cannot move it; the degree is a slope in the other, rising to 89 per cent as the bass is weighted, and the profile barely touches it. Every question is answered only in a corner of the plane — and the one place the two cues overlap is the one piece of music both can rescue.

scales · Key-relations

Named alongside it

The objects these essays reach for when they reach for this one.

Key-findingModeScale degreeTonal hierarchyKey-relationsModulationPitch-class profileProbe-toneRotationEnumerationInterval contentAmbiguity

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