Acoustics & Music Theory
why it sounds like that
Concept

Key signature — where it appears

The accidentals written at the start of a staff, which say how many steps along the chain of fifths a key sits from C. It records a position on that chain, which is why the order of sharps and flats is itself a chain of fifths.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The circle of fifths. The twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering.

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.

scales · Key-relations The key plan is the shape. The key of a classical sonata-form movement against position in the movement, measured in steps along the chain of fifths from the home key. The positions are the proportions such a movement is described by rather than bar numbers from any one score. The furthest point is 4 steps out, sharing 3 of seven notes with home.

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.

form · Key-relations The 7-note sets in which every interval occurs a different number of times. Each set of 7 notes containing C whose six interval counts are all different, with the counts printed. Every one of them is a rotation of one of two shapes, and only one of the two has steps a scale could use — the other is a run of semitones with the gap at the end.

Every interval a different number of times

Count the intervals inside a major scale and the six answers are 2, 5, 4, 3, 6 and 1 — six different numbers, no two alike. That is not decoration. It means the number of notes a key shares with a transposition of itself identifies the distance uniquely, so a listener who can only count common tones can still tell exactly how far a modulation went.

scales · The diatonic set The chain the page counts on. Every way of writing a pitch with up to one accidental, laid out along the chain of fifths and grouped by the pitch class each one sounds as. The vertical axis is what a keyboard has and the horizontal axis is what the page has: the spellings of one pitch class are seven steps of a fifth apart, which is exactly the distance a comma is measured over. Notation did not choose an arbitrary redundancy; it kept the coordinate a tuning system is built in, and equal temperament is the projection that loses it.

The stave is not a ruler

A hundred and eighty essays here draw pitch against an axis somebody computed. The one axis every reader already owns is the five lines, and it is not a pitch axis at all: it counts letters. Seven positions carry twelve pitches, so the same vertical distance is two intervals before an accidental is allowed and six after — and the accidental is not an extra symbol on a complete scale but the repair for a scale with five values missing.

scales · Notation 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 What eleven positions cover, at seven to the octave and at twelve. A clef's 11 positions, read as a range, in three systems. At seven positions to the octave they cover 1.57 octaves — an octave and a fourth, which is the seventh rung's own number. At twelve they cover 0.92. Writing a 44-semitone range then takes 3 staves and about 15 ledger positions on the staff, and 4 staves and 33 on a chromatic one. The accidentals a chromatic staff removes are paid for in vertical space, at a rate the two integers fix.

The notations invented for the overflow

Every proposal to replace the staff since the seventeenth century is a response to a specific overflow, and the commonest one — a chromatic staff with twelve positions per octave instead of seven — makes a trade computable from the same two integers the clef essay counted. It buys the accidentals outright and pays 40 per cent of the range, three ledger positions for one, and twenty-three enharmonic distinctions that are not merely absent from the page but unrecoverable.

scales · Notation

Named alongside it

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

Chain of fifthsDiatonic scaleEnharmonicInterval contentModulationNotationPivot chordRelative minorScale degreeStep patternTranspositionCircle of fifths

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