Acoustics & Music Theory
why it sounds like that
Concept

Relative pitch — where it appears

The ability to identify intervals and recognise melodies without reference to any absolute frequency. It is what nearly every listener has, and it is what makes a tune recognisable in a key it was never heard in.

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

Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded.

A memory for the note itself, and it is dated

Absolute pitch is usually described as a rare perceptual gift. It is better described as a memory for a convention — and conventions have dates. Possessors trained on A=440 mis-name Baroque pitch by a semitone, their own labels drift sharp with age, and meanwhile most listeners without it start familiar songs within a semitone of the record.

tuning · Pitch standard The arch is not a preference. Every sequence of 6 notes over 8 scale degrees — 262,144 of them, enumerated rather than sampled — classified by contour, under three constraints. With none, the nine classes are spread. Requiring the sequence to return to its starting degree leaves only the arch, the valley and the flat, at 42.2 per cent each for the first two. Requiring it to begin and end on the LOWEST degree leaves the arch alone, at 99.6 per cent. Nothing here prefers a rise followed by a fall; the constraint is that the melody comes home, and a melody that comes home from below has nowhere to go but up first.

The shape that survives everything else

Throw away a melody's key, its tuning, its instrument and the sizes of its intervals, and what is left is a string of pluses and minuses. That string is what a listener who cannot name a note still has, and it costs 37 per cent of the tune to keep. The arch that melodic shape is famous for is not in it as a preference: enumerate every six-note sequence that begins and ends on the lowest degree it uses and 99.6 per cent of them are arches, because a melody that comes home from below has nowhere to go first but up.

form · Melody What a contour costs to remember. A melody of n notes over 8 degrees carries 3 bits a note. Its contour carries fewer, and fewer than the number of distinct contours suggests, because the contours are not equally likely: at 6 notes there are 243 of them but the entropy is 6.59 bits, an effective alphabet of 96. Each further note adds 1.28 bits of contour against three of melody, so the shape keeps a stable 37 per cent of what is there however long the tune.

The part of the tune that is kept

Contour survives transposition, retuning, a change of instrument and a doubling of every interval, and the usual explanation is that it is what a listener retains. That can be counted rather than assumed. A six-note melody over eight degrees carries eighteen bits; its contour carries 6.59 — not the 7.92 the number of distinct shapes suggests, because the shapes are wildly unequal — and the effective alphabet is ninety-six out of two hundred and forty-three. Each further note adds 1.28 bits of shape against three of melody, and at about nine notes a contour is specific enough to pick one tune out of a thousand.

perception · Melody

Named alongside it

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

ContourEnumerationPitch memoryAbsolute pitchArchCritical periodDescription lengthInvarianceMelodic intervalMemory decayPerceptual presentPitch labelling

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