Acoustics & Music Theory
why it sounds like that
Concept

Krumhansl schmuckler — where it appears

The key-finding method that correlates a passage's pitch-class histogram against twenty-four profiles measured on listeners. It is the standard against which every other key-finder here is compared, and its defining property is that it throws the order of the notes away.

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

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 test a histogram cannot pass. Two passages built from the same two keys: one takes them in turn and the other sounds them together. Over a window long enough to hold both, their pitch-class histograms are 98.6 per cent alike, so they are very nearly the same object to a profile model — and it names two keys in both. The ordered model names up to four for the alternation and one for the simultaneity, because a simultaneity has no sequence in it that any single key's grammar will not fit.

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.

harmony · Key-relations Two passages, one note apart, and opposite cadences. Two passages that share eight bars drawn from the six pitch classes C major and G major have in common, and differ only in their last three bars: one cadences in C and one in G, and the single note that separates them is an F against an F sharp. The cadence evidence names C for the first and G for the second, by 9 to 3 and 6 to 4. The profile model names E minor for both, with correlations differing in the third decimal.

The cadence as evidence

Every model so far infers a key from a bag of notes, and the thing that most obviously declares a key is a cadence — an ordered pair of chords in which the order is the whole content. The essays on closure built a five-component cadence vector long before this and nobody has used it for this. Two passages differing in one note, one cadencing in C and one in G, get opposite answers from the ordered pairs and the same answer from the profile.

harmony · Progression

Named alongside it

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

Key-findingPitch-class profileBitonalityCadenceInferenceModulationAmbiguityClosureNull modelProgressionTonal functionTonic

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