Acoustics & Music Theory
why it sounds like that
Concept

Ambiguity — where it appears

A passage that supports more than one analysis, where the alternatives are equally good rather than one being merely arguable. It matters here because most of the models here return a single best answer and have no value that means two, so an ambiguity arrives as false confidence.

Named by 4 essays across 2 fields — 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 What the joint search changes, and what it never changes. Over 552 constructed passages of eight slots with rests, how often the joint reading differs from the pipeline's. The chord differs in 29 per cent and the barline in 31, with both differing in 20. The key differs in 0 per cent — never — because the key is read from a pitch-class histogram, which does not know where the bar starts or which notes are chord tones. Two of the three decisions are entangled and the third is not.

Three decisions that constrain each other

Every model here decides one thing at a time — the key from the pitch classes, the metre from the onsets, the chords from the metre — and an earlier essay ended by saying a listener does all three at once. Resolving them jointly costs a hundred and fifty-seven times the search and changes the reading of two passages in five. It never once changes the key, and the reason it cannot is the reason the whole account is built the way it is.

harmony · Progression How sure the reading is, bar by bar. Every earlier essay reports one best reading. 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 has never been printed. Here it is, in bits, for the thirty-two-bar AABA. The mean margin is 1.14 bits and 13 of 32 bars are inside one bit of a rival reading, which is where a listener would be genuinely undecided. The reading itself names C, E, B, D, G; the margin says what that naming is worth, and at the weakest bar — bar 31, C over F — it is worth 0.07.

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.

scales · Key-relations How much of the reading comes from what has not happened yet. Every margin reported earlier is two-sided: the best path through a key at a bar is the best score into it plus the best score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line, because the dynamic program already had both halves separately. The mean margin falls from 3.90 bits with hindsight to 1.79 without it, so 54 per cent of this passage's certainty is retrospective. The two passes never disagree about which key is best here, so the hindsight buys confidence rather than a different answer. This is the quantity every earlier essay has assumed and none has measured.

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.

scales · Key-relations

Named alongside it

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

Key-findingModulationInferenceInformationSegmentationAnalysisBitonalityChord toneExpectationHarmonic analysisKey colourKrumhansl schmuckler

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