Acoustics & Music Theory
why it sounds like that
Concept

Hidden markov model — where it appears

A model of a sequence in which an unobserved state changes from step to step and each state emits what is actually seen. Key-finding uses one because the key is never written down: the chords are the emissions and the key is the state that has to be inferred from them.

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

A listener has a quarter of an analyst's confidence and the same answer. The mean margin between a passage's best two key readings, against how many bars a listener's memory of the evidence takes to halve. The two-sided reading — the one that uses bars that have not happened yet — sits at 3.90; the forward pass with perfect recall at 1.79; a forward pass whose evidence halves every 6.6 bars at 1.00. The dots' size is how often that reading names the same key as the two-sided one: 100 per cent at perfect recall and 81 at a one-bar half-life. So forgetting costs a great deal of confidence and very little accuracy — the key is robust and the certainty is not.

The listener who forgets

Setting an analyst's reading of a key against a listener's measures what arrives late. Both passes assume perfect recall of their own half — which is as wrong going forward as knowing the future is going back. Put a decay on the forward pass and a listener with a memory of a few bars keeps a quarter of the confidence and nine tenths of the answers.

harmony · Key-relations Which bars the key is decided by, and which bars it is believed on. Every bar of a 32-bar scheme removed in turn, with what its absence costs. The column is how far the passage's mean margin falls without that bar — how much of the model's certainty it supplies. The dot is how many bars are then read as a different key — how much of the answer it supplies. The 18 bars an analysis would point at — a section opening or closing, a dominant, the chord a dominant resolves to — average 0.079 bits of certainty and 0.50 bars moved; the 14 ordinary bars average 0.047 and 0.07. So the structural bars carry 1.7 times as much certainty as the ordinary ones, and 7.0 times of the answer. Those are different quantities, and the second is the one an analysis is about: an ordinary bar can carry a great deal of a passage's certainty and none of its reading.

The bars a key is made of

A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.

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

Named alongside it

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

Key-findingModulationInferenceKey-relationsMemory decayMusical formCadenceLeading toneMetrical weightProbe-toneScale degreeTonic

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