Acoustics & Music Theory
why it sounds like that
Concept

Wolf fifth — where it appears

The one badly out-of-tune fifth left over when a chain of eleven good ones has to be joined up over seven octaves. Putting the whole error in one interval is a policy, and it makes eleven keys usable at the cost of making one unusable.

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

Where one interval stops being itself. Identification as a function of interval size: the probability that a listener names each category, modelled as a logistic with the boundary positions and sharpness a study reports. One category's share falls from three-quarters to one-quarter over 24 cents, against the 100 that separate adjacent categories — so the change of mind happens in 24% of the gap and the rest of it is not in doubt at all. That is what makes a mistuned third a third that is out, rather than a different interval.

The chord is still major, and that is why temperament works

A major third can be seventeen cents wrong and still be a major third. That tolerance is not a failure of hearing — it is the reason the whole subject of tuning is a discussion rather than a catastrophe. Every temperament ever proposed moves intervals around inside their categories, and the one thing none of them may do is push one across a boundary.

perception · Categorical-hearing Every keyboard temperament before 1800, on one line. The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number.

A fraction of a comma

The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.

tuning · The comma

Named alongside it

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

TemperamentCategorical perceptionCategory widthCentsEqual temperamentJust intonationMeantoneSyntonic commaTolerance

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