The local exponent — where it appears
An organ has no single exponent
The previous phase measured that a surface whose circumference grows as a power of arc length puts its transitions a fixed factor apart, and checked it on five surfaces. Every one of them had a single exponent, and no organ does — a fir cone is an ogive, whose exponent runs from 2 at the tip to nearly 0 at the shoulder.
What one exponent reports
Fit a single shape exponent to an organ that has four of them and it returns a real quantity — the harmonic mean of what its individual steps report. Harmonic means sit below arithmetic ones, so the fit understates, systematically, in a known direction, and invisibly.
How much of a cone to measure
Two rings cannot show a varying exponent — not with difficulty, but in principle, because one gap determines one exponent with nothing left to disagree. Four or five can, if each is found to within a per cent. At three per cent this specimen cannot be told from a power law however many of its rings are recorded.
Named alongside it
The objects these essays reach for when they reach for this one.
ConeLadderOgiveSpecimenTransitionsRiseCapitulumFittingHarmonic meanIdentifiabilityMeasurement errorParastichy