Where does pedal collide?

In geometry the pedal curve of a given curve, taken with respect to a fixed point called the pole, is the locus of the feet of the perpendiculars dropped from that point onto the tangent lines of the curve. Lockwood's A book of curves derives these constructions, showing how each generating curve and pole yield a new pedal curve. The referent is a curve derived from another curve by a perpendicular-foot construction, fixed by that locus, unrelated to a sustained bass note, joined only by the Latin root for foot.[1]

Unit

The locus traced by the feet of perpendiculars from a fixed point to a curve's tangents

Recovery means

Not applicable

Measured by

Constructed point by point, or derived analytically, as the foot of the perpendicular from the pole to each tangent

In music a pedal point, or pedal tone, is a single sustained note, usually in the bass, held through changing harmonies above it, so chords come and go against a fixed anchor and tension builds, often before a cadence. Gauldin's Harmonic practice in tonal music and Kostka, Payne, and Almén's Tonal harmony treat it as a non-chord tone extended in time, named for the organ pedal that could hold such a note. The referent is a sustained pitch spanning chord changes, identified by its persistence beneath moving harmony, not a curve derived from another curve, sharing only the word pedal.[2][3]

Unit

A sustained or repeated pitch held beneath changing harmonies

Recovery means

Resolution: the harmony returns to consonance with the held note, usually at the following cadence

Measured by

Identified by a pitch sustained or repeated through two or more changes of harmony above it