Theremin Trio

Concepts

Scale

A monotonically increasing trajectory through log‑frequency space, parameterized by a position t in note-space [0, 1]. The rate of increase is unconstrained — only the direction is.

Discrete scale

A scale which instantaneously switches between discrete levels and pauses on those frequencies. Piecewise constant.

Periodic scale

A scale which repeats by frequency‑doubling (or any ratio). S(t + T) = S(t) + log₂(r).

Discrete periodic scale

The intersection — discrete steps that repeat at the octave. The major scale is the canonical example.

Chord

A subset of a scale played simultaneously.

Discrete arpeggio

The same subset played sequentially.

Chord trajectory

Several note-space positions LERPing together over time. Each voice has its own time→note-space function; only the scale itself must be monotonic, the trajectories may wander.

Arpeggio trajectory

The sequential rendering of a chord trajectory — one voice at a time.

The PDF connection

A monotonic map t → log‑frequency is an inverse CDF. Treating t as uniform on [0, 1], the scale becomes a draw from a distribution over log‑frequency with density p(log f) = 1 / S'(t). Slow-rise regions are high-density (stable pitches); fast-rise regions are low-density (transition pitches).

Scale property	Distribution interpretation
Slow-rise segment	High-density — "stable" pitches
Fast-rise segment	Low-density — "transition" pitches
Plateau (discrete)	Delta spike — a point mass at one pitch
Periodic, octave ratio	Density is octave‑equivalent on log‑pitch
Continuous aperiodic	Arbitrary continuous density
Major scale	Seven delta spikes, repeating each octave
Log‑linear sweep	Uniform density over log‑frequency

In the trio below: the bass lives in slow-rise zones (it sounds anchored); the lead lives in fast-rise zones (small p-movements become wide pitch sweeps, so it reads as expressive). That alignment of voice-role to density-region wasn't planned — it's why the piece holds together.