Period

What a Period is

A Period is a continuous, bounded range with a lower bound (its minima), an origin (its midpoint), and an upper bound (its maxima). Every Period has these three anchors by default. This is the single object PPT uses to represent any range-bounded musical parameter — a pitch octave, a rhythmic bar, a dynamic swell, an effect envelope — rather than a different bespoke structure per domain.

The midpoint is the Period’s origin and is conventionally named Do. The minima and maxima are the same physical boundary approached from opposite directions (a Period is circular/octave-equivalent by default, the same way pitch space wraps at the octave), and are named ±Fi in Solfège anchor terms or ±Axis in Prime Lattice anchor terms. This dual naming of a single point is intentional, not redundant — see Anchors and Prime Lattice Coordinates for why Fi/Axis sitting at exactly the midpoint’s antipode is a structural necessity of a Do-centred circular space, not a coincidence requiring separate justification per domain.

This single model generalises two concepts that appear in more specific forms elsewhere in PPT. The external absolute that binds a rhythm hierarchy to clock time (a BPM, a reference tempo) and the one that pins a pitch hierarchy to audible frequency (a reference pitch) are both instances of the same structural requirement: every Period needs its midpoint-anchor supplied from outside — described fully under Deferred Resolution below. Equally, the reference point around which pitch comma sequences navigate is the midpoint-anchor of a pitch-octave Period. Both are the same structural role; they appear as distinct concepts only because the domains in which they appear were originally described separately.

Base vs. Reel: a real geometric property, not an authoring choice

A Period’s coordinate relationship to its parent is either:

Base — the Period’s coordinates are direct linear multiples of the parent’s. A rhythmic subdivision (a bar divided into four beats) is Base: beat 2 sits at exactly twice the position of beat 1.
Reel — the Period’s coordinates are a logarithm of the parent’s, with a named prime base. DuReel means the coordinate space is log2 relative to the parent — this is the existing pitch-cents convention (cents = 1200 × log2(ratio)), now named and generalised rather than treated as a special pitch-only rule. TriReel, QuiReel, and so on name the analogous relationship using log3, log5, etc. as the base — these are exact (logₚ(x) = log2(x)/log2(p) is a lossless change of base) but are notational conveniences for reasoning in a prime-native frame; they add no expressive power beyond what DuReel already provides, since any quantity expressed in one Reel base converts losslessly to any other.

The test for whether a property belongs in a foundational description of a Period — rather than in a discussion of how periods are specified or authored — is whether it survives resolution: whether the claim remains true of a fully-resolved coordinate structure with no memory of how it was built. Base/Reel passes this test. A resolved pitch position genuinely stands in a logarithmic relationship to its parent octave — that is a fact about auditory perception and periodicity (equal-sounding intervals are equal ratios), not a residue of how the position was specified. It would still be true if every mechanism that produced it were erased and only the final coordinates remained.

Cast: returning to the parent’s linear space

Cast is the operation that takes a Reel-typed coordinate and returns it to the parent’s linear (Base) space for a multiplicative step, before re-entering Reel space. Mechanically, Cast and its inverse are exponentiation and logarithm — exact inverses of each other. A DuReel-typed Period performing a DuCast computes 2^(position/N) to drop into linear ratio-space, applies an ordinary multiplicative step (e.g. “multiply by 4/3”), and returns via log2 — landing on exactly log2(4/3) in the DuReel coordinate, identical to adding that log value directly. Cast is a notational convenience for reasoning about a step the way a musician thinks about it (“multiply the frequency”), sitting on top of arithmetic that is exact either way — not a separate operation that could reintroduce approximation.

Implementation caveat: Cast is only lossless if it uses the true irrational value (e.g. the full-precision log2(3)) rather than a rounded rational stand-in. An implementation that rounds a Cast’d position to a fixed-precision rational before the next operation reintroduces approximation error — the same rational-versus-irrational gap that separates a Base-mode path from the exact JI position it approximates. This is a correctness requirement for any Cast implementation, not a theoretical nicety.

Deferred resolution

A Period cannot supply its own external absolute. Nothing in a Period hierarchy is bound to an absolute unit until some point outside the hierarchy — a BPM, a reference pitch, a reference dynamic level — supplies one. This is a real constraint on what a Period is, not a convention about how one is authored: a ratio, by construction, has nothing internal to it that could fix its own register. This property is sometimes stated as the Principle of Local Closure: a period’s own ratio mathematics can never resolve its own anchor.

Generalisation across domains