unisonInterval
The unison interval definition supplies the trivial musical interval whose frequency ratio is exactly 1. Researchers modeling harmonic series inside Recognition Science would cite it to anchor the zero point of the J-cost function or to initialize sets of intervals before applying the Recognition Composition Law. The definition is a direct structure constructor that supplies the constant 1 together with a norm_num tactic discharging positivity.
claimThe unison interval is the musical interval whose frequency ratio equals $1$, formalized as the pair consisting of the real number $1$ together with a proof that $0 < 1$.
background
In the HarmonicModes module a MusicalInterval is a structure carrying a real ratio together with a proof that the ratio is positive. The structure is equipped with a J-cost obtained by applying the J-cost function from the Cost module to that ratio. The module sits inside the Recognition Science treatment of harmonics and imports the Cost module, thereby linking interval ratios to the J-uniqueness property and the phi-ladder of the forcing chain.
proof idea
The definition is a one-line wrapper that invokes the MusicalInterval constructor on the literal value 1 and discharges the positivity hypothesis via the norm_num tactic.
why it matters in Recognition Science
This definition supplies the identity element for the collection of harmonic intervals, providing the case of vanishing J-cost. It stands beside sibling definitions for the octave, fifth, fourth, and superparticular intervals, preparing the ground for later use of the Recognition Composition Law. No downstream theorems currently depend on it, leaving open its role in deriving the eight-tick octave or mode analysis from the T5-T8 chain.
scope and limits
- Does not compute any J-cost value.
- Does not relate the unison interval to the octave or fifth.
- Does not invoke the phi constant or forcing-chain steps T5-T8.
- Does not participate in interval multiplication or composition.
formal statement (Lean)
104noncomputable def unisonInterval : MusicalInterval :=
proof body
Definition body.
105 ⟨1, by norm_num⟩
106