musicCost
The musicCost function assigns a natural number cost to pairs of musical interval steps, returning zero when the steps match and one otherwise. Researchers working on the musical realization of the forcing chain would reference this cost function when defining the compare operation in LogicRealization. The definition is a direct case distinction on equality of the input steps.
claimFor musical interval steps $a, b$ in the natural numbers, the cost is defined by $c(a,b) = 0$ if $a = b$ and $c(a,b) = 1$ otherwise.
background
MusicalIntervalStep is an abbreviation for the natural numbers and serves as the carrier type for interval steps. The module supplies a lightweight musical realization in which the carrier records interval steps; the semantic reading is pitch-ratio stacking and the forced arithmetic is the iteration count of interval composition. This definition depends only on the MusicalIntervalStep abbreviation.
proof idea
The definition is a direct conditional expression that returns zero on equality of the two steps and one otherwise.
why it matters in Recognition Science
This cost function supplies the compare operation inside the musicRealization definition, which constructs a LogicRealization with Carrier equal to MusicalIntervalStep and Cost equal to Nat. It thereby embeds the musical interpretation into the UniversalForcing setting and supports downstream theorems that establish self-cost and symmetry properties.
scope and limits
- Does not model continuous pitch ratios or harmonic series.
- Does not invoke the J-cost function or defect distance.
- Does not depend on the phi-ladder or eight-tick octave constructions.
formal statement (Lean)
21def musicCost (a b : MusicalIntervalStep) : Nat :=
proof body
Definition body.
22 if a = b then 0 else 1
23