fourth
plain-language theorem explainer
The fourth interval ratio is defined as the real number 4/3. Researchers building just-intonation scales or calibrating the alpha-log cost function cite this constant when constructing harmonic ladders that interface with the J-cost. It appears in 27 downstream sites including MusicalScale constructions and higher-derivative uniqueness theorems for CostAlphaLog. The definition is a direct noncomputable assignment with no further computation.
Claim. The perfect fourth has frequency ratio $4/3$.
background
The HarmonicModes module supplies rational frequency ratios for musical intervals, importing the Cost framework so that these ratios can enter derivative calibrations of the recognition cost. Sibling definitions include octave at 2 and fifth at 3/2; fourth at 4/3 completes the basic superparticular set used for scale construction. Upstream results such as J_log_second_deriv_at_zero establish the stiffness of the cost bowl that these concrete ratios must satisfy when embedded in the alpha-family.
proof idea
One-line definition that directly assigns the rational constant 4/3.
why it matters
This supplies the concrete fourth ratio required by justFifth and by the alphaCoordinateFixation theorems that pin alpha=1 via higher derivatives of CostAlphaLog. It feeds the inevitability_chain and J_uniquely_calibrated_via_higher_derivative results, allowing the Recognition Composition Law and eight-tick octave to be instantiated on auditory scales. The definition closes a small piece of scaffolding that lets the phi-ladder and D=3 spatial structure be tested against harmonic data.
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