perfectFourth
The equal-tempered perfect fourth is the frequency ratio two raised to five-twelfths. Researchers analyzing phi-derived musical scales cite this when mapping twelve-tone temperament to optimal consonance and closure. The definition is a direct assignment from the semitone division of the octave.
claimThe frequency ratio of the perfect fourth in twelve-tone equal temperament is $2^{5/12}$.
background
The Aesthetics.MusicalScale module constructs the Western twelve-tone equal temperament scale from the golden ratio φ by optimizing consonance via simple ratios and closure of the circle of fifths. It records that twelve semitones per octave arise as round(φ^5 / 2), the perfect fifth approximates 3/2 via 2^{7/12}, and the major third approximates 5/4 via 2^{4/12}. The upstream strict definition realizes the perfect fourth exactly as the rational 4/3 using octave stacking as the canonical generator.
proof idea
This is a direct definition that assigns the real value 2 raised to the power 5/12. No lemmas are applied; the expression follows immediately from dividing the octave into twelve equal semitones.
why it matters in Recognition Science
The definition supplies the equal-temperament numerical value required by the module's RS mechanism, where twelve emerges from φ-scaling and the relation (3/2)^12 ≈ 2^7. It supports the parent strict music realization by furnishing the approximation used in predictions for scale optimality. The construction aligns with the eight-tick octave period and self-similar fixed-point structure in the forcing chain.
scope and limits
- Does not claim equality between 2^{5/12} and the just interval 4/3.
- Does not derive the exponent 5/12 from the Recognition Composition Law.
- Does not address closure or consonance for scales other than twelve tones.
formal statement (Lean)
52def perfectFourth : ℝ := 2 ^ (5 / 12 : ℝ)
proof body
Definition body.
53
54/-- Major third in equal temperament: 2^(4/12). -/