scale_fibonacci
The pentatonic scale size plus the diatonic scale size equals the semitone count per octave, confirming the 5 + 7 = 12 Fibonacci relation in the 12-tone system. Researchers deriving musical scales from the golden ratio in Recognition Science would cite this identity to link scale structure to phi-ladder arithmetic. The proof is a one-line native computation that evaluates the three constant definitions directly.
claimThe pentatonic scale size (5 notes) plus the diatonic scale size (7 notes) equals the number of semitones per octave (12).
background
The module derives the Western 12-tone equal temperament scale from the golden ratio phi by optimizing consonance and closure under frequency ratios. PentatonicSize is defined as the constant 5, diatonicSize as the constant 7, and semitonesPerOctave as the constant 12. These sizes are presented as consecutive Fibonacci numbers whose sum recovers the octave division, consistent with the module observation that 12 emerges from rounding phi^5 / 2 and that the circle of fifths closes after 12 steps with (3/2)^12 approximately 2^7.
proof idea
The proof is a one-line wrapper that applies native_decide to the three constant definitions. Native_decide evaluates the arithmetic equality 5 + 7 = 12 by direct computation without further lemmas.
why it matters in Recognition Science
This identity anchors the Fibonacci-like structure 5, 7, 12 inside the phi-derived musical scale construction. It supports the module claim that 12 semitones arise from phi^5 scaling and optimal frequency ratios, placing the result within the Recognition Science aesthetics framework that connects phi-ladder arithmetic to observable harmonic structure. No downstream uses are recorded, so the declaration functions as a local consistency check rather than a lemma for further theorems.
scope and limits
- Does not derive the scale sizes from the Recognition Composition Law or J-cost.
- Does not prove that 12 is the unique optimal division for consonance.
- Does not address alternative scale sizes such as 19 or 31.
- Does not connect the equality to spatial dimension D=3 or the eight-tick octave.
formal statement (Lean)
177theorem scale_fibonacci : pentatonicSize + diatonicSize = semitonesPerOctave := by native_decide
proof body
Term-mode proof.
178
179/-! ## Falsification Criteria
180
181The musical scale derivation is falsifiable:
182
1831. **12 not optimal**: If a different number gives better consonance/closure
184
1852. **φ connection spurious**: If φ^5 ≈ 11 is coincidental
186
1873. **Circle of fifths**: If (3/2)^n ≠ 2^m for any small n, m
188-/
189
190end
191
192end MusicalScale
193end Aesthetics
194end IndisputableMonolith