ScaleCountCert
This structure packages three facts into a certificate: exactly five canonical scale types exist, diatonic note count equals pentatonic plus two, and chromatic count is positive. Ethnomusicologists and Recognition Science researchers cite it to connect configDim D = 5 to the observed five-scale classification across cultures. The definition assembles the three properties as record fields with no further computation.
claimA structure $S$ whose fields require that the canonical scale count equals 5, the diatonic note count equals the pentatonic note count plus 2, and the chromatic note count is strictly positive.
background
The module states that cross-cultural ethnomusicology identifies five canonical scale types in virtually every tradition: pentatonic, diatonic, hexatonic, octatonic, and chromatic. Recognition Science predicts this count from configDim D = 5, using the same template as other five-element systems. Note counts follow the phi-ladder, with pentatonic at 5, diatonic at 7, and chromatic at 12.
proof idea
The structure is defined by directly requiring the three properties to hold simultaneously as record fields. It collects the canonical scale count equality, the diatonic-pentatonic relation, and the chromatic positivity without invoking tactics or lemmas beyond the upstream definitions.
why it matters in Recognition Science
This certificate is used to construct the inhabitedness theorem cert_inhabited, confirming the structure is nonempty. It realizes the module claim that five scale types are forced by configDim D = 5, consistent with the phi-ladder and eight-tick octave in the Recognition framework. The supplied falsifier is any survey finding fewer than five or more than seven canonical types across at least fifty cultures.
scope and limits
- Does not derive the scale counts from the J-function or forcing chain T0-T8.
- Does not verify the counts against specific empirical data sets.
- Does not address non-canonical or microtonal scales.
- Does not connect the counts to acoustic physics beyond the phi-ladder.
formal statement (Lean)
56structure ScaleCountCert where
57 scale_count : canonicalScaleCount = 5
58 diatonic_eq : diatonicCount = pentatonicCount + 2
59 chromatic_pos : 0 < chromaticCount
60