Acoustics Music Consonance From Jcost

The consonance ranking

In acoustics, consonance is the quality that makes some pairs of notes sound stable and agreeable together, while dissonance makes others sound tense or clashing. The classical ranking of intervals from most to least consonant is well known: the unison, then the octave, the perfect fifth, the perfect fourth, the major third, the minor third, the major second, and finally the tritone, which sounds the most dissonant. Recognition Science offers a framework for deriving this ranking from a single mathematical function rather than from listening tests or cultural convention.

The function is the cost of recognition, a number that measures how expensive it is for a system to recognize one value as matching another. The framework's central theorem proves that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. In the acoustics module, the cost of recognizing a musical interval is defined as J(m/e), where m and e are two frequencies. When the frequencies are equal, the ratio is 1 and the cost is 0, meaning perfect recognition at the unison. For any positive frequencies, the cost is never negative, so the ranking is always well defined.

The module proves three general facts about this cost function: it vanishes when the two frequencies are equal, it is nonnegative for all positive inputs, and the threshold phi - 3/2 is positive, where phi is the golden ratio. These facts are proved in the machine-checked library of formal theorems. However, the module itself proves nothing specific to music, because the definition of the cost function does not reference any particular physical property of sound. The ranking of intervals, from unison at J=0 through octave, fifth, fourth, major third, minor third, major second, and tritone at the largest J, is a research note recording where the idea was meant to go, not a proved result.

What the module does establish is the mathematical skeleton: a cost function that is zero at the unison, never negative, and has a positive threshold. The consonance ranking is the intended interpretation, but the bridge from the mathematics to the acoustics remains open. The framework proves the cost function's general properties; the specific claim that this cost function ranks musical intervals as described is a hypothesis that would need a definition of m and e in terms of actual sound waves to become a theorem.

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem domainCost_at_equilibrium (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module proves the consonance ranking itself; it proves only general properties of the cost function. The ranking of intervals is a research note, not a theorem, because m and e are not defined in acoustical terms. The cost function J is derived from five conditions, but its application to music is a definitional choice, not a forced result.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Acoustics/MusicConsonanceFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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