Acoustics Musical Note A4 Exact Rs

The A4 standard and its cost

In acoustics, the musical note A4 is the standard tuning pitch. The International Organization for Standardization set it at exactly 440 hertz in 1955, and that value remains the reference for tuning most instruments today. A hertz is one vibration cycle per second, so A4 means the air pressure oscillates 440 times each second. Musicians use this as the anchor note: orchestras tune to it, and it gives a common reference point for all other pitches.

The Recognition Science framework builds on a cost function, a way of measuring how expensive it is for one quantity to be recognized in terms of another. The framework's central result, proved in a machine-checked library of formal theorems, is that any cost function satisfying five plain conditions must equal J(x) = (x + 1/x)/2 - 1. This function measures the gap between two positive numbers: it is zero when they are equal, and it grows as they move apart. The framework applies this cost function to ratios of quantities across many domains.

In Recognition Science, the module for A4 begins with the same observation as the standard: A4 is 440 Hz exactly. The framework's research note records that 440 is close to phi^17 times 0.123, where phi is the golden ratio, about 1.618. That product gives 439.2 Hz, within about two tenths of a percent of the standard. But the module itself proves nothing about this specific match. The cost function is defined on the ratio of two real numbers, m over e, without any definition of what m and e mean for this musical note.

What the module does establish, in plain language, are three general properties of the cost function. First, the cost is zero when the two numbers are equal: if m equals e, then J(m/e) equals zero. Second, the cost is never negative when both inputs are positive. Third, the golden ratio minus 1.5 is a positive number. These are true for any positive inputs, not for A4 specifically. The module packages these three facts into a structure it calls A4ExactRS, and it proves that this structure exists. But the structure would only become a theorem about the musical note if the module defined m and e in acoustic terms, such as a measured frequency and a target frequency.

The consequence is a clear boundary. The framework's cost function is a general tool that applies to ratios of positive numbers, and it has the useful property of measuring how far apart two quantities are. The A4 module demonstrates that the tool works on any ratio, including one that could represent a frequency ratio. But it does not yet connect the golden ratio to the 440 Hz standard in a proved way. That connection remains a research note, not a result.

theorem domainCost_at_eq (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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

This module proves that A4 equals 440 Hz through the golden ratio. The golden ratio approximation to 440 Hz is a derived result rather than a research note. The cost function has any specific acoustic interpretation for the note A4.

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/Musical_Note_A4_Exact_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: