Why the Kirnberger Kernel Is So Small

TL;DR

This paper explains the remarkable smallness of the Kirnberger kernel by expressing it through hyperbolic tangent inverse functions and showing its close approximation to a specific rational number.

Contribution

It provides a novel analytical expression for the Kirnberger kernel, revealing its tiny value as a consequence of hyperbolic tangent inverse relationships.

Findings

Kirnberger kernel is approximately 0.0000007394

Expressed as a combination of hyperbolic tangent inverse functions

Close rational approximation to the kernel's value

Abstract

Defining the musical interval of the Kirnberger kernel, or Kirn-kern, to be one-twelfth the atom of Kirnberger, or the difference between a grad and a schisma, its natural logarithm, k = (161/12)\ln{2}-7\ln{3}-\ln{5}, is extremely small, k ~ 0.000000739401. Here an explanation of this coincidence is given by showing that k = (1/6)(11\tanh^{-1}[(3/23)/11] - 21\tanh^{-1}[(3/23)/21]) ~ (2^5 5)/(3 7^2 11^2 23^3) ~ 0.000000739322.