Acoustic phi

 Expression $\varphi = \frac{ 1 + \sqrt{5} }{2}$ Size in cents 833.0903¢ Name acoustic phi Special properties reduced Harmonic entropy(Shannon, $\sqrt{nd}$) ~4.22684 bits

ϕ taken as a frequency ratio (ϕ*f where f = 1/1) is about 833.1 cents. This metastable interval is sometimes called acoustic phi, or the phi neutral sixth. It is wider than a 12edo minor sixth (800 cents) by about a sixth-tone (33.3... cents).

ϕ is the most difficult interval to approximate by rational numbers, as its continued fraction consists entirely of 1's. The convergents (rational number approximations, obtained from the continued fractions) are the ratios of successive terms of the Fibonacci sequence converge on ϕ, the just intonation intervals 3/2, 5/3 (~884.4¢), 8/5 (~814.7¢), 13/8 (~840.5¢), 21/13 (~830.3¢), … converge on ~833.1 cents.

Erv Wilson accordingly described ϕ as "the worstest of the worst — and yet somehow with divinity imbued, Lord have mercy!", inspiring the term merciful intonation.

Acoustic phi is not to be confused with logarithmic phi, which is 1941.6¢ (741.6¢ octave-reduced).