Acoustic phi

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Interval information
Expression [math]\varphi = \frac{ 1 + \sqrt{5} }{2}[/math]
Size in cents 833.0903¢
Name acoustic phi
Special properties reduced
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~4.57036 bits

Phi taken as a musical 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).

Phi 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 phi, 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 phi 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).

Additional reading