Acoustic phi

Interval information

Expression	[math]\displaystyle{ \varphi = \frac{ 1 + \sqrt{5} }{2} }[/math]
Size in cents	833.0903¢
Name	acoustic phi
Special properties	reduced

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).

See also

• 833 Cent Golden Scale (Bohlen)
• Edφ, tunings created by dividing acoustic phi into equally sized smaller steps
• Phi as a generator
• sqrtphi, a temperament based on the square root of phi (~416.5 cents) as a generator.
• photosynthesis, a temperament using phi as a “prime” in its subgroup