Acoustic phi: Difference between revisions
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[[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. | [[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 [[wikipedia:Golden_ratio#Continued_fraction_and_square_root|its continued fraction]] consists entirely of 1's. The [[wikipedia:Convergent (continued fraction)|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 741.6¢. | Acoustic phi is not to be confused with [[logarithmic phi]], which is 741.6¢. | ||
Revision as of 15:45, 21 October 2022
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 741.6¢.
Additional reading
- 833 Cent Golden Scale (Bohlen)
- Phi as a Generator
- sqrtphi, a temperament based on the square root of phi (~416.5 cents) as a generator.