Acoustic phi: Difference between revisions
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| Name = acoustic phi | | Name = acoustic phi | ||
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ϕ taken as a [[frequency ratio]] ( | ϕ taken as a [[frequency ratio]] (ϕ⋅''f'' where {{nowrap|''f'' {{=}} 1/1}}) is about 833.1 [[cent]]s. 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 {{w|Golden ratio #Continued fraction and square root|its continued fraction}} consists entirely of 1's. The | ϕ is the most difficult interval to approximate by rational numbers, as {{w|Golden ratio #Continued fraction and square root|its continued fraction}} consists entirely of 1's. The {{w|Convergent (continued fraction)|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]]. | [[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]]. | ||