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).

As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 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¢.

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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).
-As the ratios of successive terms of the Fibonacci sequence converge on phi, the just intonation intervals 3/2, 5/3, 8/5, 13/8, 21/13, ... converge on ~833.1 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¢.