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. It is wider than a [[12edo]] [[minor sixth]] (800 cents) by about a [[sixth-tone]] (33.~~3...~~ cents).

{{Infobox Interval

| Ratio = \varphi = \frac{ 1 + \sqrt{5} }{2}

| Cents = 833.0902963567409

| Name = acoustic phi

}}

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).

~~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.

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 ~~phi~~ 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]].

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 1941.6¢ (741.6¢ octave-reduced).

== ~~Additional reading~~ ==

== See also ==

* [[833 Cent Golden Scale (Bohlen)]]

* [[Phi as a ~~Generator~~]]

* [[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

~~[[Category:Irrational intervals]]~~

[[Category:Golden ratio]]

[[Category:Supraminor sixth]]

@@ Line 1: / Line 1: @@
-[[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).
+{{Infobox Interval
+| Ratio = \varphi = \frac{ 1 + \sqrt{5} }{2}
+| Cents = 833.0902963567409
+| Name = acoustic phi
+}}
+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).
-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.
+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 phi 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]].
-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 1941.6¢ (741.6¢ octave-reduced).
-== Additional reading ==
+== See also ==
 * [[833 Cent Golden Scale (Bohlen)]]
-* [[Phi as a Generator]]
+* [[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
-[[Category:Irrational intervals]]
 [[Category:Golden ratio]]
+[[Category:Supraminor sixth]]