Acoustic phi: Difference between revisions

(9 intermediate revisions by 3 users not shown)

Line 1:

{{Infobox Interval

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

| Cents = ~~{{#expr: 1200 * ln((1 + sqrt(5)) / 2) / ln(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).

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 ~~[[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 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.

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]].

Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).

The [[phith root of phi]] is another interval with interesting properties, that divides acoustic phi logarithmically by phi, which creates self similar, fractal-like scales.

== Approximation ==

{{Interval edo approximation|interval = 1618/1000 | interval_name = ϕ}}

== 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.

* [[Sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.

* [[~~Edφ~~]], ~~tunings created by dividing acoustic~~ phi ~~into equally sized smaller steps~~

* [[Photosynthesis]], a temperament using phi as a “prime” in its subgroup

[[Category:Golden ratio]]

[[Category:Supraminor sixth]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = \varphi = \frac{ 1 + \sqrt{5} }{2}
-| Cents = {{#expr: 1200 * ln((1 + sqrt(5)) / 2) / ln(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).
+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 [[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 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.
+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]].
 Acoustic phi is not to be confused with [[logarithmic phi]], which is 1941.6¢ (741.6¢ octave-reduced).
+The [[phith root of phi]] is another interval with interesting properties, that divides acoustic phi logarithmically by phi, which creates self similar, fractal-like scales.
+== Approximation ==
+{{Interval edo approximation|interval = 1618/1000 | interval_name = ϕ}}
 == 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.
+* [[Sqrtphi]], a temperament based on the square root of phi (~416.5 cents) as a generator.
-* [[Edφ]], tunings created by dividing acoustic phi into equally sized smaller steps
+* [[Photosynthesis]], a temperament using phi as a “prime” in its subgroup
 [[Category:Golden ratio]]
 [[Category:Supraminor sixth]]