Acoustic phi: Difference between revisions

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{{Infobox Interval

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

| Cents = ~~833.0902963567409~~

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

| Name = acoustic phi

}}

ϕ taken as a [[frequency ratio]] (ϕ*''f'' where ''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.

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Ratio = \varphi = \frac{ 1 + \sqrt{5} }{2}
-| Cents = 833.0902963567409
+| Cents = {{#expr: 1200 * ln((1 + sqrt(5)) / 2) / ln(2)}}
 | Name = acoustic phi
 }}
-ϕ taken as a [[frequency ratio]] (ϕ*''f'' where ''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.