Golden ratio: Difference between revisions
m Added Wikipedia box, removed capital phi |
mNo edit summary |
||
| Line 1: | Line 1: | ||
{{Wikipedia}} | {{Wikipedia}} | ||
The '''golden ratio''' or '''phi''' (Greek letter | The '''golden ratio''' or '''phi''' (Greek letter <math>\phi</math> / <math>\varphi</math>) may be defined by <math>\frac{a}{b}</math> such that <math>\frac{a}{b} = \frac{a+b}{a}</math>. It follows that <math>\varphi - 1 = 1 / \varphi</math>, and also that <math>\varphi = \frac{1+\sqrt{5}}{2}</math>, or approximately 1.6180339887... <math>\varphi</math> is an irrational number that appears in many branches of mathematics. | ||
== Musical applications == | == Musical applications == | ||
The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as [[acoustic phi]] and in the latter case it is known as [[logarithmic phi]]. [[Lemba]] is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. [[Triforce]] is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions. | |||
[[Category:Golden ratio| ]] | |||
[[Category:Theory]] | |||
Revision as of 03:52, 24 January 2022
The golden ratio or phi (Greek letter [math]\displaystyle{ \phi }[/math] / [math]\displaystyle{ \varphi }[/math]) may be defined by [math]\displaystyle{ \frac{a}{b} }[/math] such that [math]\displaystyle{ \frac{a}{b} = \frac{a+b}{a} }[/math]. It follows that [math]\displaystyle{ \varphi - 1 = 1 / \varphi }[/math], and also that [math]\displaystyle{ \varphi = \frac{1+\sqrt{5}}{2} }[/math], or approximately 1.6180339887... [math]\displaystyle{ \varphi }[/math] is an irrational number that appears in many branches of mathematics.
Musical applications
The golden ratio can be used as a frequency multiplier or as a pitch fraction; in the former case it is known as acoustic phi and in the latter case it is known as logarithmic phi. Lemba is particularly notable for approximating both simply and accurately simultaneously, at a generator + a period for acoustic and 2 generators for logarithmic, making it an excellent choice for experimenting with phi based composition. Triforce is also essentially based on dividing the 1/3 octave period into logarithmic phi sized fractions.