Golden ratio

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Golden ratio

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.

The phith root of phi ([math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math] or approximately 514.878 cents) is another useful interval for generating golden scales. Similarly to logarithmic phi ([math]\displaystyle{ \sqrt[2]{\varphi} }[/math]), [math]\displaystyle{ \sqrt[\varphi]{\varphi} }[/math] can be used as a generator interval to produce MOS scales whose sizes are Fibonacci numbers, where the equave is the acoustic phi instead of the octave. In this way it is a useful generator if you wish to avoid octaves and maximize the golden properties of the resulting scale.