Golden Times

This a golden ratio calculator for time durations of a musically useful length.

The golden ratio, or ϕ (phi), is a number often understood as what the Fibonacci sequence eventually expresses. Two numbers have the golden ratio if $a/b = (a+b)/a$. It has properties such that $ϕ^2 = ϕ+1$ and $1/ϕ = ϕ-1$.

Why this?

It’s a number that has fascinated people for millennia, and musical composers are no exception. I have used it to plan out sequences and compositions for my own music.

Unfortunately, the process of taking a time in my workstation, converting it to seconds, and then experimentally running some calculations on it is tedius. How many total seconds is 4:31.8? How many minutes is 261 seconds, again? What if it were 3:14.15 instead? I wanted a calculator that could take a time value c for input in an “hours, minutes, seconds” format, and provide a, b, d, and e, such that $a+b = c$, $b+c = d$, and $c + d = e$.

There are other ways the golden ratio might be used in music, but this calculator is about time values expressible in hours, minutes, and seconds.