I am looking for the mathematics that can be used to calculate a diatonic scale.
It is my understanding that a number of musical scales can be represented in logarithmic form using mathematical expressions.
From what I have read, our hearing and vision are logarithmic in nature and so I am intrigued by the mathematics behind it and wish to also apply the same mathematics to color shades.
Could someone enlighten me as to a method for calculating geometric, diatonic, tritonic, tetronic, typographic, and pentatonic scales or point me to where I can continue my research?
The standard Western music system, which is also used in many other places around the world, contains 12 distinct pitches. In Pythagorean tuning, a pitch has double the frequency of the pitch an octave below it, $\frac{3}{2}$ the frequency of the pitch a fifth below it, $\frac{4}{3}$ the frequency of the pitch a fourth below it, $\frac{5}{4}$ the frequency of the pitch a major third below it, and $\frac{6}{5}$ the frequency of the pitch a minor third below it. This is known as the harmonic series. The problem is, counting notes (half steps), 12 fifths is equal to 7 octaves, but $2^7$ = 128 while $(\frac{3}{2})^{12} \approx$ 129.75 . To reconcile these differences, tuning systems (temperament) were created. In equal temperament, each octave still has its upper note double the frequency of its lower note, and all the half steps in between are made to be the same frequency ratio, so each half step has its upper note $\sqrt[12]{2}$ times higher in frequency than its lower note. To add intervals, multiply their ratios. So a major third (which contains 4 half steps) plus a half step equals a perfect fourth (which contains 5 half steps) --> $2^\frac{4}{12} * 2^\frac{1}{12} = 2^\frac{5}{12}$. In other temperaments (tuning systems) the intervals are not equal ratios of frequencies apart, so each scale has to be calculated separately depending on which particular notes it uses. I'd stick with equal temperament for that reason.