Green Dungeon Alchemist Laboratories
y=2^x-1

What is that inane jibberish in the title? I may hear you cry, why, it is the exponential function! I reply.

Yesterday I got a book from my electronics tutor called “The physics of music” by Alexander Wood, revisions by J M Bowsher. This book both greatly infuriated and assisted me in developing my own musical scale over the last day. I think I spent about 10 hours reading it on and off and doing sums. The damn thing kept going on and on about history (next section to be read in posh accent) <posh> how such and such a musician in 1864 used 436hertz for A while france was still using 437 hertz, oh what a situation that would cause should they play an orchestra together hahahahaha! </posh>

All I know is that it got to 20 past midnight and I was still doing sums and reading things. Like Sheldon in the Big Bang Theory I had become victim to my own determination, sleep deprivation deprived me of my wits and thus an answer. 

I slept and this morning I figured it out with help from the exponential function and this equation:

f=440*2^(n/12) 

That’s the equation for figuring out the notes in the standard 12 note musical scale, gleaned from amongst the history and factoids in “the physics of music” last night. At the time I was either too sleep deprived or the book was too vague for me too understand it fully. I just got my head around it this morning with the help of a graph and an example which the book was sadly missing, I think it was a little bit too sparse on this area for me to fully comprehend. Anyway the point is, I do now and A 400Hz = 0 = n, the next B up would be n = 1, G# or Ab would be n = -1. A 880 an octave up is n = 12. If only the book had portrayed it that simply!

No matter, in my sleep deprived stated I scrawled next to it: 2^n-1?

The chior of understanding sounded and it is now understood as a variant of 2^(n-1) or graphulacily y=2^(x-1) That’ll give you the traditional exponential curve.

My goal was to make a musical scale which started at a base frequency of 1Hz. Then as it increases in octaves you will get nice even numbers; 2, 4, 8, 16, 32, 64, 128, etc. Do they look familiar? ;) It’s the ol’ binary values!

I noticed that using a base frequency of 1, f=1*(2^n/(notes in octave)) gives the same values as y=2^(x-1) if you divide the integer x by the number of notes in an octave; e.g. if you have 8 notes per octave: 0, 1/8, 2/8, 3/8 … 7/8, 1. Add 1 for each additional octave or multiply octave 0 by 2, then that by 2 and so on to get the values for additional octaves. The two equations are very similar.

It strikes me that someone somewhere has probably done all this and I could have just read it somewhere else, but I didn’t, I number crunched intermittently for somewhere between 5 and 10 hours while reading this unintelligable, but ultimately helpful book, and I came up with good results at the end! Hell yes!