EFD: Difference between revisions

An '''EFD''' ('''equal frequency division''') or '''AD''' ('''arithmetic division''') is a [[period]]ic and [[Arithmetic tuning|arithmetic]] [[tuning system]] in which each period is divided to a number of steps of equal frequency difference.  

Its full specification is ''n''-efd-''p'' or ''n''-ad-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic divisions of ''p'' .  

To find the steps for an ''n''-efd-''p'', begin by recognizing that while the multiplicative interval relating your root position to the end position is <math>p</math> (or <math>\frac p1</math>), if you are going to move arithmetically (by repeated addition) from <math>1</math> to <math>p</math>, then the difference in frequency space that you are dividing up is not actually <math>p</math>, but <math>p - 1</math>. And because you are dividing it into <math>n</math> parts, each step will have a size of <math>\frac{p-1}{n}</math>. So within each period, the ratio ''c'' of the ''k''-th step of an ''n''-efd-''p'' is:

c = 1 + (\frac kn)(p-1)

This way, when <math>k</math> is <math>0</math>, <math>f(k)</math> is simply <math>1</math>. And when <math>k</math> is <math>n</math>, <math>f(k)</math> is simply <math>1 + (p-1) = p</math>.