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:

Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12epdo, or 12edo (because pitch can be assumed), standard tuning, you could divide it into 12 equal parts by ''frequency''. This would give you 12efdo.  

OD is equivalent to EFD, except that the period of OD is rational.  

An EFD will be equivalent to some [[AFS|AFS, or arithmetic frequency sequence]], which has had its count of pitches specified by prefixing "''n''-"; specifically, ''n''-efd-''p'' = ''n''-AFS((''p'' - 1)/''n'').

|+Example: 4efdφ

! Steps

! 0

! Frequency Ratios (''r'')

| 1 + (0/4)(φ - 1)<br>= (0φ + 4)/4<br>= 1

! Octaves (log₂''r'')

| 0

! Length Ratios (1/''r'')

| 1

| 4/(φ + 3)

| 2/(φ + 1)

| 4/(3φ + 1)

| 1/φ