EFD: Difference between revisions
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An '''EFD''' | An '''EFD''' ('''equal frequency division''') or '''AFD''' ('''arithmetic frequency division''') is a kind of [[Arithmetic tunings|arithmetic]] and [[period]]ic [[tuning]] in which each period is divided to a number of steps of equal frequency difference. | ||
== Specification == | |||
Its full specification is ''n''-EFD-''p'' or ''n''-AFD-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic frequency divisions of ''p'' . | |||
== Formula == | |||
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: | |||
<math> | |||
c = 1 + (\frac kn)(p-1) | |||
</math> | |||
This way, when <math>k</math> is <math>0</math>, <math>c</math> is simply <math>1</math>. And when <math>k</math> is <math>n</math>, <math>c</math> is simply <math>1 + (p-1) = p</math>. | |||
== Relationship to other tunings == | |||
=== Vs. EPD === | |||
Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12-EPDO, or 12-EDO (because pitch can be assumed), standard tuning, you could divide it into 12 equal parts by ''frequency''. This would give you 12-EFDO. | |||
=== Vs. OD === | |||
An [[OD|''n''-OD-''p'']] is equivalent to an ''n''-EFD-''p'' except that the period <math>p</math> of the OD must be rational. | |||
=== Vs. ELD === | |||
The analogous utonal equivalent of an EFD is an [[ELD|ELD (equal length division)]]. | |||
=== Vs. AFS === | |||
One period of 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''). | |||
== Examples == | |||
{| class="wikitable" | |||
|+Example: 4-EFDφ | |||
|- | |||
! quantity | |||
! (0) | |||
! 1 | |||
! 2 | |||
! 3 | |||
! 4 | |||
|- | |||
! frequency (''f'', ratio) | |||
| (1 + (0/4)(φ - 1))<br>= (0φ + 4)/4<br>= 1 | |||
| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4 | |||
| 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4 | |||
| 1 + (3/4)(φ - 1)<br>= (3φ + 1)/4 | |||
| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ | |||
|- | |||
! pitch (log₂''f'', octaves) | |||
| (0) | |||
| 0.21 | |||
| 0.39 | |||
| 0.55 | |||
| 0.69 | |||
|- | |||
! length (1/''f'', ratio) | |||
| (1) | |||
| 4/(φ + 3) = 0.87 | |||
| 2/(φ + 1) = 0.76 | |||
| 4/(3φ + 1) = 0.68 | |||
| 1/φ = 0.62 | |||
|} | |||
[[Category:Otonality]] | |||
[[Category:Harmonic]] | |||
[[Category:Harmonic series]] | |||