EFD: Difference between revisions
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An '''EFD''' ('''equal frequency division''') or ''' | 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 == | == Specification == | ||
Its full specification is ''n''-EFD-''p'' or ''n''- | 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 == | == Formula == | ||
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=== Vs. OD === | === Vs. OD === | ||
An [[OD|n- | 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 === | === Vs. ELD === | ||
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=== Vs. AFS === | === 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 == | == Examples == | ||
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|- | |- | ||
! quantity | ! quantity | ||
! 0 | ! (0) | ||
! 1 | ! 1 | ||
! 2 | ! 2 | ||
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! 4 | ! 4 | ||
|- | |- | ||
! frequency (f) | ! frequency (''f'', ratio) | ||
| 1 + (0/4)(φ - 1)<br>= (0φ + 4)/4<br>= 1 | | (1 + (0/4)(φ - 1))<br>= (0φ + 4)/4<br>= 1 | ||
| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4 | | 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4 | ||
| 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4 | | 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4 | ||
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| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ | | 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ | ||
|- | |- | ||
! pitch ( | ! pitch (log₂''f'', octaves) | ||
| 0 | | (0) | ||
| 0.21 | | 0.21 | ||
| 0.39 | | 0.39 | ||
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| 0.69 | | 0.69 | ||
|- | |- | ||
! length (1/f) | ! length (1/''f'', ratio) | ||
| 1 | | (1) | ||
| 4/(φ + 3) | | 4/(φ + 3) = 0.87 | ||
| 2/(φ + 1) | | 2/(φ + 1) = 0.76 | ||
| 4/(3φ + 1) | | 4/(3φ + 1) = 0.68 | ||
| 1/φ | | 1/φ = 0.62 | ||
|} | |} | ||