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EFD: Difference between revisions

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An '''EFD''' ('''equal frequency division''') or '''AD''' ('''arithmetic division''') is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning in which each period is divided to a number of steps of equal frequency difference. It is a [[period]]ic [[tuning system]].
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''-AD-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic divisions of ''p'' .  
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 ===


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


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'').
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 ==


{| class="wikitable"
{| class="wikitable"
|+Example: 4efdφ
|+Example: 4-EFDφ
|-
|-
! Steps
! quantity
! 0
! (0)
! 1
! 1
! 2
! 2
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! 4
! 4
|-
|-
! Frequency Ratios (''r'')
! 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
|-
|-
! Octaves (log<sub>2</sub>''r'')
! pitch (log₂''f'', octaves)
| 0
| (0)
| 0.21
| 0.21
| 0.39
| 0.39
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| 0.69
| 0.69
|-
|-
! Length Ratios (1/''r'')
! 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
|}
|}


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