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An '''EFD''', or '''equal frequency division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Monotonic tunings|monotonic]] tuning.
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.  


n-EFDp: n equal frequency divisions of interval p
== Specification ==


7EFDπ is dividing the frequency space between 1 and π by 7, so that makes each step (π-1)/7. So the 2nd step is 1+(π-1)/7 = 7/7+(π-1)/7 = (π+6)/7, the 3rd step would be 1+2(π-1)/7 = (2π+5)/7, then I see the pattern forming so it goes (3π+4)/7, (4π+3)/7, (5π+2)/7, (6π+1)/7, and (7π+0)/7 = π at the end.
Its full specification is ''n''-EFD-''p'' or ''n''-AFD-''p'': ''n'' equal frequency divisions of ''p'', or ''n'' arithmetic frequency divisions of ''p'' .  


So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
== Formula ==


The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
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:


But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. However, that's not exactly ideal because, as with arithmetic sequences, different acronyms are used to distinguish rational (JI) tunings from irrational (non-JI) tunings, and so EFD and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.
<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"
{| class="wikitable"
|+example: 4-EFDφ
|+Example: 4-EFDφ
|-
|-
! quantity
! quantity
! (0)
! 1
! 1
! 2
! 2
! 3
! 3
! 4
! 4
! (5)
|-
|-
! frequency (f)
! frequency (''f'', ratio)
|1
| (1 + (0/4)(φ - 1))<br>= (0φ + 4)/4<br>= 1
|1.15
| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4
|1.31
| 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4
|1.46
| 1 + (3/4)(φ - 1)<br>= (3φ + 1)/4
|(φ)
| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ
|-
|-
! pitch (log₂f)
! pitch (log₂''f'', octaves)
|0
| (0)
|0.21
| 0.21
|0.39
| 0.39
|0.55
| 0.55
|(0.69)
| 0.69
|-
|-
! length (1/f)
! length (1/''f'', ratio)
|1
| (1)
|0.87
| 4/(φ + 3) = 0.87
|0.76
| 2/(φ + 1) = 0.76
|0.68
| 4/(3φ + 1) = 0.68
|(1/φ)
| 1/φ = 0.62
|}
|}
An [[OD|OD (or otonal division)]] is a specific (rational) type of EFD.
 
[[Category:Otonality]]
[[Category:Harmonic]]
[[Category:Harmonic series‏‎]]

Latest revision as of 20:35, 19 October 2023

An EFD (equal frequency division) or AFD (arithmetic frequency division) is a kind of arithmetic and periodic 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]\displaystyle{ p }[/math] (or [math]\displaystyle{ \frac p1 }[/math]), if you are going to move arithmetically (by repeated addition) from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ p }[/math], then the difference in frequency space that you are dividing up is not actually [math]\displaystyle{ p }[/math], but [math]\displaystyle{ p - 1 }[/math]. And because you are dividing it into [math]\displaystyle{ n }[/math] parts, each step will have a size of [math]\displaystyle{ \frac{p-1}{n} }[/math]. So within each period, the ratio c of the k-th step of an n-EFD-p is:

[math]\displaystyle{ c = 1 + (\frac kn)(p-1) }[/math]

This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ c }[/math] is simply [math]\displaystyle{ 1 }[/math]. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ c }[/math] is simply [math]\displaystyle{ 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 n-OD-p is equivalent to an n-EFD-p except that the period [math]\displaystyle{ p }[/math] of the OD must be rational.

Vs. ELD

The analogous utonal equivalent of an EFD is an ELD (equal length division).

Vs. AFS

One period of an EFD will be equivalent to some 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

Example: 4-EFDφ
quantity (0) 1 2 3 4
frequency (f, ratio) (1 + (0/4)(φ - 1))
= (0φ + 4)/4
= 1 		1 + (1/4)(φ - 1)
= (1φ + 3)/4 		1 + (2/4)(φ - 1)
= (2φ + 2)/4 		1 + (3/4)(φ - 1)
= (3φ + 1)/4 		1 + (4/4)(φ - 1)
= (4φ + 0)/4
= φ
pitch (log₂f, 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
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