EFD: Difference between revisions

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</math>
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This way, when <math>k</math> is <math>0</math>, <math>f(k)</math> is simply <math>1</math>. And when <math>k</math> is <math>n</math>, <math>f(k)</math> is simply <math>1 + (p-1) = p</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  ==
== Relationship to other tunings  ==
=== Vs. EPD ===
=== 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.
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.  
 
=== Vs. ODO ===


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 is typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name this tuning 12-ODO, for [[OD|otonal divisions]] of the octave.
=== Vs. OD ===


The only difference between [[OD|n-ODp]] and n-EFDp is that the p for an EFD is irrational.
OD is equivalent to EFD, except that the period of OD is 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-EFDp = n-AFS((p-1)/n).
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: 4-EFDφ
|+Example: 4efdφ
|-
|-
! quantity
! Steps
! (0)
! 0
! 1
! 1
! 2
! 2
Line 47: Line 44:
! 4
! 4
|-
|-
! frequency (f)
! Frequency Ratios (''r'')
|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
| 1 + (0/4)(φ - 1)<br>= (0φ + 4)/4<br>= 1
|1+(1/4)(φ-1) = (1φ + 3)/4
| 1 + (1/4)(φ - 1)<br>= (1φ + 3)/4
|1+(2/4)(φ-1) = (2φ + 2)/4
| 1 + (2/4)(φ - 1)<br>= (2φ + 2)/4
|1+(3/4)(φ-1) = (3φ + 1)/4
| 1 + (3/4)(φ - 1)<br>= (3φ + 1)/4
|1+(4/4)(φ-1) = (4φ + 0)/4 = φ
| 1 + (4/4)(φ - 1)<br>= (4φ + 0)/4<br>= φ
|-
|-
! pitch (log₂f)
! Octaves (log<sub>2</sub>''r'')
|(0)
| 0
|0.21
| 0.21
|0.39
| 0.39
|0.55
| 0.55
|0.69
| 0.69
|-
|-
! length (1/f)
! Length Ratios (1/''r'')
|(1)
| 1
|0.87
| 4/(φ + 3)
|0.76
| 2/(φ + 1)
|0.68
| 4/(3φ + 1)
|1/φ
| 1/φ
|}
|}


Revision as of 05:15, 22 March 2023

An EFD (equal frequency division) or AD (arithmetic division) is a periodic and arithmetic tuning system 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-ad-p: n equal frequency divisions of p, or n arithmetic 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 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.

Vs. OD

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

Vs. ELD

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

Vs. AFS

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: 4efdφ
Steps 0 1 2 3 4
Frequency Ratios (r) 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
Octaves (log2r) 0 0.21 0.39 0.55 0.69
Length Ratios (1/r) 1 4/(φ + 3) 2/(φ + 1) 4/(3φ + 1) 1/φ
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