EFD: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Rework (1/2, improve intro, +new terms, and formatting)
Cmloegcmluin (talk | contribs)
autopatrolled, emailconfirmed
5,170 edits
→Examples: update row headers per agreement at https://en.xen.wiki/w/Talk:APS
 
(12 intermediate revisions by 2 users not shown)
Line 1: Line 1:
An '''EFD''' ('''equal frequency division''') or '''AD''' ('''arithmetic division''') is a [[period]]ic and [[Arithmetic tuning|arithmetic]] [[tuning system]] in which each period is divided to a number of steps of equal frequency difference.  
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 ==


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:
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>
<math>
Line 13: Line 13:
</math>
</math>


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 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. 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.
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 ===
Line 33: Line 30:
=== 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).
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: 4-EFDφ
|+Example: 4-EFDφ
|-
|-
! quantity
! quantity
Line 47: Line 44:
! 4
! 4
|-
|-
! frequency (f)
! frequency (''f'', ratio)
|(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)
! 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
|}
|}


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
Retrieved from "https://en.xen.wiki/index.php?title=EFD&oldid=126176"