EFD: Difference between revisions

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An '''EFD''', or '''~~equal~~ frequency division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[~~Haronotonic tunings|harmonotonic~~]] 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.

~~Its full specification is n-EFDp: n equal frequency divisions of irrational interval p. The only difference between [[OD|n-ODp]] and n-EFDp is that the p for an EFD is irrational.~~

== Specification ==

Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12-EDO, standard tuning, you could divide it into 12 equal parts by '''frequency'''. This would give you 12-EFDO. ~~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 otonal divisions of the octave~~.

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>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>

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"

|+~~example~~: 4-EFDφ

|+Example: 4-EFDφ

|-

! quantity

Line 15:

Line 44:

! 4

|-

! frequency (f)

! frequency (''f'', ratio)

|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1

| (1 + (0/4)(φ - 1)) = (0φ + 4)/4 = 1

|1+(1/4)(φ-1) = (1φ + 3)/4

| 1 + (1/4)(φ - 1) = (1φ + 3)/4

|1+(2/4)(φ-1) = (2φ + 2)/4

| 1 + (2/4)(φ - 1) = (2φ + 2)/4

|1+(3/4)(φ-1) = (3φ + 1)/4

| 1 + (3/4)(φ - 1) = (3φ + 1)/4

|1+(4/4)(φ-1) = (4φ + 0)/4 = φ

| 1 + (4/4)(φ - 1) = (4φ + 0)/4 = 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

|}

~~[[Category:Overtone]]~~

~~[[Category:Overtone‏‎ series]]~~

[[Category:Otonality]]

[[Category:Harmonic]]

[[Category:Harmonic series‏‎]]

@@ Line 1: / Line 1: @@
-An '''EFD''', or '''equal frequency division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Haronotonic tunings|harmonotonic]] 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.
-Its full specification is n-EFDp: n equal frequency divisions of irrational interval p. The only difference between [[OD|n-ODp]] and n-EFDp is that the p for an EFD is irrational.
+== Specification ==
-Instead of equally dividing the octave into 12 equal parts by pitch, as is done for 12-EDO, standard tuning, you could divide it into 12 equal parts by '''frequency'''. This would give you 12-EFDO. 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 otonal divisions of the octave.
+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>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>
+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"
-|+example: 4-EFDφ
+|+Example: 4-EFDφ
 |-
 ! quantity
@@ Line 15: / Line 44: @@
 ! 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
 |}
-[[Category:Overtone]]
-[[Category:Overtone‏‎ series]]
 [[Category:Otonality]]
 [[Category:Harmonic]]
 [[Category:Harmonic series‏‎]]