EFD: Difference between revisions

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

~~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-EFDp: n equal frequency divisions of irrational interval p.

~~The analogous utonal equivalent of an EFD is an [[ELD|ELD (equal length division)]].~~

== Formula ==

~~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).~~

To find the steps for an n-EFDp, 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, the formula for the frequency of step <math>k</math> of an n-EFDp is:

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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>.

== 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. 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.

The only difference between [[OD|n-ODp]] and n-EFDp is that the p for an EFD is irrational.

=== vs. ELD ===

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

=== 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).

== Examples ==

{| class="wikitable"

@@ Line 1: / Line 1: @@
 An '''EFD''', or '''equal frequency division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.
-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-EFDp: n equal frequency divisions of irrational interval p.
-The analogous utonal equivalent of an EFD is an [[ELD|ELD (equal length division)]].
+== Formula ==
-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).
 To find the steps for an n-EFDp, begin by recognizing that while the multiplicative interval relating your root position to the end position is <span><math>p</math></span> (or <span><math>\frac p1</math></span>), if you are going to move arithmetically (by repeated addition) from <span><math>1</math></span> to <span><math>p</math></span>, then the difference in frequency space that you are dividing up is not actually <span><math>p</math></span>, but <span><math>p - 1</math></span>. And because you are dividing it into <span><math>n</math></span> parts, each step will have a size of <span><math>\frac{p-1}{n}</math></span>. So, the formula for the frequency of step <span><math>k</math></span> of an n-EFDp is:
@@ Line 16: / Line 14: @@
 This way, when <span><math>k</math></span> is <span><math>0</math></span>, <span><math>f(k)</math></span> is simply <span><math>1</math></span>. And when <span><math>k</math></span> is <span><math>n</math></span>, <span><math>f(k)</math></span> is simply <span><math>1 + (p-1) = p</math></span>.
+== 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. 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.
+The only difference between [[OD|n-ODp]] and n-EFDp is that the p for an EFD is irrational.
+=== vs. ELD ===
+The analogous utonal equivalent of an EFD is an [[ELD|ELD (equal length division)]].
+=== 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).
+== Examples ==
 {| class="wikitable"