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. | An '''EFD''', or '''equal frequency division''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning. | ||
== Specification == | |||
Its full specification is n-EFDp: n equal frequency divisions of irrational interval p. | |||
== Formula == | |||
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: | 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: | ||
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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>. | 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" | {| class="wikitable" | ||
Revision as of 20:38, 24 March 2021
An EFD, or equal frequency division, is a kind of arithmetic and harmonotonic tuning.
Specification
Its full specification is n-EFDp: n equal frequency divisions of irrational interval p.
Formula
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]\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, the formula for the frequency of step [math]\displaystyle{ k }[/math] of an n-EFDp is:
[math]\displaystyle{ f(k) = 1 + (\frac kn)(p-1) }[/math]
This way, when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ 0 }[/math], [math]\displaystyle{ f(k) }[/math] is simply [math]\displaystyle{ 1 }[/math]. And when [math]\displaystyle{ k }[/math] is [math]\displaystyle{ n }[/math], [math]\displaystyle{ f(k) }[/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. 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 otonal divisions of the octave.
The only difference between 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 (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-EFDp = n-AFS((p-1)/n).
Examples
| quantity | (0) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| frequency (f) | (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) | (0) | 0.21 | 0.39 | 0.55 | 0.69 |
| length (1/f) | (1) | 0.87 | 0.76 | 0.68 | 1/φ |