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


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


7EFDπ is dividing the frequency space between 1 and π by 7, so that makes each step (π-1)/7. So the 2nd step is 1+(π-1)/7 = 7/7+(π-1)/7 = (π+6)/7, the 3rd step would be 1+2(π-1)/7 = (2π+5)/7, then I see the pattern forming so it goes (3π+4)/7, (4π+3)/7, (5π+2)/7, (6π+1)/7, and (7π+0)/7 = π at the end.
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.
 
So far we've looked at arithmetic tunings produced by sequencing a single step repeatedly. But if an arithmetic tuning is defined by having equal step sizes of some kind of quantity (frequency, pitch, or length), then it also follows that they can be produced by taking a larger interval and equally dividing it according to that kind of quantity.
 
The most common example of this type of tuning is 12-EDO, standard tuning, which takes the interval of the octave, and equally divides its pitch into 12 parts. For long, we could call this 12-EPDO, for 12 equal '''pitch''' divisions of the octave (whenever pitch is the chosen kind of quality, we can assume it, and skip pointing it out; that's why 12-EDO is the better name).
 
But it is also possible to — instead of equally dividing the octave in 12 equal parts by pitch — divide it into 12 equal parts by '''frequency''', or '''length'''. In the former case, you will have 12-EFDO, and in the latter case, you will have 12-ELDO. 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 and ELD are typically reserved for irrational tunings, such as 12-EFDφ. So it would be more appropriate to name these two tunings 12-ODO and 12-UDO, for otonal divisions of the octave and utonal divisions of the octave, respectively.


{| class="wikitable"
{| class="wikitable"
Line 22: Line 16:
|-
|-
! frequency (f)
! frequency (f)
|(1+(0/4)(φ-1)) = 1
|(1+(0/4)(φ-1)) = (0φ + 4)/4 = 1
|1+(1/4)(φ-1)
|1+(1/4)(φ-1) = (1φ + 3)/4
|1+(2/4)(φ-1)
|1+(2/4)(φ-1) = (2φ + 2)/4
|1+(3/4)(φ-1)
|1+(3/4)(φ-1) = (3φ + 1)/4
|1+(4/4)(φ-1) = φ
|1+(4/4)(φ-1) = (4φ + 0)/4 = φ
|-
|-
! pitch (log₂f)
! pitch (log₂f)
Line 33: Line 27:
|0.39
|0.39
|0.55
|0.55
|(0.69)
|0.69
|-
|-
! length (1/f)
! length (1/f)
Line 40: Line 34:
|0.76
|0.76
|0.68
|0.68
|(1/φ)
|1/φ
|}
|}
An [[OD|OD (or otonal division)]] is a specific (rational) type of EFD.


[[Category:Overtone]]
[[Category:Overtone]]

Revision as of 22:22, 22 March 2021

An EFD, or equal frequency division, is a kind of arithmetic and monotonic tuning.

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

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.

example: 4-EFDφ
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
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