EFD: Difference between revisions

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 </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  ==
 === 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 12epdo, or 12edo (because pitch can be assumed), standard tuning, you could divide it into 12 equal parts by ''frequency''. This would give you 12efdo.
-=== 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.
+OD is equivalent to EFD, except that the period of OD is rational.
 === Vs. ELD ===
@@ Line 33: / Line 30: @@
 === 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).
+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: 4efdφ
 |-
-! quantity
+! Steps
-! (0)
+! 0
 ! 1
 ! 2
@@ Line 47: / Line 44: @@
 ! 4
 |-
-! frequency (f)
+! Frequency Ratios (''r'')
-|(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)
+! Octaves (log<sub>2</sub>''r'')
-|(0)
+| 0
-|0.21
+| 0.21
-|0.39
+| 0.39
-|0.55
+| 0.55
-|0.69
+| 0.69
 |-
-! length (1/f)
+! Length Ratios (1/''r'')
-|(1)
+| 1
-|0.87
+| 4/(φ + 3)
-|0.76
+| 2/(φ + 1)
-|0.68
+| 4/(3φ + 1)
-|1/φ
+| 1/φ
 |}