Direct approximation: Difference between revisions

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A '''direct approximation''' of an interval in a given [[~~EDO~~]] is the number of ~~EDO steps~~ that most closely approximates it, found by [[rounding]] to the nearest integer the ~~EDO~~ number times the [[~~Wikipedia: binary logarithm~~|binary logarithm]] of the interval: ~~<math>⌈n_{\text{edo}}·\log_2(i)⌋</math>.~~

{{interwiki

| en = Direct approximation

| ja = 直接近似

}}

A '''direct approximation''' of an interval in a given [[edo]] is the number of edosteps that most closely approximates it, found by [[rounding]] to the nearest integer the edo number times the [[log2|binary logarithm]] of the interval:

== Examples ~~of direct approximations~~ ==

<math>\operatorname {round} (n\log_2(i))</math>

for ratio ''i'' in ''n''-edo.

== Examples ==

{| class="wikitable center-all"

Line 7:

Line 15:

! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]]

|-

! ~~Just perfect~~ fifth, [[3/2]]

! Perfect fifth, [[3/2]]

| 7 || 10 || 11 || 15

|-

! Just ~~classic~~ major third, [[5/4]]

! Just major third, [[5/4]]

| 4 || 5 || 6 || 8

|-

! Just ~~classic~~ minor third, [[6/5]]

! Just minor third, [[6/5]]

| 3 || 4 || 5 || 7

|-

Line 20:

Line 28:

|}

Of these intervals, the fifth plays an important role for characterizing [[~~EDO~~]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given ~~EDO~~ system, because for this the sizes of fifth and octave must be relatively prime.

Of these intervals, the fifth plays an important role for characterizing [[edo]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given edo system, because for this the sizes of fifth and octave must be relatively prime.

== Problems ==

Unless one sticks to one or two notes at a time, direct approximation is not always practical in harmony. For example, it is impossible to construct a [[4:5:6|just major triad]] using the direct approximations of 3/2, 5/4, and 6/5 in 17edo since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps). The closest 3/2 and 5/4 imply the second closest 6/5; the closest 3/2 and 6/5 imply the second closest 5/4; and the closest 5/4 and 6/5 imply the second closest 3/2. We see one of the direct approximations must be given up. This is called [[consistency|inconsistency]], and chords like this exists in every edo.

In [[regular temperament theory]], intervals are mapped through [[val]]s. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals {{val| 17 27 39 }}, {{val| 17 27 40 }}, and {{val| 17 26 39 }}, respectively.

[[Category:Interval]]

[[Category:Terms]]

[[Category:Method]]

~~[[Category:Val]]~~

~~[[Category:Todo]]~~

@@ Line 1: / Line 1: @@
-A '''direct approximation''' of an interval in a given [[EDO]] is the number of EDO steps that most closely approximates it, found by [[rounding]] to the nearest integer the EDO number times the [[Wikipedia: binary logarithm|binary logarithm]] of the interval: <math>⌈n_{\text{edo}}·\log_2(i)⌋</math>.
+{{interwiki
+| en = Direct approximation
+| ja = 直接近似
+}}
+A '''direct approximation''' of an interval in a given [[edo]] is the number of edosteps that most closely approximates it, found by [[rounding]] to the nearest integer the edo number times the [[log2|binary logarithm]] of the interval:
-== Examples of direct approximations ==
+<math>\operatorname {round} (n\log_2(i))</math>
+for ratio ''i'' in ''n''-edo.
+== Examples ==
 {| class="wikitable center-all"
@@ Line 7: / Line 15: @@
 ! [[12edo]] || [[17edo]] || [[19edo]] || [[26edo]]
 |-
-! Just perfect fifth, [[3/2]]
+! Perfect fifth, [[3/2]]
 |   7   ||   10  ||   11  || 15
 |-
-! Just classic major third, [[5/4]]
+! Just major third, [[5/4]]
 |   4   ||   5   ||   6   || 8
 |-
-! Just classic minor third, [[6/5]]
+! Just minor third, [[6/5]]
 |   3   ||   4   ||   5   || 7
 |-
@@ Line 20: / Line 28: @@
 |}
-Of these intervals, the fifth plays an important role for characterizing [[EDO]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given EDO system, because for this the sizes of fifth and octave must be relatively prime.
+Of these intervals, the fifth plays an important role for characterizing [[edo]] systems (as it defines the size of M2, m2, A1). Also, a simple test can show if [[circle-of-fifths notation]] can be applied to a given edo system, because for this the sizes of fifth and octave must be relatively prime.
+== Problems ==
+Unless one sticks to one or two notes at a time, direct approximation is not always practical in harmony. For example, it is impossible to construct a [[4:5:6|just major triad]] using the direct approximations of 3/2, 5/4, and 6/5 in 17edo since the step numbers do not add up (5 steps + 4 steps ≠ 10 steps). The closest 3/2 and 5/4 imply the second closest 6/5; the closest 3/2 and 6/5 imply the second closest 5/4; and the closest 5/4 and 6/5 imply the second closest 3/2. We see one of the direct approximations must be given up. This is called [[consistency|inconsistency]], and chords like this exists in every edo.
+In [[regular temperament theory]], intervals are mapped through [[val]]s. Although more complex, it recognizes the fact that intervals like 3/2, 5/4, and 6/5 are related, as the number of steps of one interval is determined once the other two have been determined. The three situations in the above example correspond to using vals {{val| 17 27 39 }}, {{val| 17 27 40 }}, and {{val| 17 26 39 }}, respectively.
+[[Category:Interval]]
 [[Category:Terms]]
 [[Category:Method]]
-[[Category:Val]]
-[[Category:Todo]]