Direct approximation: Difference between revisions

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{{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:

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

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

== Examples ==

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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]]

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+{{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:
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 for ratio ''i'' in ''n''-edo.
-== Examples of direct approximations ==
+== Examples ==
 {| class="wikitable center-all"
@@ Line 26: / Line 30: @@
 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]]