Direct approximation: Difference between revisions

Line 1:

A '''~~patent interval~~''' in a given [[EDO]] is the number of EDO steps ~~needed to reach the best approximation of a given interval – usually, but not necessarily just – in~~ that ~~EDO. The method for calculating patent intervals is referred to as '''direct mapping'''~~, ~~and it involves~~ [[rounding]] the ~~product of~~ the [[Wikipedia: binary logarithm|binary logarithm]] ~~(''log2'')~~ of the interval ~~ratio~~ (~~''r'') and the EDO number (''nEdo''~~).

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>.

~~round(log2(r)*nEdo)~~

== Examples of direct approximations ==

A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s.

==== Examples of ~~Patent Intervals =~~===

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.

{| class="wikitable center-all"

Line 24:

Line 19:

| 10 || 14 || 15 || 21

|}

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.

[[Category:Terms]]

@@ Line 1: / Line 1: @@
-A '''patent interval''' in a given [[EDO]] is the number of EDO steps needed to reach the best approximation of a given interval – usually, but not necessarily just – in that EDO.  The method for calculating patent intervals is referred to as '''direct mapping''', and it involves [[rounding]] the product of the [[Wikipedia: binary logarithm|binary logarithm]] (''log2'') of the interval ratio (''r'') and the EDO number (''nEdo'').
+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>.
- round(log2(r)*nEdo)
+== Examples of direct approximations ==
-A [[patent val]] is the best mapping of a representative set of intervals (taken to be [[generator]]s for a [[JI subgroup]]) in a given EDO; for the ''p''-[[prime limit]] this set consists of [[prime interval]]s.
-==== Examples of Patent Intervals ====
-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.
 {| class="wikitable center-all"
@@ Line 24: / Line 19: @@
 |   10  ||   14  ||  15   || 21
 |}
+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.
 [[Category:Terms]]