Relative interval error: Difference between revisions

Line 1:

''This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see [[Tenney-Euclidean temperament measures #TE simple badness]].''

The '''relative error''' of an [[interval]] in an [[edo]] is the error in cents ~~approximating a targeted interval~~ divided by the ~~size~~ of an [[edostep]], or equivalently stated, the error in [[Relative cent|relative cents]]. The formula for closest mapping of any [[JI]] interval is

The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[Relative cent|relative cents]].

For example, in 24-edo, 3/2 has an '''absolute error''' of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12-edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12-edo's absolute and relative errors are always identical.)

== Formula ==

The formula for closest mapping of any [[JI]] interval is

<math>e(n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>

Line 9:

Line 14:

The unit of relative error is ''relative cent'' or ''percent''.

With ~~closest~~ mapping, the relative error ranges from -50% to +50%. With [[patent val]] ~~mapping~~, it can be farther from zero. To obtain the relative error in patent val mapping, first find ~~that~~ of ~~relevant~~ prime ~~harmonics~~, and then ~~apply~~ the ~~additive rule (see below)~~.

With a direct mapping via the ratio's cents, the relative error ranges from -50% to +50%. With an indirect mapping via [[patent val]] or other val, it can be farther from zero. To obtain the relative error in patent val mapping, first find the relative errors of each prime, and then find the dot product of this vector with the ratio's monzo.

== Additivity ==

Line 26:

Line 31:

== See also ==

* [[Relative cent]]

* [[Relative Errors of Small Edos]]

* [[Relative errors of small edos|Relative Errors of Small Edos]]

[[Category:Terms]]

@@ Line 1: / Line 1: @@
 ''This article is about the error of intervals measured in relative cents. For the relative error of temperaments, see [[Tenney-Euclidean temperament measures #TE simple badness]].''
-The '''relative error''' of an [[interval]] in an [[edo]] is the error in cents approximating a targeted interval divided by the size of an [[edostep]], or equivalently stated, the error in [[Relative cent|relative cents]]. The formula for closest mapping of any [[JI]] interval is
+The '''relative error''' of an [[interval]] in an [[edo]] is the interval's error in cents divided by the cents of an edostep, or equivalently stated, the error in [[Relative cent|relative cents]].
+For example, in 24-edo, 3/2 has an '''absolute error''' of about -2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and -2 / 50 = -0.04, therefore the relative error is about -4% or -4 relative cents. In contrast, 12-edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12-edo's absolute and relative errors are always identical.)
+== Formula ==
+The formula for closest mapping of any [[JI]] interval is
 <math>e(n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>
@@ Line 9: / Line 14: @@
 The unit of relative error is ''relative cent'' or ''percent''.
-With closest mapping, the relative error ranges from -50% to +50%. With [[patent val]] mapping, it can be farther from zero. To obtain the relative error in patent val mapping, first find that of relevant prime harmonics, and then apply the additive rule (see below).
+With a direct mapping via the ratio's cents, the relative error ranges from -50% to +50%. With an indirect mapping via [[patent val]] or other val, it can be farther from zero. To obtain the relative error in patent val mapping, first find the relative errors of each prime, and then find the dot product of this vector with the ratio's monzo.
 == Additivity ==
@@ Line 26: / Line 31: @@
 == See also ==
 * [[Relative cent]]
-* [[Relative Errors of Small Edos]]
+* [[Relative errors of small edos|Relative Errors of Small Edos]]
 [[Category:Terms]]