Relative interval error: Difference between revisions

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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]]s.

For example, in 24edo, 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, 12edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12edo's absolute ~~and relative errors are always identical~~.)

For example, in 24edo, 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, 12edo has the same absolute error, but a smaller relative error of −2%. (In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.)

== Computation ==

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

With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.

With direct approximation via the ratio's cents, the relative error ranges from −50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.

=== In val mapping ===

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=== Example ===

Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, −11.43% and −11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (−11.43%) − (−11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.

== Linearity ==

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We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.

It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of −50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.

== See also ==

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 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]]s.
-For example, in 24edo, 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, 12edo has the same absolute error, but a smaller relative error of -2%. (In fact, 12edo's absolute and relative errors are always identical.)
+For example, in 24edo, 3/2 has an absolute error of about &minus;2¢, meaning that the nearest interval in the edo is about 2¢ flat of 3/2. One edostep is 50¢, and &minus;2 / 50 = &minus;0.04, therefore the relative error is about &minus;4% or &minus;4 relative cents. In contrast, 12edo has the same absolute error, but a smaller relative error of &minus;2%. (In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.)
 == Computation ==
@@ Line 15: / Line 15: @@
 The unit of relative error is ''relative cent'' or ''percent''.
-With direct approximation via the ratio's cents, the relative error ranges from -50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.
+With direct approximation via the ratio's cents, the relative error ranges from &minus;50% to +50%. With a [[val]] mapping via [[patent val]] or other vals, it can be greater.
 === In val mapping ===
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 === Example ===
-Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, -11.43% and -11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (-11.43%) - (-11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.
+Let us try finding the relative error of 6/5 in 19edo's patent val. We may first find the errors of 2/1, 3/1 and 5/1 in 19edo. They are 0, &minus;11.43% and &minus;11.66%, respectively. Since 6/5 = (2/1)(3/1) / (5/1), its error is 0 + (&minus;11.43%) &minus; (&minus;11.66%) = 0.23%. That shows 19edo has fairly flat fifths and major thirds, yet they cancel out when it comes to minor thirds and results in a very accurate approximation.
 == Linearity ==
@@ Line 57: / Line 57: @@
 We see how the errors of a sharp tending system and a flat tending system cancel out each other by the sum, and result in a much more accurate equal temperament.
-It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of -50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.
+It is somewhat applicable to direct approximation, but with some quirks. If the error exceeds the range of &minus;50% to +50%, it indicates that there is a discrepancy in val mapping and direct approximation. In this case, you need to modulo the result by 100%.
 == See also ==