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{{c}}, meaning that the nearest interval in the edo is about 2{{c}} flat of 3/2. One edostep is 50{{c}}, and {{nowrap| −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%<ref group="note">In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.</ref>.

== 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.~~

=== 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 ==

In val mapping, the relative error space {''Ɛ''r} is linear. That is, if ''n'' = ''αn''1 + ''βn''2 and V = ''α''V1 + ''β''V2, then

In val mapping, the relative error space {''Ɛ''r} is linear. That is, if ''n'' = ''αn''1 + ''βn''2 and ''V'' = ''αV''1 + ''βV''2, then

<math>

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</math>

For example, the relative error map of 26edo using its 5-limit patent val is

In direct approximation, the relative error space is also linear, but modulo 100%. Furthermore, we can show the relative error of any individual interval is {{w|equidistributed sequence|equidistributed}} from −50% to +50%, according to the {{w|equidistribution theorem}}.

An application of these properties concerns the fact that we can add the relative error maps of two edos together to form the relative error map of their sum. For example, the relative error map of 26edo using its 5-limit patent val is

<math>\mathcal {E}_\text {r} (26) = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>

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<math>\mathcal {E}_\text {r} (27) = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>

As 53 = 26 + 27, the relative error map of 53edo using its 5-limit patent val is

As {{nowrap| 53 {{=}} 26 + 27 }}, the relative error map of 53edo using its 5-limit patent val is

<math>\mathcal {E}_\text {r} (53) = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>

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%.

== See also ==

* [[Relative cent]]

* [[Relative errors of small EDOs]]

== Notes ==

[[Category:Terms]]

@@ Line 3: / Line 3: @@
 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{{c}}, meaning that the nearest interval in the edo is about 2{{c}} flat of 3/2. One edostep is 50{{c}}, and {{nowrap| −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%<ref group="note">In fact, 12edo always has the same relative and absolute interval error since it is the basis for the cent.</ref>.
 == Computation ==
@@ Line 14: / Line 14: @@
 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.
 === In val mapping ===
@@ Line 29: / Line 27: @@
 === 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 ==
-In val mapping, the relative error space {''Ɛ''<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and V = ''α''V<sub>1</sub> + ''β''V<sub>2</sub>, then
+In val mapping, the relative error space {''Ɛ''<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and ''V'' = ''αV''<sub>1</sub> + ''βV''<sub>2</sub>, then
 <math>
@@ Line 43: / Line 41: @@
 </math>
-For example, the relative error map of 26edo using its 5-limit patent val is
+In direct approximation, the relative error space is also linear, but modulo 100%. Furthermore, we can show the relative error of any individual interval is {{w|equidistributed sequence|equidistributed}} from −50% to +50%, according to the {{w|equidistribution theorem}}.
+An application of these properties concerns the fact that we can add the relative error maps of two edos together to form the relative error map of their sum. For example, the relative error map of 26edo using its 5-limit patent val is
 <math>\mathcal {E}_\text {r} (26) = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>
@@ Line 51: / Line 51: @@
 <math>\mathcal {E}_\text {r} (27) = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>
-As 53 = 26 + 27, the relative error map of 53edo using its 5-limit patent val is
+As {{nowrap| 53 {{=}} 26 + 27 }}, the relative error map of 53edo using its 5-limit patent val is
 <math>\mathcal {E}_\text {r} (53) = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>
 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%.
 == See also ==
 * [[Relative cent]]
 * [[Relative errors of small EDOs]]
+== Notes ==
+<references group="note"/>
 [[Category:Terms]]