Relative interval error: Difference between revisions

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<math>E_\text {r} = (A - nJ) \times 100\%</math>

where J is the [[JIP]].

where J = {{val| 1 log23 … log2''p'' }} is the [[JIP]].

~~The~~ relative error for any monzo b is given by

Thanks to the [[Monzos and interval space|linearity of the interval space]], the relative error for any monzo b is given by

=== 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.

== Linearity ==

~~=== Linearity of the relative error space ===~~

In val mapping, the relative error space {Er} is linear. That is, if ''n'' = ''αn''1 + ''βn''2 and A = ''α''A1 + ''β''A2, then

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It is somewhat applicable to direct approximation, too, but 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%.

~~=== Linearity of the interval space ===~~

~~{{See also| Monzos and interval space }}~~

~~Another linearity is actually about the interval space {b}. This enables us to find the relative error of any ratio in a given tuning of an equal temperament.~~

Let us try finding the relative error of 6/5 in 19edo's patent val. We first find the errors of 2/1, 3/1 and 5/1 in 19edo 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.

== See also ==

@@ Line 22: / Line 22: @@
 <math>E_\text {r} = (A - nJ) \times 100\%</math>
-where J is the [[JIP]].
+where J = {{val| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[JIP]].
-The relative error for any monzo b is given by
+Thanks to the [[Monzos and interval space|linearity of the interval space]], the relative error for any monzo b is given by
 <math>E_\text {r} \vec b</math>
+=== 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.
 == Linearity ==
-=== Linearity of the relative error space ===
 In val mapping, the relative error space {E<sub>r</sub>} is linear. That is, if ''n'' = ''αn''<sub>1</sub> + ''βn''<sub>2</sub> and A = ''α''A<sub>1</sub> + ''β''A<sub>2</sub>, then
@@ Line 54: / Line 56: @@
 It is somewhat applicable to direct approximation, too, but 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%.
-=== Linearity of the interval space ===
-{{See also| Monzos and interval space }}
-Another linearity is actually about the interval space {b}. This enables us to find the relative error of any ratio in a given tuning of an equal temperament.
-Let us try finding the relative error of 6/5 in 19edo's patent val. We first find the errors of 2/1, 3/1 and 5/1 in 19edo 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.
 == See also ==