Relative interval error: Difference between revisions

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== Computation ==

To find the relative error of any [[JI]] ~~interval~~ in direct approximation:

=== In direct approximation ===

To find the relative error of any [[JI]] ratio in direct approximation:

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

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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 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~~.

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 farther.

== ~~Additivity~~ ==

=== In val mapping ===

In val ~~mapping~~, ~~there are two additivities~~ of ~~relative errors.~~

Given ''n''-edo equipped with ''p''-limit val A = {{val| ''a''1 ''a''2 … ''a''π (''p'') }}, the relative error map Er of each prime harmonic is given by

~~First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''3 = ''r''1~~<~~/sub~~>''r~~''2 for ''n'', then ''e'' (''n'', ''r''3)~~ = ~~''e''~~ (~~''n'', ''r''1~~) ~~+ ''e'' (''n'', ''r''2~~</~~sub~~>).

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

~~Second, for the same ratio, an edo which~~ is the sum of some other edos have their relative errors additive, that is, if ''n''3 = ''n''1 + ''n''2 for ''r'', then ''e'' (''n''3, ''r'') = ''e'' (''n''1, ''r'') + ''e'' (''n''2'','' ''r'').

where J is the [[JIP]].

~~In either case, if the~~ error ~~exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there~~ is ~~a discrepancy in val mapping and direct approximation, so is the error. To find the error in direct approximation, modulo the previous result~~ by ~~100%.~~

The relative error for any monzo b is given by

~~An example of the first additivity is shown as follows. 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.

Here is an example ~~for the second additivity~~. The errors of ~~3/1 for~~ 26edo ~~and 27edo are~~ -20.90% ~~and~~ +20.60%, ~~respectively~~, and ~~their~~ sum -0.30% is the error of 3/1 ~~for 53edo~~.

== 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

<math>

E_\text {r} = (A - nJ) \times 100\% \\

= ((\alpha A_1 + \beta A_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\

= \alpha (A_1 - n_1 J) \times 100\% + (\beta (A_2 - n_2 J) \times 100\% \\

= \alpha E_\text {r1} + \beta E_\text {r2}

</math>

Here is an example. The relative errors of 26edo in its 7-limit patent val is

<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>

That of 27edo in its 5-limit patent val is

<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>

As 53 = 26 + 27, the relative errors of 53edo in its 5-limit patent val is

<math>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, 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 ===

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 6: / Line 6: @@
 == Computation ==
-To find the relative error of any [[JI]] interval in direct approximation:
+=== In direct approximation ===
+To find the relative error of any [[JI]] ratio in direct approximation:
 <math>e (n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>
@@ Line 14: / 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 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.
+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 farther.
-== Additivity ==
+=== In val mapping ===
-In val mapping, there are two additivities of relative errors.
+Given ''n''-edo equipped with ''p''-limit val A = {{val| ''a''<sub>1</sub> ''a''<sub>2</sub> … ''a''<sub>π (''p'')</sub> }}, the relative error map E<sub>r</sub> of each prime harmonic is given by
-First, for the same edo, a ratio which is the product of some other ratios have their relative errors additive, that is, if ''r''<sub>3</sub> = ''r''<sub>1</sub>''r''<sub>2</sub> for ''n'', then ''e'' (''n'', ''r''<sub>3</sub>) = ''e'' (''n'', ''r''<sub>1</sub>) + ''e'' (''n'', ''r''<sub>2</sub>).
+<math>E_\text {r} = (A - nJ) \times 100\%</math>
-Second, for the same ratio, an edo which is the sum of some other edos have their relative errors additive, that is, if ''n''<sub>3</sub> = ''n''<sub>1</sub> + ''n''<sub>2</sub> for ''r'', then ''e'' (''n''<sub>3</sub>, ''r'') = ''e'' (''n''<sub>1</sub>, ''r'') + ''e'' (''n''<sub>2</sub>'','' ''r'').
+where J is the [[JIP]].
-In either case, if the error exceeds the range -50% to +50%, it indicates that an inconsistency occurs, and there is a discrepancy in val mapping and direct approximation, so is the error. To find the error in direct approximation, modulo the previous result by 100%.
+The relative error for any monzo b is given by
-An example of the first additivity is shown as follows. 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.
+<math>E_\text {r} \vec b</math>
-Here is an example for the second additivity. The errors of 3/1 for 26edo and 27edo are -20.90% and +20.60%, respectively, and their sum -0.30% is the error of 3/1 for 53edo.
+== 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
+<math>
+E_\text {r} = (A - nJ) \times 100\% \\
+= ((\alpha A_1 + \beta A_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
+= \alpha (A_1 - n_1 J) \times 100\% + (\beta (A_2 - n_2 J) \times 100\% \\
+= \alpha E_\text {r1} + \beta E_\text {r2}
+</math>
+Here is an example. The relative errors of 26edo in its 7-limit patent val is
+<math>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>
+That of 27edo in its 5-limit patent val is
+<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>
+As 53 = 26 + 27, the relative errors of 53edo in its 5-limit patent val is
+<math>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, 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 ==