Relative interval error: Difference between revisions

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~~''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~~]].''~~

{{About|the error of intervals measured in relative cents|the relative error of temperaments|Tenney-Euclidean temperament measures #TE simple badness}}

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.

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 ==

=== In direct approximation ===

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

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

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

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

where ''n'' is the edo number and ''r'' is the ~~targeted [[frequency~~ ratio]].

where ''n'' is the edo number and ''r'' is the ratio in question.

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.~~

=== In val mapping ===

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

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

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

<math>\mathcal {E}_\text {r} = (V - nJ) \times 100\%</math>

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

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

Thanks to the [[Monzos and interval space|linearity of the interval space]], 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 '''m''' is given by

<math>\mathcal {E}_\text {r} \cdot \vec m</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.

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 {Er} is linear. That is, if ''n'' = ''αn''1 + ''βn''2 and A = ''α''A1 + ''β''A2, 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>

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

\begin{align}

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

\mathcal {E}_\text {r} &= (V - nJ) \times 100\% \\

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

&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\

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

&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\

&= \alpha \mathcal {E}_\text {r1} + \beta \mathcal {E}_\text {r2}

\end{align}

</math>

~~Here~~ is an example~~. The~~ relative ~~errors~~ of 26edo in 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>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>

<math>\mathcal {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

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

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

<math>\mathcal {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

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

<math>E_\text {r, 53} = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>

<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, 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%.

== See also ==

* [[Relative cent]]

* [[Relative errors of small EDOs]]

== Notes ==

[[Category:Terms]]

[[Category:Error]]

[[Category:Approximation]]

~~[[Category:Measure]]~~

[[Category:Relative measures]]

[[Category:Relative ~~measure~~]]

@@ 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]].''
+{{About|the error of intervals measured in relative cents|the relative error of temperaments|Tenney-Euclidean temperament measures #TE simple badness}}
-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.
+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 ==
 === In direct approximation ===
-To find the relative error of any [[JI]] ratio in direct approximation:
+To find the relative error ''ε'' of any [[JI]] [[ratio]] in direct approximation:
-<math>e (n, r) = (\text{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>
+<math>\varepsilon (n, r) = (\operatorname{round} (n \log_2 r) - n \log_2 r) \times 100\%</math>
-where ''n'' is the edo number and ''r'' is the targeted [[frequency ratio]].
+where ''n'' is the edo number and ''r'' is the ratio in question.
 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.
 === In val mapping ===
-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
+Given ''n''-edo equipped with ''p''-limit val ''V'' = {{val| ''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>π (''p'')</sub> }}, the relative error map ''Ɛ''<sub>r</sub> of each prime harmonic is given by
-<math>E_\text {r} = (A - nJ) \times 100\%</math>
+<math>\mathcal {E}_\text {r} = (V - nJ) \times 100\%</math>
-where J = {{val| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[JIP]].
+where ''J'' = {{val| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[just tuning map]].
-Thanks to the [[Monzos and interval space|linearity of the interval space]], 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 '''m''' is given by
-<math>E_\text {r} \vec b</math>
+<math>\mathcal {E}_\text {r} \cdot \vec m</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.
+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 {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
+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>
-E_\text {r} = (A - nJ) \times 100\% \\
+\begin{align}
-= ((\alpha A_1 + \beta A_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
+\mathcal {E}_\text {r} &= (V - nJ) \times 100\% \\
-= \alpha (A_1 - n_1 J) \times 100\% + (\beta (A_2 - n_2 J) \times 100\% \\
+&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
-= \alpha E_\text {r1} + \beta E_\text {r2}
+&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\
+&= \alpha \mathcal {E}_\text {r1} + \beta \mathcal {E}_\text {r2}
+\end{align}
 </math>
-Here is an example. The relative errors of 26edo in 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>E_\text {r, 26} = \langle \begin{matrix} 0.00\% & -20.90\% & -37.01\% \end{matrix} ]</math>
+<math>\mathcal {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
+That of 27edo using its 5-limit patent val is
-<math>E_\text {r, 27} = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>
+<math>\mathcal {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
+As {{nowrap| 53 {{=}} 26 + 27 }}, the relative error map of 53edo using its 5-limit patent val is
-<math>E_\text {r, 53} = \langle \begin{matrix} 0.00\% & -0.30\% & -6.22\% \end{matrix} ]</math>
+<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, 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%.
 == See also ==
 * [[Relative cent]]
 * [[Relative errors of small EDOs]]
+== Notes ==
+<references group="note"/>
 [[Category:Terms]]
 [[Category:Error]]
 [[Category:Approximation]]
-[[Category:Measure]]
+[[Category:Relative measures]]
-[[Category:Relative measure]]