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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¢, 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.)


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


<math>e (n, r) = (\operatorname{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 ratio in question.  
where ''n'' is the edo number and ''r'' is the ratio in question.  
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=== In val mapping ===
=== In val mapping ===
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 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} = (V - 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 m 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} \cdot \vec m</math>
<math>\mathcal {E}_\text {r} \cdot \vec m</math>


=== Example ===
=== Example ===
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== Linearity ==
== 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 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 = ''α''V<sub>1</sub> + ''β''V<sub>2</sub>, then


<math>
<math>
\begin{align}
\begin{align}
E_\text {r} &= (V - nJ) \times 100\% \\
\mathcal {E}_\text {r} &= (V - nJ) \times 100\% \\
&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
&= ((\alpha V_1 + \beta V_2) - (\alpha n_1 + \beta n_2)J) \times 100\% \\
&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\
&= \alpha (V_1 - n_1 J) \times 100\% + \beta (V_2 - n_2 J) \times 100\% \\
&= \alpha E_\text {r1} + \beta E_\text {r2}
&= \alpha \mathcal {E}_\text {r1} + \beta \mathcal {E}_\text {r2}
\end{align}
\end{align}
</math>
</math>
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For example, the relative error map of 26edo using its 5-limit patent val is  
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 using 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 error map of 53edo using its 5-limit patent val is
As 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.  
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.  
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[[Category:Error]]
[[Category:Error]]
[[Category:Approximation]]
[[Category:Approximation]]
[[Category:Relative measure]]
[[Category:Relative measures]]
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