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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 ==
== Computation ==
To find the relative error of any [[JI]] interval in direct approximation:  
=== 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''.  
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.
=== 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 ''Ɛ''<sub>r</sub> of each prime harmonic is given by


== Additivity ==
<math>\mathcal {E}_\text {r} = (V - nJ) \times 100\%</math>
In val mapping, there are two additivities of relative errors. 


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>).
where ''J'' = {{val| 1 log<sub>2</sub>3 … log<sub>2</sub>''p'' }} is the [[just tuning map]].  


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


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


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.  
=== 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, &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.


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 ==
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>
\begin{align}
\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 - 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>
 
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>
 
That of 27edo using its 5-limit patent val is
 
<math>\mathcal {E}_\text {r} (27) = \langle \begin{matrix} 0.00\% & +20.60\% & +30.79\% \end{matrix} ]</math>
 
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.  


== See also ==
== See also ==
* [[Relative cent]]
* [[Relative cent]]
* [[Relative errors of small EDOs]]
* [[Relative errors of small EDOs]]
== Notes ==
<references group="note"/>


[[Category:Terms]]
[[Category:Terms]]
[[Category:Error]]
[[Category:Approximation]]
[[Category:Approximation]]
[[Category:Measure]]
[[Category:Relative measures]]
[[Category:Relative measure]]
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