# Relative interval error

*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.*

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 cents.

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

### 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]

where *n* is the edo number and *r* is the targeted frequency ratio.

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 = ⟨*a*_{1} *a*_{2} … *a*_{π (p)}], the relative error map E_{r} of each prime harmonic is given by

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

where J = ⟨1 log_{2}3 … log_{2}*p*] is the JIP.

Thanks to the 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

In val mapping, the relative error space {E_{r}} is linear. That is, if *n* = *αn*_{1} + *βn*_{2} and A = *α*A_{1} + *β*A_{2}, 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 5-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%.