Relative interval error
- This page 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 ε of any JI ratio in direct approximation:
[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.
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 greater.
In val mapping
Given n-edo equipped with p-limit val V = ⟨v1 v2 … vπ (p)], the relative error map Ɛr of each prime harmonic is given by
[math]\mathcal {E}_\text {r} = (V - nJ) \times 100\%[/math]
where J = ⟨1 log23 … log2p] is the just tuning map.
Thanks to the 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.
Linearity
In val mapping, the relative error space {Ɛr} is linear. That is, if n = αn1 + βn2 and V = αV1 + βV2, 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]
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 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.
It is somewhat applicable to direct approximation, but with some quirks. 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%.