Relative interval error: Difference between revisions

''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]].''

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]].  

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.  

Given ''n''-edo equipped with ''p''-limit val A = {{val| ''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>

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

<math>E_\text {r} \vec b</math>

In val mapping, the relative error space {E_r} is linear. That is, if ''n'' = ''αn''₁ + ''βn''₂ and A = ''α''A₁ + ''β''A₂, then

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}

Here is an example. The relative errors of 26edo in its 5-limit patent val is  

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

As 53 = 26 + 27, the relative errors of 53edo in its 5-limit patent val is

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%.