User:Overthink/The circle of relative error: Difference between revisions
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[[File:Graph Harmonic Error 12et.png|alt=12edo 3: +2.0% 5: +13.7% 7: +31.2% 9: +3.9% 11: +48.7% 13: -40.5% 15: +11.7%|thumb|383x383px|Relative error of harmonics in 12edo]] | [[File:Graph Harmonic Error 12et.png|alt=12edo 3: +2.0% 5: +13.7% 7: +31.2% 9: +3.9% 11: +48.7% 13: -40.5% 15: +11.7%|thumb|383x383px|Relative error of harmonics in 12edo]]In an [[equal temperament]], [[relative error]] is the amount by which the [[mapping]] of an interval or harmonic deviates from its [[Just intonation point|just value]]. For example, in [[12edo]] the relative error of [[3/2]] is -2.0%, and the relative error of [[5/4]] is +13.7%. We can plot the relative error of harmonics in a graph like the one on the right. This graph lets us calculate the relative error of intervals. For example, the relative error of [[6/5]] is 13.7%-(-2.0%)=15.6% (not 15.7% due to rounding error). Note that absolute error can be found by multiplying the relative error by the equal division's step size, and in 12edo absolute and relative error are identical. However, one may not always want to use the nearest approximation of every harmonic. For example, in 12edo, using the second best approximation for harmonic 13 (relative error +59.5%) actually gives us less error overall due to cancellation of errors between harmonics. As an example, the relative error of 13/11 in this mapping is 59.5%-48.7%=10.8%. Compare this to the [[patent val]], where all primes use their nearest mappings, where 13/11 has an error of -40.5%-48.7%=-89.2%. If a prime is near perfectly off, the sharp and flat mappings are each the best about equally often, depending on the errors of the other harmonics. Therefore, it is natural to plot relative error not on a range from -50% to +50%, but on a circle. | ||
== | == The circle of relative error == | ||
[[File:Relative Error Circle 12et.png|alt=Relative error on a circle, Clockwise: sharp. Counterclockwise: flat.|thumb|Circle of relative error of 12edo]] | |||
We plot relative error on a circle as follows: The top corresponds to zero error, far left is 25% flat, far right is 25% sharp, and the bottom is ±50% error. The plot for 12edo is on the right. | |||
Revision as of 20:47, 28 September 2025
In an equal temperament, relative error is the amount by which the mapping of an interval or harmonic deviates from its just value. For example, in 12edo the relative error of 3/2 is -2.0%, and the relative error of 5/4 is +13.7%. We can plot the relative error of harmonics in a graph like the one on the right. This graph lets us calculate the relative error of intervals. For example, the relative error of 6/5 is 13.7%-(-2.0%)=15.6% (not 15.7% due to rounding error). Note that absolute error can be found by multiplying the relative error by the equal division's step size, and in 12edo absolute and relative error are identical. However, one may not always want to use the nearest approximation of every harmonic. For example, in 12edo, using the second best approximation for harmonic 13 (relative error +59.5%) actually gives us less error overall due to cancellation of errors between harmonics. As an example, the relative error of 13/11 in this mapping is 59.5%-48.7%=10.8%. Compare this to the patent val, where all primes use their nearest mappings, where 13/11 has an error of -40.5%-48.7%=-89.2%. If a prime is near perfectly off, the sharp and flat mappings are each the best about equally often, depending on the errors of the other harmonics. Therefore, it is natural to plot relative error not on a range from -50% to +50%, but on a circle.
The circle of relative error
We plot relative error on a circle as follows: The top corresponds to zero error, far left is 25% flat, far right is 25% sharp, and the bottom is ±50% error. The plot for 12edo is on the right.