User:Userminusone/SoB Approximation

"Sum of Best approximation" or "SoB approximation" is a method of approximating a root scale, R, with the best approximations out of a set of other scales, S. This method of approximation was invented by Userminusone. (If anyone else came up with this first, please tell me in the discussion section of this page)

Example

For this example, I'm going to make a SoB approximation of harmonics 8-16 using 10edo, 12edo, and 22edo. The root scale has the cent values 203.910, 386.314, 551.318, 701.955, 840.528, 968.826, 1088.269, 1200. (For the purposes of this example, all cent values will be rounded to the nearest thousandth of a cent)

For the approximation part, the first cent value of 203.910 gets compared to the nearest cent values in 10edo, 12edo, and 22edo. These values are 240, 200, and 218.182, respectively. In this case, 12edo's 200 cent interval provides the best approximation, so this is what gets used in the final scale. For the second cent value in the root scale, which is 386.314, the best approximations from 10edo, 12edo, and 22edo are 360, 400, and 381.818, respectively. In this case, 22edo's 381.818 cent interval provides the best approximation and is used in the final scale along with the 200 cent interval from 12edo. After doing this process for all cent values in the root scale, the final SoB approximation is 200, 381.818, 545.455, 700, 840, 960, 1090.909, 1200. In this particular example, there are intervals of 10edo, 12edo, and 22edo in the approximation.

(Todo: add diagrams to make the explanation more intuitive)