User:Overthink/159edo well temperament

A lot of people consider the 23-limit as an ideal stopping point for prime limits, due to the record prime gap between 23 and 29. Unfortunately, no edo up less than 282, aside from 94, is consistent in the 23-odd-limit. An edo that comes close, however, is 159edo, which is consistent to the 17-odd-limit and nearly to the 27-odd-limit, only missing 19/17 with a relative error of 51.4%. We will construct a well temperament that attempts to mitigate that error as much as possible, by making 19/17 "consistent" in most keys, without harming consistency of intervals like 17/14 and 17/13.

The well temperament

Before we create our well temperament, we must consider this problem: Consistency is only defined for equal-step tunings; for example, 19/17 is inconsistent in 159edo because the patent val maps it to 25 steps, while the nearest interval by direct approximation is 26 steps. However, in an unequal well temperament, there isn't an agreed definition for consistency. In this article we consider our well temperament to be a MOS scale, where each number of steps has at most two distinct sizes. We want the larger 25-step interval to be closer to 19/17 than any 26-step intervals, have as many 25-step intervals be large as possible, and not harm the consistency of any other intervals. For example, 17/14 is mapped to 45 steps, with its second closest being 44 steps, and we want any 45-step interval to be closer to 17/14 than any 44-step interval. In a MOS scale, all step counts that are a multiple of the period have one size, and all other step counts have two. Since 25 and 26 are both coprime with 159, neither can be a multiple of the period, and therefore both have two step sizes. Therefore, we want the major* 25-step to be closer to 19/17 than the minor* 26-step.

*Besides multiples of the period, generator steps are considered perfect, augmented, or diminished, rather than major or minor. This agrees with diatonic naming, but is more of a nuisance in our article, so pretend they are major and minor as well.

We want 19/17 to be closer to the major 25-step than the minor 26-step, which means the major 25-step must be flat of 19/17 by less than the minor 26-step is sharp of 19/17. Therefore the sum of the major 25-step and minor 26-step, which is some kind of 51-step, must be mapped sharp of the just value of 2 19/17's, or 361/289. This interval, at 385.11521 cents, is wider than 51\159, 384.9057 cents, by 0.2096 cents, so the 51-step interval that is the sum of the major-25 step and minor 26-step interval must be sharper than its closest 159edo degree, and therefore must be mapped to a positive number of bright generators, so must be major. We must therefore map the major 51-step wider than a just 361/289, and therefore 51 degrees can't be a multiple of the period, so the period can't be 3, and therefore we don't need to consider period-53 MOSes.