abc, high quality commas, and epimericity
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Todo: intro |
Epimericity
If n/d > 1 is a rational number in simplest form, we may define the epimericity of n/d in terms of cents as cents(n - d)/cents(d) - that is, for an example 9/7, n-d is 2 and d is 7, so we end up with roughly 1200/3369 or 0.356. (Note that using other logarithms is possible). Then it appears to be true that Størmer's theorem generalizes to a claim that for any prime p, only finitely many rational numbers in the p-limit exist with epimericity less than or equal to any constant c less than one. Hence "interesting" commas in any p-limit can be defined as those below a given epimericity, such as the 7-limit commas under 0.5 in epimericity, or the 11-limit commas under 0.3.
abc conjecture
This conjecture is related to the abc conjecture, and a related claim is in fact precisely the abc conjecture, which defines what we may call a high quality comma. Define the radical rad (n/d) of n/d as the product of all the primes dividing n, d, and n - d; so that rad (128/125) = 2×3×5 = 30. Then define the quality q (n/d) of n/d as log (n)/log (rad (n/d)). Then the abc conjecture, a very powerful conjecture, says that for any ϵ > 0 there are only finitely many commas such that q (n/d) > 1 + ϵ, where we may assume without loss of generality that n/d < 2 so that it is an actual comma. Any comma with q (n/d) > 1 we may call "high quality"; there are an infinite number of these, but the conjecture is that there are only finitely many above any value greater than 1. High quality commas in the 5-limit include 250/243, 128/125, 3125/3072, 81/80 and 2048/2025. It should be noted that while every superparticular ratio has an epimericity of 0, merely being superparticular is by no means enough to make a comma high quality; the list of high-quality superparticulars starts 9/8, 49/48, 64/63, 81/80, 225/224, 243/242 ... .
DoReMi conjecture
Since not much musical meaning seems to attach to the commas dividing n - d, it makes sense for our purposes to modify the definition of quality. Let doremi (n/d) = log (n)/log ((n - d) radical(nd)), where radical (nd) is the product of the primes dividing nd. Then q (n/d) ≤ doremi (n/d), so that the condition that doremi (n/d) > 1 + ϵ is stronger than q (n/d) > 1 + ϵ, and there will be fewer intervals which qualify. This means that if the list of q (n/d) > 1 + ϵ is finite, so is the list of doremi (n/d) > 1 + ϵ. So abc implies DoReMi but the converse is not true; DoReMi is a slightly weaker conjecture that is also unproven (according to Noam Elkies and Stack Overflow). Aside from its clearer music theory implications, DoReMi is enormously easier to compute if n/d is in some small p-limit, as then the computation of radical (nd) involves only small primes. A comma n/d with doremi (n/d) > 1 may be called a DoReMi comma.
The DoReMi commas with numerator less than 1000 are 32/27, 9/8, 256/243, 250/243, 128/125, 49/48, 64/63, 81/80, 512/507, 245/243, 225/224, 243/242, 289/288, 513/512, 625/624, 676/675, 729/728 and 961/960. Note that large powers of small primes are favored, so that commas such as 512/507 are on the list.