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, but the result is still the same due to the bases cancelling out). 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 a particular finite subset of 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; these are considered "interesting" according to a value judgment whose nature is not known.
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.