*abc*, high quality commas, and epimericity

## Epimericity

If *n*/*d* > 1 is a rational number with positive integers *n* and *d* relatively prime, we may define the *epimericity* of *n*/*d* as log (*n* - *d*)/log (*d*). Which logarithm we use is irrelevant; we can if we like use cents and so the epimericity is also cents (*n* - *d* )/cents (*d*). 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.