Abc, high quality commas, and epimericity: Difference between revisions
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== 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 [[wikipedia: størmer's theorem|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. | ||
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 [ | |||
= | == ''abc'' conjecture == | ||
This conjecture is related to the [ | This conjecture is related to the [[Wikipedia: abc conjecture|''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 | 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 [http://math.stackexchange.com/a/235373 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. | 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. | ||
= | == See also == | ||
[ | * [[Superpartient]] | ||
[ | == External links == | ||
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_4458.html Seven and eleven limit comma lists] | |||
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5556.html An 11-limit linear temperament top 100 list] | |||
[[ | [[Category:Comma]] | ||
[[Category:Math]] | |||
Revision as of 21:04, 13 November 2021
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.