Borcherdsma: Difference between revisions

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It is named after the Fields medalist mathematician {{w|Richard Borcherds}}, in reference to his contributions in the theory of the {{w|monstrous moonshine}}.

Notable [[edo]]s that temper it out include:

Notable [[edo]]s that temper it out by [[patent val]] include:

[[6edo]] - the smallest edo that does so~~. Although 6p does indeed temper the borcherdsma with its patent val, there's a lot of doubt whether one would seriously use it to tune the 71-limit~~.

[[6edo]] - the smallest edo that does so.

[[7edo]] - the second smallest edo that does so. 7edo is a [[strict zeta edo]], but that's not a lot of progress from 6edo yet.

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[[8539edo]] - the third strict zeta edo that does so.

[[2901533edo]] - the minimal edo ~~with~~ [[distinctly consistent~~|distinct~~ odd~~-consistency~~-limit~~]] 79~~ (and also all the way to 131)

[[2901533edo]] - the minimal edo [[distinctly consistent]] in the 79-odd-limit (and also all the way to 131-odd-limit).

[[70910024edo]] - the minimal edo ~~with distinct~~ odd~~-consistency~~-limit ~~133 (and also 135)~~

[[70910024edo]] - the minimal edo [[distinctly consistent]] in the 133- and 135-odd-limit.

The largest edo to temper out the borcherdsma is not known, although it is known to be above 6.61 × 1011 ~~and conjectured~~ to be below 1012.

(The last two edos are taken from the list of [[minimal consistent edos]].)

The largest edo to temper out the borcherdsma by patent val is not known, although it is known to be above 6.61 × 1011. It is also known to be below 9.46 × 1011, as beyond that, a prime would need to have 50% or more relative error in order to map the comma to 0 steps. This is because the sum of the absolute values of the monzo entries for primes greater than 2 is 17, so there must be some prime with an absolute error at least 1/17th this commas size, or around 6.343 × 10-10{{C}}, and the largest EDO where this is less than half the step size is around 9.459 × 1011.

[[Category: Superparticular ratios]]

[[Category:Commas named after mathematicians]]

@@ Line 12: / Line 12: @@
 It is named after the Fields medalist mathematician {{w|Richard Borcherds}}, in reference to his contributions in the theory of the {{w|monstrous moonshine}}.
-Notable [[edo]]s that temper it out include:
+Notable [[edo]]s that temper it out by [[patent val]] include:
-[[6edo]] - the smallest edo that does so. Although 6p does indeed temper the borcherdsma with its patent val, there's a lot of doubt whether one would seriously use it to tune the 71-limit.
+[[6edo]] - the smallest edo that does so.
 [[7edo]] - the second smallest edo that does so. 7edo is a [[strict zeta edo]], but that's not a lot of progress from 6edo yet.
@@ Line 22: / Line 22: @@
 [[8539edo]] - the third strict zeta edo that does so.
-[[2901533edo]] - the minimal edo with [[distinctly consistent|distinct odd-consistency-limit]] 79 (and also all the way to 131)
+[[2901533edo]] - the minimal edo [[distinctly consistent]] in the 79-odd-limit (and also all the way to 131-odd-limit).
-[[70910024edo]] - the minimal edo with distinct odd-consistency-limit 133 (and also 135)
+[[70910024edo]] - the minimal edo [[distinctly consistent]] in the 133- and 135-odd-limit.
-The largest edo to temper out the borcherdsma is not known, although it is known to be above 6.61 × 10<sup>11</sup> and conjectured to be below 10<sup>12</sup>.
+(The last two edos are taken from the list of [[minimal consistent edos]].)
+The largest edo to temper out the borcherdsma by patent val is not known, although it is known to be above 6.61&nbsp;×&nbsp;10<sup>11</sup>. It is also known to be below 9.46&nbsp;×&nbsp;10<sup>11</sup>, as beyond that, a prime would need to have 50% or more relative error in order to map the comma to 0&nbsp;steps. This is because the sum of the absolute values of the monzo entries for primes greater than 2 is 17, so there must be some prime with an absolute error at least 1/17th this commas size, or around 6.343&nbsp;×&nbsp;10<sup>-10</sup>{{C}}, and the largest EDO where this is less than half the step size is around 9.459&nbsp;×&nbsp;10<sup>11</sup>.
 [[Category: Superparticular ratios]]
 [[Category:Commas named after mathematicians]]