Borcherdsma

This page presents a novelty topic. It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks.

160561400000/160561399999, the borcherdsma, is a 71-limit superparticular comma, measuring 1.078 × 10^-8 cents. It is the smallest superparticular interval in the 2.3.5.7.11.13.17.19.23.29.31.41.47.59.71 subgroup, which consists of all the supersingular primes - primes dividing the order of the monster group.

Interval information

Factorization	26 × 55 × 7-1 × 11-2 × 13-1 × 19 × 29 × 31 × 47 × 59-3 × 71-1
Monzo	[6 0 5 -1 -2 -1 0 1 0 1 1 0 0 0 1 0 -3 0 0 -1⟩
Size in cents	1.078284e-08¢
Name	borcherdsma
Color name	71u59u347o31o29o19o3u1uury5-2
FJS name	[math]\displaystyle{ \text{d}{-2}^{5,5,5,5,5,19,29,31,47}_{7,11,11,13,59,59,59,71} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	74.4487
Weil norm (log2 max(n, d))	74.4487
Wilson norm (sopfr(nd))	453
Comma size	unnoticeable
Open this interval in xen-calc

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

Notable edos that temper it out by patent val include:

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.

1578edo - the second strict zeta edo that does so, after 7edo.

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

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

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

(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 ¢, and the largest EDO where this is less than half the step size is around 9.459 × 10¹¹.