Stern-Brocot commas

There has been some discussion of what should count as a comma, and
what doesn't. One possible definition, which we might call an SB-
comma, is that something is an SB-comma if it is in the image of the
comma map I defined, meaning take the ratio of the intevals which are
ratios of the intervals between a node of the Stern-Brocot tree and
its two subnodes; this gives a map from nodes to SB-commas, or in
other words from positive rationals to SB-commas.

Here is a list of 11-limit SB-commas for numbers with denominators
less than 52:

9801/9800, 4375/4374, 6250/6237, 1375/1372, 441/440, 8019/8000,
5120/5103, 243/242, 225/224, 2200/2187, 2835/2816, 1728/1715,
126/125, 245/243, 1944/1925, 81/80, 875/864, 2079/2048, 64/63,
405/392, 126/121, 135/128, 77/72, 27/25, 35/32, 10/9, 9/8

It looks a little cheesy, now that I look at it; I'd better check my
program. :)

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> It looks a little cheesy, now that I look at it; I'd better check
my
> program. :)

I think the problem is my definition; I should confine it to the
branch of the tree coming from 3/2.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > It looks a little cheesy, now that I look at it; I'd better check
> my
> > program. :)

> I think the problem is my definition; I should confine it to the
> branch of the tree coming from 3/2.

The first definition was too soft, and this one too hard. "Just
right" seems to be taking the comma function of every rational number
greater than 1 which is not an integer. This gives the following
11-limit list of SB commas:

9801/9800, 4375/4374, 41503/41472, 6250/6237, 1375/1372, 441/440,
8019/8000, 5120/5103, 243/242, 225/224, 2200/2187, 2835/2816,
1728/1715, 126/125, 245/243, 1944/1925, 81/80, 875/864, 64/63

It seems that SB commas are good commas, but unfortunately not every
good comma is an SB comma. We can make the definition recursive, and
define an SB comma of level n to be the ratio of the SB commas of
level n-1 at each of the subnodes of a given node. If we do that, for
level 2 11-limit SB commas we may add the following: 2401/2376 and
4000/3969. It was nice to see my old friend 4000/3969 again, but I
would have been much happier to get 2401/2400 than 2401/2376. I
didn't get any new 11-limit commas from level 3.