19-limit miracle

I was pondering the 351/350 comma, which allows the 1--9/7--20/13 triad
to be considered as having a 6/5 from 9/7 to 20/13. This is a feature
of 13-limit meanpop, and hence my "ratwolf" and what one might call
"wilwolf", which is meanpop[12] using a Wilson fifth. It's also a
feature of the logical extension of 11-limit miracle to 13-limit; one
adds 351/350 to <225/224, 243/242, 385/384> and gets a 13-limit
version of miracle which 72-et supports, but for which 175--which is
poptimal--is a much better choice. We pick the second-best tuning for
13, and [175, 277, 406, 491, 605, 647] works out nicely. It also
extends naturally to a 19-limit version,
[175, 277, 406, 491, 605, 647, 715, 743]. A comma basis for this is
[225/224, 243/242, 273/272, 324/323, 351/350, 375/374, 1445/1444]. If
we leave off the 1445/1444 we get a basis for the 19-limit miracle,
but really there seems little point in trying to separate this from
175, any more than much is to be gained by prying 11-limit miracle
away from 72.

It might be interesting to explore some of the Fokker blocks these
comma sets lead to.

So Gene,

What is the mapping from generators to primes and what are the errors?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:

> What is the mapping from generators to primes and what are the errors?

Mapping to primes:
[[1, 1, 3, 3, 2, 7, 7, -1], [0, 6, -7, -2, 15, -34, -30, 54]]

Errors of the primes, in cents:
[-2.52, -2.32, -1.97, -2.76, -3.96, -2.09, -2.66]

commas: [225/224, 243/242, 273/272, 324/323, 351/350, 375/374]
(Since this is in effect 175-et, we may want to add 1445/1444)

175-et val: [175, 277, 406, 491, 605, 647, 715, 743] (not "standard")

rms secor: .09714280747968 = 16.999991308944 / 175

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > What is the mapping from generators to primes and what are the errors?
>
> Mapping to primes:
> [[1, 1, 3, 3, 2, 7, 7, -1], [0, 6, -7, -2, 15, -34, -30, 54]]
>
> Errors of the primes, in cents:
> [-2.52, -2.32, -1.97, -2.76, -3.96, -2.09, -2.66]
>
> commas: [225/224, 243/242, 273/272, 324/323, 351/350, 375/374]
> (Since this is in effect 175-et, we may want to add 1445/1444)
>
> 175-et val: [175, 277, 406, 491, 605, 647, 715, 743] (not "standard")
>
> rms secor: .09714280747968 = 16.999991308944 / 175

I'm afraid that jump in complexity from 22 to 49, in going from
11-limit to 13-limit, (and then 88 for 19-limit) makes it fairly
uninteresting, except perhaps as a way of mapping higher primes to a
miracle (decimal) keyboard.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <D.KEENAN@U...>
wrote:

> I'm afraid that jump in complexity from 22 to 49, in going from
> 11-limit to 13-limit, (and then 88 for 19-limit) makes it fairly
> uninteresting, except perhaps as a way of mapping higher primes to a
> miracle (decimal) keyboard.

This more or less says that 19-limit temperaments are never
interesting.