Brats, beats, and 2/7-comma meantone

This got bounced as email by Yahoo, on the grounds that I don't have
an account. Does anyone know how to cure that?

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> >> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
> >> minor [-3, -2/3, -1/2]
> >>
> >> ... We see 4/15-comma is good for minor triads
> >> and marginal for major ones
>
>
>
> OK, so i can see that the numbers for minor are
> both smaller and less complex than those for major.
> but what *exactly* does "minor [-3, -2/3, -1/2]"
> *signify*?
>
> can i please get a real explanation of that?
>
> and what's the "-9/2"?

The ratios are, for a closed position major *or* minor triad:

[minor/major, fifth/minor, major/fifth]

These are (signed) ratios of beats. The "-9/2", which I've been
calling the "brat", is minor/major in the major triad, or

brat = (6t-5f)/(4t-5)

where t is the major third and f is the minor third. These definitions
are general, and have nothing specific to do with meantone. For
meantone, or any other linear temperament, we have a specific function
which goes from the fraction of a comma together with the canonical
5-limit generator to the brat. In the case of 2/7-comma meantone, that
gives a brat of -1.497, or nearly -1.5. The major beats are

[-3/2, 5/6, -4/5]

the minor beats

[-1, 1, -1]

The synchronized beating of minor triads in 2/7-comma meantone is
striking. To get the same beats for major triads, we can use
5/17-comma meantone, a system worth investigating.

>This got bounced as email by Yahoo, on the grounds that I don't have
>an account. Does anyone know how to cure that?

You have to send from the address that you're subscribed under.

-Carl

>The ratios are, for a closed position major *or* minor triad:

So what happens as one inverts and re-voices these chords?

>The synchronized beating of minor triads in 2/7-comma meantone is
>striking.

All the beats happen at the same rate? Does that also mean they'll
happen at the same *time* (wouldn't this depend on phase)?

Bob says he's empirically verified that beat *ratios* of 2/1 or
3/2 are pleasant. Is the same true for 1/1?

Is 1/1 what is traditionally called "equal beating"?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >This got bounced as email by Yahoo, on the grounds that I don't
have
> >an account. Does anyone know how to cure that?
>
> You have to send from the address that you're subscribed under.

I sent it from my primary email addess. How do I change the one I am
subscribed under?

>I sent it from my primary email addess. How do I change the one I am
>subscribed under?

Go to http://groups.yahoo.com/mygroups/ and click 'e-mail prefs' and
make sure the address is listed there. Then click 'edit my groups'
and assign that address to the affected groups.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I sent it from my primary email addess. How do I change the one I am
> >subscribed under?
>
> Go to http://groups.yahoo.com/mygroups/ and click 'e-mail prefs' and
> make sure the address is listed there. Then click 'edit my groups'
> and assign that address to the affected groups.

Did that some time back. With all of the amazing computer problems I
have been having, it's small potates anyway aqt this point.