More mutant well-temperaments

Here is a version of Vallotti, adjuseted for brats:

Vallotti-Smith

[1, 66760/62973, 94103/83964, 124661/104955, 633295/503784, 4/3,
267040/188919, 94103/62973, 498644/314865, 633295/377838, 16/9,
44655184/23614875]

brats:

[5/2, 5/2, 2, 2, 1782208/1863495, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 5/2]

Here are some mutated Kirnbergers, which I wouldn't call "versions",
but which might be of interest to someone:

Kirnberger2-Smith

[1, 135/128, 9/8, 161/135, 5/4, 4/3, 45/32, 3/2, 43/27, 5/3,
16/9, 15/8]

brats:

[undefined, 128/207, undefined, 3, 3/2, undefined, 3/2, undefined,
2,3/2, 3/2, 3/2]

Kirnberger3-Smith:

[1, 135/128, 107/96, 427/360, 5/4, 4/3, 45/32, 107/72, 1001/640, 5/3,
16/9, 15/8]

Brats:

[infinity, 3, 2, 2, 3/2, undefined, 3/2, 3/2, 160/207, 3/2, 3/2, 3/2]

My apologies to Erv; I was thinking of the wrong convergence issue. He
meant convergence of the ratios of successive terms, which do converge
because there is a single largest root in terms of absolute value
which is real and positive. The terms of the recurrence do not
converge to exponential growth, but wiggle increasingly around an
expontially growing function, but the wiggles themselves, though they
are growing in absolute terms, grow smaller *in proportion* to the
size of the terms of the sequence.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

/tuning/topicId_42756.html#42756

> Here is a version of Vallotti, adjuseted for brats:
>

***I have to ask, since I'm obviously not "getting it..."

Are "brats" "beat-rates??" or some such??

Thanks!

J. Pehrson

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> /tuning/topicId_42756.html#42756
>
> > Here is a version of Vallotti, adjuseted for brats:
> >
>
> ***I have to ask, since I'm obviously not "getting it..."
>
> Are "brats" "beat-rates??" or some such??
>
> Thanks!
>
> J. Pehrson

yes, it seems to be the ratio of

the rate of beating of the minor third

to

the rate of beating of the major third

when

the close-voiced root-position major triad is played.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"

/tuning/topicId_42756.html#42849

<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > /tuning/topicId_42756.html#42756
> >
> > > Here is a version of Vallotti, adjuseted for brats:
> > >
> >
> > ***I have to ask, since I'm obviously not "getting it..."
> >
> > Are "brats" "beat-rates??" or some such??
> >
> > Thanks!
> >
> > J. Pehrson
>
> yes, it seems to be the ratio of
>
> the rate of beating of the minor third
>
> to
>
> the rate of beating of the major third
>
> when
>
> the close-voiced root-position major triad is played.

***Got it. Tx!

JP

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

Speaking of mutant well-temperaments, do you have any more which you
think might be worth mutating? Both Werckmeister III and Vallotti gave
interesting results. What are the "important" well temperaments?

> yes, it seems to be the ratio of
> the rate of beating of the minor third to
> the rate of beating of the major third
> when the close-voiced root-position major triad is played.

I've updated Scala to show this, if you do
SHOW/RELATIVE BEATS 5/4 3/2
then the last column contains the brats.

Manuel

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> Speaking of mutant well-temperaments, do you have any more which you
> think might be worth mutating? Both Werckmeister III and Vallotti
gave
> interesting results. What are the "important" well temperaments?

if you add kirberger ii and kirnberger iii, and every possible
interpolation between these four, you pretty much have
the "important" space covered.