Another oddball system

Here's another system which turned up in the survey I mentioned; I'd
like to know if this has been considered. The generator is the tenth
root of 3, 3^(1/10), which is whole tone. As a consequence it has
perfect fifths; the third is two generators, and is flat, and 7/4
takes five generators, and is even flatter. However neither is as
flat as they are in the 19-et using the same generator, and of course
the fifth is much better, to say the least.

It can be approximated fairly well in the 82-et, and in terms of 82
et steps, where the generator is a 13, we have a six note scale
(13)^5 17, a seven note scale (13)^6 4, a thirteen note scale
(49)^5 449, and a tempering of 19 notes, (445)^5 4445.

--- In tuning@y..., genewardsmith@j... wrote:

One tempting possibility related to this system is to concentrate on
getting the 5/4 and 7/4 right; if we do that, we get 68+31 in the
99-et; or the <3136/3125, 2401/2400> linear temperament. It now takes
16 generators to get to 6, which is no longer exact, but the 5/4 and
7/4 are both nearly pure, and with the small number of generators
required, quite usable.

In-Reply-To: <9tl9lt+dn1i@eGroups.com>
Gene wrote:

> Here's another system which turned up in the survey I mentioned; I'd
> like to know if this has been considered. The generator is the tenth
> root of 3, 3^(1/10), which is whole tone. As a consequence it has
> perfect fifths; the third is two generators, and is flat, and 7/4
> takes five generators, and is even flatter. However neither is as
> flat as they are in the 19-et using the same generator, and of course
> the fifth is much better, to say the least.

I get a worst 7-limit error of 17.8 cents, so it won't make the charts.
It might be useful otherwise, but we don't have a shortage of temperaments
these days.

Graham

--- In tuning@y..., genewardsmith@j... wrote:
> Here's another system which turned up in the survey I mentioned;
I'd
> like to know if this has been considered. The generator is the
tenth
> root of 3, 3^(1/10),

Wow -- was that the exact least-squares optimum??

which is whole tone. As a consequence it has
> perfect fifths; the third is two generators, and is flat, and 7/4
> takes five generators, and is even flatter. However neither is as
> flat as they are in the 19-et using the same generator, and of
course
> the fifth is much better, to say the least.

Randy Winchester has used every third note in 19-tET (a non-octave
whole-tone scale) -- very different, but related.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:
> > Here's another system which turned up in the survey I mentioned;
> I'd
> > like to know if this has been considered. The generator is the
> tenth
> > root of 3, 3^(1/10),
>
> Wow -- was that the exact least-squares optimum??

It happens. :)

> Randy Winchester has used every third note in 19-tET (a non-octave
> whole-tone scale) -- very different, but related.

Then there's every sixteenth note of the 99-et to try.