Osmium-Orwell-Secor

One way of doing something along the lines Dan suggested would be to
use the Osmium versions of the orwell and the secor, together with
the octave, to express notes. These are pretty far off their usual
values, but looking at the 9-note scale I gave we find a lot of 16/15
(secor) and some 75/64 (orwell) relationships, so it might be just
the ticket. If we invert the matrix for <2,16/15,75/64> we get

[1 0 0]
[2 -1 -2]
[2 1 1]

From this we see we can express notes in the Osmium system by
q ~ 2^f1(q) * o^f2(q) * s^f3(q), where f1=v2+2v3+2v5,f2=-v3+v5,
f3=-2v3+v5. Here o is the Osmium orwell, (40-11z-14z^2)/241, of
268.1254776 cents, and s is the Osmium secor, (43-11z-3z^2)/241, of
115.3367774 cents. Of course given our 225/224~1 and 385/384~1 we
have the 11-limit expressed in these terms. In these terms, we have

3 ~ 2^2 o^(-1) s^(-2)
5 ~ 2^2 o s
7 ~ 2^3 s^(-2)
11 ~ 2^4 o^(-2) s^(-1)

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

> Here o is the Osmium orwell, (40-11z-14z^2)/241, of
> 268.1254776 cents, and s is the Osmium secor, (43-11z-3z^2)/241, of
> 115.3367774 cents.

I calculated the 11-limit least squares versions of these, and got
the following temperament: o = 267.1445284 cents, s = 116.0783521
cents. The same map to the primes obtains, of course:

> 3 ~ 2^2 o^(-1) s^(-2)
> 5 ~ 2^2 o s
> 7 ~ 2^3 s^(-2)
> 11 ~ 2^4 o^(-2) s^(-1)

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
>
> > ... and s is the Osmium secor, (43-11z-3z^2)/241, of
> > 115.3367774 cents.

"Osmium secor" sounds like a good record album title for an
alternative tuning heavy metal band performing in the Miracle
temperament. :)

--George

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> >
> > > ... and s is the Osmium secor, (43-11z-3z^2)/241, of
> > > 115.3367774 cents.
>
> "Osmium secor" sounds like a good record album title for an
> alternative tuning heavy metal band performing in the Miracle
> temperament. :)
>
> --George

Did anybody get this joke? Osmium is one of the heavy metals in the
platinum group.

--GS

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> > > --- In tuning-math@y..., genewardsmith@j... wrote:
> > >
> > > > ... and s is the Osmium secor, (43-11z-3z^2)/241, of
> > > > 115.3367774 cents.
> >
> > "Osmium secor" sounds like a good record album title for an
> > alternative tuning heavy metal band performing in the Miracle
> > temperament. :)
> >
> > --George
>
> Did anybody get this joke? Osmium is one of the heavy metals in
the
> platinum group.

yup, gene mentioned that osmium is the densest metal, so i got it.
now, if gene would help me with the graph density problem . . .

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

> yup, gene mentioned that osmium is the densest metal, so i got it.
> now, if gene would help me with the graph density problem . . .

Which graph are you talking about?

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > yup, gene mentioned that osmium is the densest metal, so i got
it.
> > now, if gene would help me with the graph density problem . . .
>
> Which graph are you talking about?

it was my last post before this one:

/tuning-math/message/5003