Ennealimmal 72

The focus has been on seeing the 72-et either in terms of the secor
or as 6x12; however the ennealimmal picture which focuses on 8x9 has
a lot to be said for it. The 72-et can be tempered via the
ennealimmal temperament for those people who really want to be sure
they are "effectively just."

I might also mention that aside from the approximations for 2,3/2 and
5/4 in terms of 27/25 and 3635, the 11-limit is available
also, for we have the approximation (27/25)^(-1) (36/35)^14 ~ 11/8,
which is 1.77 cents flat.

Hi, Gene. I haven't been focusing on your threads lately. Little time
these days, but this stuff is definitly catching my eye. I apologize
if I'm too far behind on this to ask anything but very naive
questions, but I have a couple:

1. How are two generators employed? Are they just separately iterated
from a reference pitch some arbitrary number of times, like the
secor, for example, then perhaps closed using some really tight
approximation as a unison vector? Then you have two parallel scales
that interleave to form one scale, each generator seeding half the
pitches as subsets of the full scale?

2. What do you mean by "The 72-et can be tempered via the
> ennealimmal temperament", since 72-tET is already a temperament?
Are you modifying it to transform it into an irregular temperament
with uneven steps in order to exploit the nearer-JI advantages of
ennealimmal?

Thanks,

Bob

--- In tuning@y..., genewardsmith@j... wrote:
> The focus has been on seeing the 72-et either in terms of the secor
> or as 6x12; however the ennealimmal picture which focuses on 8x9
has
> a lot to be said for it. The 72-et can be tempered via the
> ennealimmal temperament for those people who really want to be sure
> they are "effectively just."
>
>
> I might also mention that aside from the approximations for 2,3/2
and
> 5/4 in terms of 27/25 and 3635, the 11-limit is available
> also, for we have the approximation (27/25)^(-1) (36/35)^14 ~ 11/8,
> which is 1.77 cents flat.

--- In tuning@y..., BobWendell@t... wrote:

> 1. How are two generators employed? Are they just separately
iterated
> from a reference pitch some arbitrary number of times, like the
> secor, for example, then perhaps closed using some really tight
> approximation as a unison vector?

The obvious thing for the limma (27/25) generator is to take it to be
2^(1/9) and close it up after exactly nine steps. The 36/35 or
quarter-tone generator we can now regard as creating stacks of 9-ets.
There are various ways of closing that up to make the whole thing
into an et if you like--8 makes the 72 et, 11 the 99 et, 19 the 171
et, 30 the 270 et, 49 the 441 et and 68 the 612 et. Obviously, this
involves too many notes for most purposes.

> 2. What do you mean by "The 72-et can be tempered via the
> > ennealimmal temperament", since 72-tET is already a temperament?

The 12-et is also a temperament, by your definition, but that does
not stop us from tempering it via meantone. If and when people
finally decide on a 72-et notation I'll show what an ennealimmal
tempered version would look like. We can choose a value of exactly 49
cents for the quarter-tone (3 72-et steps) generator, which is as
good as anything and very close to the 441 and 612 values. We then
end up adding exactly two cents to the 3/2 and 7/4 of the 72-et, and
three cents to the 5/4; this makes these values very, very, very
good. We may use the ennealimmal temperament in conjunction with the
secor, getting more than one size of secor, and slightly altered
versions of Blackjack and Canasta with some effectively just
harmonies.