What are these 17-limit temperaments called?

These are a few more "modalities" consistent with 12-equal, and I'm
not sure if they have names.
3.5.17-limit rank 3 temperaments, combining the 17-axis with the 5-axis:
- The temperament equivocating 135/128 and 17/16, meaning 136/135 is
tempered out
- The temperament equivocating 27/25 and 17/16, meaning 432/425 is tempered out
- The temperament equivocating 25/24 and 17/16, meaning 51/50 is tempered out

Is there like a fundamental 5.17-limit comma here that you can combine
with 81/80 and 128/125 to yield the other ones? I just spent like 20
minutes crunching numbers and am convinced at this point I am simply
not sure how to solve this problem.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Is there like a fundamental 5.17-limit comma here that you can combine
> with 81/80 and 128/125 to yield the other ones? I just spent like 20
> minutes crunching numbers and am convinced at this point I am simply
> not sure how to solve this problem.

You mean 83521/78125 I think.

> > Is there like a fundamental 5.17-limit comma here that you can combine
> > with 81/80 and 128/125 to yield the other ones? I just spent like 20
> > minutes crunching numbers and am convinced at this point I am simply
> > not sure how to solve this problem.
>
> You mean 83521/78125 I think.

83521/78125 * 81/80 = 6765201/6250000
83521/78125 * 128/125 = 10690688/9765625

Can't be that...

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > Is there like a fundamental 5.17-limit comma here that you can combine
> > > with 81/80 and 128/125 to yield the other ones? I just spent like 20
> > > minutes crunching numbers and am convinced at this point I am simply
> > > not sure how to solve this problem.
> >
> > You mean 83521/78125 I think.
>
> 83521/78125 * 81/80 = 6765201/6250000
> 83521/78125 * 128/125 = 10690688/9765625

I'm not sure what you are asking, nor what the point of the above is. Obviously, three commas composed from 2, 5, and 17 can't reduce 432/425 or 136/135 to unity because they have a 3 in their factorization. They can, however, reduce both to 51/50.

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

> I'm not sure what you are asking, nor what the point of the above is. Obviously, three commas composed from 2, 5, and 17 can't reduce 432/425 or 136/135 to unity because they have a 3 in their factorization. They can, however, reduce both to 51/50.

Sorry, I guess you wanted 81/80 in the mix. But the problem is the same, as the 3 only appears to the fourth power. Maybe you can explain what it is, more precisely, you want?

All of the temperaments listed are types of semitones that generally
only differ by 81/80 and/or 128/125 - 16/15, 25/24, 27/25, and
135/128. And all of the temperaments listed basically just involve the
concept of merging each semitone with 17/16.

After typing this I realize that I left out 16/15 in the original
example. Merging 17/16 and 16/15 would yield 256/255 as the comma
being tempered out.

I worded it poorly, but I suppose all I'm trying to figure out is what
the set of different common commas is for 3.5.17 limit temperaments of
whatever rank would be. Something analogous to 64/63 for 3.5.7 limit
stuff - that shows up as the difference between the septimal and
pythagorean tunings of the same generic interval (7/4 vs 16/9 for
example), and then 36/35 shows up as the difference between the
septimal and 5-limit tunings of the same generic interval (7/4 vs 9/5
for example). Furthermore, 36/35 is actually made up of 64/63 * 81/80.

So I'm trying to figure out what an analogous 17-limit construct might
be - some "generic" 17-limit comma that appears often, and which can
be combined with 81/80 to yield some "generic" 17-limit diesis that
also appears often, and etc.

Or even simpler, there are 4 commas here being tempered out:
256/255
51/50
136/135
432/425

The four of them I think are based on one "fundamental" 17 limit comma
with the rest resulting from tacking on 81/80's and 128/125's, and I'm
trying to figure out what that is. (Or a good way to figure it out...)

-Mike

On Wed, Apr 28, 2010 at 12:50 AM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > > > Is there like a fundamental 5.17-limit comma here that you can combine
> > > > with 81/80 and 128/125 to yield the other ones? I just spent like 20
> > > > minutes crunching numbers and am convinced at this point I am simply
> > > > not sure how to solve this problem.
> > >
> > > You mean 83521/78125 I think.
> >
> > 83521/78125 * 81/80 = 6765201/6250000
> > 83521/78125 * 128/125 = 10690688/9765625
>
> I'm not sure what you are asking, nor what the point of the above is. Obviously, three commas composed from 2, 5, and 17 can't reduce 432/425 or 136/135 to unity because they have a 3 in their factorization. They can, however, reduce both to 51/50.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Or even simpler, there are 4 commas here being tempered out:
> 256/255
> 51/50
> 136/135
> 432/425
>
> The four of them I think are based on one "fundamental" 17 limit comma
> with the rest resulting from tacking on 81/80's and 128/125's, and I'm
> trying to figure out what that is.

You can write all of the commas in terms of 51/50, 136/135 and 432/425. Using the process of "Hermite reduction", we can find the equivalent set 531441/524288, 32805/32768, 4131/4096. This may be what you are looking for.