Need help analyzing a linear temperament

This is an 11-note MOS found in 12-equal. A few years back, I wrote a
composition in it. A recording of it can be found here:

http://rabbit.eng.miami.edu/students/mbattaglia/Senior_Recital/1-01%20Sand%20Prism.mp3

(a little messy, but oh well, this was a few years ago and I was
pretty nervous :) )

The MOS is, as far as 12-equal is concerned, every note except for
4/3. So it's this:

100.0
200.0
300.0
400.0
600.0
700.0
800.0
900.0
1000.0
1100.0
1200.0

I've been assuming the generator here is 100 cents and I've been using
it as something like a 2.3.5.7.17.19 subgroup temperament.
- You get 16:17:18:19:20 in there, but the 19/16 also doubles as
75/64, because you can use it in chords like C-E-B-D#. This means that
76/75 vanishes.
- The 17/16 also doubles as 135/128, because you can use it in chords
like C-E-G-B-D-F#-A-C#, so 136/135 vanishes.
- If two generators gets you to 9/8, and four gets you to 5/4, then
81/80 vanishes.
- If three generators gets you to 6/5, which it does between the 15/8
and the 9/4, then that means that 19/16 and 6/5 are equated, so 96/95
vanishes.
- If two 4/3's gets you to 7/4, which was kind of how I was sort of
trying to use it then 64/63 vanishes.

The problem: we're at rank 1 now. Whoops. So let's ditch the last
comma, and say that that isn't 7/4. OK. But now we have a rank 2
temperament with period 100 cents. Oops. Wait a sec, let's try this
again. If four generators going up gets us 5/4, and three generators
going up gets us 6/5, then 7 generators gets us 3/2. But in the
initial MOS I gave, five generators going down gets us 3/2. So we're
in 12-equal again.

So is there no useful way to derive a linear temperament from all of
this that I'm missing? Or is this just a case of a scale that needs to
be rank-1 to actually work as advertised?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So is there no useful way to derive a linear temperament from all of
> this that I'm missing? Or is this just a case of a scale that needs to
> be rank-1 to actually work as advertised?

What does it matter? Do you expect to find a better tuning for it? The way you're using the subgroup, I'm pretty sure 12-TET is going to be the optimal (POTE) tuning for it.

-Igs

On Tue, Feb 22, 2011 at 3:31 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > So is there no useful way to derive a linear temperament from all of
> > this that I'm missing? Or is this just a case of a scale that needs to
> > be rank-1 to actually work as advertised?
>
> What does it matter? Do you expect to find a better tuning for it? The way you're using the subgroup, I'm pretty sure 12-TET is going to be the optimal (POTE) tuning for it.

It matters because I want to come to a better understanding of what,
conceptually, this system is.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> It matters because I want to come to a better understanding of what,
> conceptually, this system is.

But didn't you just say exactly what it was? I mean, you know what you were using it for, right? What more do you need to know?

-Igs

On Tue, Feb 22, 2011 at 4:19 PM, cityoftheasleep
<igliashon@...> wrote:

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > It matters because I want to come to a better understanding of what,
> > conceptually, this system is.
>
> But didn't you just say exactly what it was? I mean, you know what you were using it for, right? What more do you need to know?

There are all kinds of mysterious things that I need to know, most of
which I'm unfortunately not able to divulge on a public forum such as
this.

-Mike

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

> So is there no useful way to derive a linear temperament from all of
> this that I'm missing? Or is this just a case of a scale that needs to
> be rank-1 to actually work as advertised?

I don't know if you'll find it useful, but svals for the 2.3.5.7.17.19 subgroup are [<12 19 28 34 49 51|, <0 0 0 1 0 0|]; you get 12edo, and then a generator (which could be 7 or 8/7, etc, but which above is an approximate 85/84.) The POTE tuning for the generator is 28.893 cents, and 2/84 for the generator, with a period of 7/84, looks good to me, though 1/36 might help you escape the question of why in hell don't you use compton temperament instead.