Some rank 2 temperaments based off of Rod Poole's scale

Hey all,

Ron Sword pointed me towards Rod Poole's apparently famous, no-5's 17
note scale here:

http://anaphoriasouth.blogspot.com/2009/10/scale-develops-rod-pooles-tuning.html

I decided to test myself and see if I could come up with a decent rank
2 scale for it. I came up with the 2.3.7.11.13 rank 2 temperament that
eliminates 729/728, 896/891, and 352/351. I also thought that it might
be worthwhile to bring this down into the full 13-limit by eliminating
105/104 as well. There is probably a secret trick to represent these
commas as a set of simpler ones using Smith normal form, but I'm not
in on the secret.

The obvious choice here was to equate 81/64 and 14/11, which means
that 896/891 vanishes. I'm tempted to say that should be a feature of
every 11-limit temperament, ever. I did the rest from there by hand.

I don't think that Graham's temperament finder supports subgroup
temperaments, but here's the resulting full 13-limit temperament with
105/104 vanishing:

http://x31eq.com/cgi-bin/rt.cgi?ets=17_24&error=8.069&limit=13&invariant=2_-16_13_5_-1_1_1_7_-1_2_4

Can anyone do better here?

-Mike

On 21 March 2011 12:07, Mike Battaglia <battaglia01@...> wrote:

> I don't think that Graham's temperament finder supports subgroup
> temperaments, but here's the resulting full 13-limit temperament with
> 105/104 vanishing:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=17_24&error=8.069&limit=13&invariant=2_-16_13_5_-1_1_1_7_-1_2_4

It does support subgroup temperaments, but not from the ratio search.
You can lter the limit in your link though:

http://x31eq.com/cgi-bin/rt.cgi?ets=17_24&limit=2.3.7.11.13

Graham

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hey all,
>
> Ron Sword pointed me towards Rod Poole's apparently famous, no-5's 17
> note scale here:
>
> http://anaphoriasouth.blogspot.com/2009/10/scale-develops-rod-pooles-tuning.html

This apparently famous scale is not in the Scala directory, so here it is:

! rodpoole.scl
!
Rod Poole's 13-limit scale
! http://anaphoriasouth.blogspot.com/2009/10/scale-develops-rod-pooles-tuning.html
17
!
33/32
13/12
9/8
7/6
11/9
14/11
4/3
11/8
13/9
3/2
14/9
44/27
27/16
7/4
11/6
21/11
2/1

I wish people could be induced to include the Scala file when talking about a scale.

> I decided to test myself and see if I could come up with a decent rank
> 2 scale for it. I came up with the 2.3.7.11.13 rank 2 temperament that
> eliminates 729/728, 896/891, and 352/351.

Good choice; it could also be called 144/143, 243/242, 364/363. Either way, the no-fives temperament you get is hemif:

http://xenharmonic.wikispaces.com/Chromatic+pairs#Hemif

I also thought that it might
> be worthwhile to bring this down into the full 13-limit by eliminating
> 105/104 as well.

Extending hemif to hemififhs is an obvious move. That would mean adding 196/195 to the comma mix.

On Mon, Mar 21, 2011 at 7:26 AM, Graham Breed <gbreed@...> wrote:
>
> On 21 March 2011 12:07, Mike Battaglia <battaglia01@...> wrote:
>
> > I don't think that Graham's temperament finder supports subgroup
> > temperaments, but here's the resulting full 13-limit temperament with
> > 105/104 vanishing:
> >
> > http://x31eq.com/cgi-bin/rt.cgi?ets=17_24&error=8.069&limit=13&invariant=2_-16_13_5_-1_1_1_7_-1_2_4
>
> It does support subgroup temperaments, but not from the ratio search.
> You can lter the limit in your link though:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=17_24&limit=2.3.7.11.13
>
> Graham

Oh, nice! I never realized this before. Does the &invariant=xxxxxx thing matter?

-Mike

On 22 March 2011 10:18, Mike Battaglia <battaglia01@...> wrote:

> Oh, nice! I never realized this before. Does the &invariant=xxxxxx thing matter?

It matters if the defining ETs don't give the right mapping. You have
to alter it to fit the subgroups -- and be careful with this because a
crazy mapping might bog down the ET search. It comes from the
(Hermite) reduced mapping. I forget what happens when the Hermite
normal form isn't the one shown.

Graham

On Mon, Mar 21, 2011 at 12:52 PM, genewardsmith
<genewardsmith@...> wrote:
>
> I also thought that it might
> > be worthwhile to bring this down into the full 13-limit by eliminating
> > 105/104 as well.
>
> Extending hemif to hemififhs is an obvious move. That would mean adding 196/195 to the comma mix.

Is eliminating 196/195 the same here as eliminating 105/104, or is
this a separate suggestion?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Mar 21, 2011 at 12:52 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > I also thought that it might
> > > be worthwhile to bring this down into the full 13-limit by eliminating
> > > 105/104 as well.
> >
> > Extending hemif to hemififhs is an obvious move. That would mean adding 196/195 to the comma mix.
>
> Is eliminating 196/195 the same here as eliminating 105/104, or is
> this a separate suggestion?

It leads to a temperament which is the same in 41et but not in general, and which has a higher badness figure. I don't know a name for it.