The 9-limit diamond in keemic

If you look at the lattices for the 9-(odd)-limit diamond in various 7-limit planar temperaments, they tend to be disconnected, and in any event not convex. The marvel version is interesting because it is connected but has two holes; it can be made convex by adding a secor and an octave less a secor. Keemic (875/864 planar) does even better: the 19 notes of the diamond are connected and convex. While there's not much reason to tune keemic in anything but 41edo, below I give the keemic tempering of the diamond in POTE tuning. That makes the 11-limit intervals lurking about slightly irregular, which is what they deserve for lurking in a 7-limit temperament in the first place.

! diamond9keemic.scl
Keemic (875/864) tempering of 9-limit diamond, POTE tuning
19
!
176.72185
203.74653
235.78536
262.34137
321.40488
380.46839
439.53189
498.12673
583.74625
616.25375
701.87327
760.46811
819.53161
878.59512
937.65863
964.21464
996.25347
1023.27815
1200.00000
!
! diamond9keemictrans.scl
!
!Keemic (875/864) transversal of the 9-limit diamond
! 19
!!
! 10/9
! 9/8
! 144/125
! 125/108
! 6/5
! 5/4
! 125/96
! 4/3
! 864/625
! 625/432
! 3/2
! 192/125
! 8/5
! 5/3
! 216/125
! 125/72
! 16/9
! 9/5
! 2/1

On Tue, Aug 23, 2011 at 12:22 PM, genewardsmith
<genewardsmith@...t> wrote:
>
> If you look at the lattices for the 9-(odd)-limit diamond in various 7-limit planar temperaments, they tend to be disconnected, and in any event not convex. The marvel version is interesting because it is connected but has two holes; it can be made convex by adding a secor and an octave less a secor. Keemic (875/864 planar) does even better: the 19 notes of the diamond are connected and convex. While there's not much reason to tune keemic in anything but 41edo, below I give the keemic tempering of the diamond in POTE tuning. That makes the 11-limit intervals lurking about slightly irregular, which is what they deserve for lurking in a 7-limit temperament in the first place.

This is sort of related to when you marvel-tempered the eikosany to
fill the hole in the middle, yes? It was apparently manifesting itself
perceptually as the structure "being symmetric around a tonal center
that isn't there," if I remember correctly. This is a very neat idea.

BTW, how do you generate a 9-limit tonality diamond in scala? The
"Tonality Diamond" menu option only lets me work in prime-limit, for
some stupid reason.

-Mike

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

> This is sort of related to when you marvel-tempered the eikosany to
> fill the hole in the middle, yes?

Sort of, but this is the marvel tempering of the 9-limit diamond.

It was apparently manifesting itself
> perceptually as the structure "being symmetric around a tonal center
> that isn't there," if I remember correctly. This is a very neat idea.

Right, that was the eikosany. The marvel 9-limit diamond has two holes.

> BTW, how do you generate a 9-limit tonality diamond in scala? The
> "Tonality Diamond" menu option only lets me work in prime-limit, for
> some stupid reason.

Yeah, I don't like that either, but what you do instead is easy enough--you use "New Rectangular Scale" under the "New" pull-down menu.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > BTW, how do you generate a 9-limit tonality diamond in scala? The
> > "Tonality Diamond" menu option only lets me work in prime-limit, for
> > some stupid reason.
>
> Yeah, I don't like that either, but what you do instead is easy enough--you use "New Rectangular Scale" under the "New" pull-down menu.
>

How did you guys get this impression? You can enter any factor in the File:New:Square tonality diamond dialog.

By the way there's a new feature that's not very visible. In the File:New:Linear temperament dialog you can now type a temperament name in the generator field, for example Amity, and then a completion list drops down where you can select a particular temperament.

Manuel

--- In tuning@yahoogroups.com, "kleisma" <manuel.op.de.coul@...> wrote:

> How did you guys get this impression? You can enter any factor in the File:New:Square tonality diamond dialog.

Apparently because you must enter 1 as a "factor".

On Wed, Aug 24, 2011 at 5:39 AM, kleisma
<manuel.op.de.coul@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> >
> > > BTW, how do you generate a 9-limit tonality diamond in scala? The
> > > "Tonality Diamond" menu option only lets me work in prime-limit, for
> > > some stupid reason.
> >
> > Yeah, I don't like that either, but what you do instead is easy enough--you use "New Rectangular Scale" under the "New" pull-down menu.
> >
>
> How did you guys get this impression? You can enter any factor in the File:New:Square tonality diamond dialog.

So how do I get the full 9-integer-limit tonality diamond? Do I just
type 1 2 3 4 5 6 7 8 9 for factors? Or 1 2 3 5 7 9 or something like
that?

> By the way there's a new feature that's not very visible. In the File:New:Linear temperament dialog you can now type a temperament name in the generator field, for example Amity, and then a completion list drops down where you can select a particular temperament.

Oh nice! That's handy.

I've been trying to get a hold of you for a while actually, to ask -
is there any way to make custom keyboard layouts in the chromatic
clavier? For instance, there's a group of us on Facebook who are
taking a lot of time to exploring mavila, and we tend to like the
Armodue layout, which you call "A16". We've also started spending some
time with mavila in 25-equal lately, but there's no armodue layout for
it. If we load up 25-equal and just set the notation to A16, that sort
of works, but then it craps out entirely for 23-equal. If we use "LT"
and type in generator 14, division 25, it sets the diatonic scale to
mavila[7] instead of mavila[9]. Is there no way around this?

Thanks,
Mike

Gene wrote:
>Apparently because you must enter 1 as a "factor".

You don't have to if you don't want, you will get another diamond then.

Mike wrote:
>So how do I get the full 9-integer-limit tonality diamond? Do I just
>type 1 2 3 4 5 6 7 8 9 for factors? Or 1 2 3 5 7 9 or something like that?

1 3 5 7 9 is sufficient, with the Normalize checkbox on.

>I've been trying to get a hold of you for a while actually, to ask -
>is there any way to make custom keyboard layouts in the chromatic
>clavier? For instance, there's a group of us on Facebook who are
>taking a lot of time to exploring mavila, and we tend to like the
>Armodue layout, which you call "A16". We've also started spending some
>time with mavila in 25-equal lately, but there's no armodue layout for
>it. If we load up 25-equal and just set the notation to A16, that sort
>of works, but then it craps out entirely for 23-equal. If we use "LT"
>and type in generator 14, division 25, it sets the diatonic scale to
>mavila[7] instead of mavila[9]. Is there no way around this?

There are no custom keyboard layouts. And "LT" always gives a layout with 7 white keys. I'll see if this can be changed.
You could try other notations with 9 nominals, like W16, R20, O31, EL72, EL99 and EL441.

Manuel

--- In tuning@yahoogroups.com, "kleisma" <manuel.op.de.coul@...> wrote:

> There are no custom keyboard layouts. And "LT" always gives a layout with 7 white keys. I'll see if this can be changed.

If you are making changes, I'll repeat here my claim that adding fractional monzos as a data type would be much more useful than 64 bit integers. Anyone else care to comment on that notion?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> If you are making changes, I'll repeat here my claim that adding fractional monzos as a data type would be much more useful than 64 bit integers. Anyone else care to comment on that notion?

Yeah! Fractional monzos, wooo!!

I want to be able to put in a temperament like quarter-comma meantone and have all the JI 5/4s appear as "5/4", not as a cents approximation. Fractional monzos make that possible, right?

Keenan

On Thu, Aug 25, 2011 at 6:23 AM, kleisma
<manuel.op.de.coul@...> wrote:
>
> There are no custom keyboard layouts. And "LT" always gives a layout with 7 white keys. I'll see if this can be changed.

If that's not too much trouble - please! If it's not too hard to do
I'd much like to request the following -

1) For the LT notation, to be able to type in the diatonic scale size
that you want
2) To be able to type in how many generators up vs generators down
that you want, for the purposes of specifying which mode gets to be
the "diatonic" scale

This would be so extremely helpful that it's almost
impossible to describe.

Thanks,
Mike

PS: Also, if this isn't too difficult -
3) For the nominals to get names like "1, 2, 3, 4, 5, 6, 7," etc, with
the "C" note getting 1
4) For the accidentals to be generalized to the "#" and "b" glyphs,
where each accidental is defined as the chroma c= L-s

But less important than #1 and #2. I'd be more than willing to write
the code for you (is Scala
open-source?) or to write a pseudocode algorithm or something in C++
if that'd make it easier.

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> I want to be able to put in a temperament like quarter-comma meantone and have all the JI 5/4s appear as "5/4", not as a cents approximation. Fractional monzos make that possible, right?

Indeed they do--you can do any minimax or least squares tuning that way. You can also deal with the problem that numerators and denominators of rational numbers can grow very large in certain operations, such as continual modulation in just intonation or projection of a scale onto a subgroup scale.

So 25/16 & 32/25 are not required for convexity because 25 does not divide any of the numerators or denominators in the diamond to yield any other, whereas 125 & 5 are because they do?

Just Curious
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> If you look at the lattices for the 9-(odd)-limit diamond in various 7-limit planar temperaments, they tend to be disconnected, and in any event not convex. The marvel version is interesting because it is connected but has two holes; it can be made convex by adding a secor and an octave less a secor. Keemic (875/864 planar) does even better: the 19 notes of the diamond are connected and convex. While there's not much reason to tune keemic in anything but 41edo, below I give the keemic tempering of the diamond in POTE tuning. That makes the 11-limit intervals lurking about slightly irregular, which is what they deserve for lurking in a 7-limit temperament in the first place.
>
>
> ! diamond9keemic.scl
> Keemic (875/864) tempering of 9-limit diamond, POTE tuning
> 19
> !
> 176.72185
> 203.74653
> 235.78536
> 262.34137
> 321.40488
> 380.46839
> 439.53189
> 498.12673
> 583.74625
> 616.25375
> 701.87327
> 760.46811
> 819.53161
> 878.59512
> 937.65863
> 964.21464
> 996.25347
> 1023.27815
> 1200.00000
> !
> ! diamond9keemictrans.scl
> !
> !Keemic (875/864) transversal of the 9-limit diamond
> ! 19
> !!
> ! 10/9
> ! 9/8
> ! 144/125
> ! 125/108
> ! 6/5
> ! 5/4
> ! 125/96
> ! 4/3
> ! 864/625
> ! 625/432
> ! 3/2
> ! 192/125
> ! 8/5
> ! 5/3
> ! 216/125
> ! 125/72
> ! 16/9
> ! 9/5
> ! 2/1
>

--- In tuning@yahoogroups.com, "manuphonic" <manuphonic@...> wrote:
>
> So 25/16 & 32/25 are not required for convexity because 25 does not divide any of the numerators or denominators in the diamond to yield any other, whereas 125 & 5 are because they do?

I think I screwed up last time; the lattice diagram I get this time is completely different.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:
> >
> > So 25/16 & 32/25 are not required for convexity because 25 does not divide any of the numerators or denominators in the diamond to yield any other, whereas 125 & 5 are because they do?
>
> I think I screwed up last time; the lattice diagram I get this time is completely different.

Nah, I screwed up the second time when I said I screwed up. But I'm a little bummed, since it looks convex using the rectangular array but not the triangular, which suggests I can't eyeball this stuff very well.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "manuphonic" <manuphonic@> wrote:
> > >
> > > So 25/16 & 32/25 are not required for convexity because 25
> > > does not divide any of the numerators or denominators in the
> > > diamond to yield any other, whereas 125 & 5 are because they do?
> >
> > I think I screwed up last time; the lattice diagram I get this time is
> > completely different.
>
> Nah, I screwed up the second time when I said I screwed up.

Then your beautiful diamond9keemictrans.scl really is convex, just as you started off saying. I wasn't trying to suggest otherwise; I was trying to understand what is meant in this context by convexity.

The high-powered 27/624 ratio in 864/625 would (according to Euler-Fokker generic rules about composite intervals "guiding" chords by summoning all descending prime power factors, harmonics from the numerator, subharmonics from the denominator) require 27/16 & 32/25 (among other intervals) for completion, & it would require them in the same key signature where 864/625 occurs. Yet such intervals can be omitted without loss of convexity? Or is convexity satisfied by their presence in a different key signature of the same tuning?

Then again, in practice, can you tell 864/625 apart from 25/18 with its low-powered 25/3 composite inside it?

> But I'm a little bummed, since it looks convex using the rectangular
> array but not the triangular, which suggests I can't eyeball this stuff
> very well.

I'm still fumbling instead of grasping, so you're way ahead of me.

Cheers!
==
MLV aka Manu Phonic

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Aug 25, 2011 at 6:23 AM, kleisma
> <manuel.op.de.coul@...> wrote:
> >
> > There are no custom keyboard layouts. And "LT" always gives a layout with 7 white keys. I'll see if this can be changed.
>
> If that's not too much trouble - please! If it's not too hard to do
> I'd much like to request the following -
>
> 1) For the LT notation, to be able to type in the diatonic scale size
> that you want

I have a new upload that supports this, let me know if this is what you mean. Go to the Chromatic Clavier, select the LT notation and click on the wizard button. At the bottom there's a "white keys" entry.

> 2) To be able to type in how many generators up vs generators down
> that you want, for the purposes of specifying which mode gets to be
> the "diatonic" scale

This is already possible, see the "lower bound" entry there. Or do
HELP SET LT_PARS.

> This would be so extremely helpful that it's almost
> impossible to describe.
>
> Thanks,
> Mike
>
> PS: Also, if this isn't too difficult -
> 3) For the nominals to get names like "1, 2, 3, 4, 5, 6, 7," etc, with
> the "C" note getting 1
> 4) For the accidentals to be generalized to the "#" and "b" glyphs,
> where each accidental is defined as the chroma c= L-s

I'd have to think about it.

> But less important than #1 and #2. I'd be more than willing to write
> the code for you (is Scala
> open-source?) or to write a pseudocode algorithm or something in C++
> if that'd make it easier.

Yes, in the sense that anyone may read and compile the source. But not in the GPL sense.

Manuel

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

> > Nah, I screwed up the second time when I said I screwed up.
>
> Then your beautiful diamond9keemictrans.scl really is convex, just as you started off saying. I wasn't trying to suggest otherwise; I was trying to understand what is meant in this context by convexity.

Sorry; it turns out I can't eyeball it accurately in all cases using Scala, so I wrote a program for finding convex closures and am planning on presenting a Gallery of Convex Closures. So far 225/224 is still the winner when it comes to the 9-limit diamond.

Hi Manuel,

On Tue, Aug 30, 2011 at 11:40 AM, kleisma
<manuel.op.de.coul@...> wrote:
>
> I have a new upload that supports this, let me know if this is what you mean. Go to the Chromatic Clavier, select the LT notation and click on the wizard button. At the bottom there's a "white keys" entry.

Is the upload released only for Windows? I'm on Mac but the latest
version there is the 2.26y version which is the same one I have.

Thanks,
Mike