need a particular lattice basis for 37-edo

Someone (Gene? Graham? anyone?) please help me with this:

I have discovered that 37-edo, except for its fairly poor
approximation of prime-factor 3, has very good approximations
thru the 13-limit (i.e., 5, 7, 11, and 13 are all very good),
and is not too bad for the higher prime-factors either,
all the way thru the 41-limit.

I used the TM-reduced lattice basis in Tonescape
to create a 37-edo tuning latticed in 2,3,5,7,11,13-prime-space.
The unison-vectors tempered out in the TM-basis calculated
by Tonescape are:

[-1 -3, 1 0 1, 0> = 55:54
[-1 -2, -1 1 0, 1> = 91:90
[2 -2, 2 0 -1, 0> = 100:99
[2 -1, -1 2 0, -1> = 196:195
[6 -2, 0 -1 0, 0> = 64:63

But the TM-basis gives me too many ratios along the 3-axes
(which i don't want, since the approximation is bad),
and doesn't give even the single notes which map 7, 11, and 13.
In detail:

. the best mapping of 7:4 is to 30 degrees of 37,
but the TM-basis instead maps 30deg37 to the 16:9 ratio;

. the best mapping of 11:8 is 17deg37, which TM maps to 27:20 ratio;

. the best mapping of 13:8 is 26deg37, which TM maps to 45:28.

So which unison-vectors will give me all of the mappings to 7^+/-1,
11^+/-1, and 13^+/-1?

Specifically, i would like to have a mapping which reflects the
accuracy of 37-edo's approximations to the primes. That accuracy
is tabulated below, where the third column * 100 gives the
percent error in terms of a degree of 37-edo:

2 = 37.000000 +0.00 --> 37
3 = 58.643613 +0.36 --> 59
5 = 85.911340 +0.09 --> 86
7 = 103.872132 +0.13 --> 104
11 = 127.998970 +0.00 --> 128
13 = 136.916270 +0.08 --> 137

Prime-factor 11 has essentially no error, 13 has ~8%, 5 has ~9%,
7 has ~13%, and 3 has a whopping ~36%. Of course 2 has zero error
by definition (i'm not doing TOP here).

So my ideal choice of unison-vectors would give only one
exponent along any of the 3-axes, so that 30deg37 maps to 7:4
instead of 16:9 on the Lattice, and so on.

Thanks.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

On Thu, Apr 5, 2012 at 3:41 AM, joemonz <joemonz@yahoo.com> wrote:
>
> . the best mapping of 7:4 is to 30 degrees of 37,
> but the TM-basis instead maps 30deg37 to the 16:9 ratio;

I'm confused - when you say you're looking for a basis, do you mean
that you're looking for a JI Fokker block that corresponds to 37-EDO?
Otherwise I'm not sure I understand your question - since 64/63 is
tempered out, 16/9 and 7/4 are the same thing.

-Mike

Yes, exactly: a Fokker block which corresponds to 37-edo.
I need to choose unison-vectors which leave certain pitches
inside the block, in particular, 7:4, 8:7, 11:8, 16:11, 13:8,
16:13, etc.

When Tonescape creates an equal-temperament, the Lattice
only shows the JI pitches which fall inside the Fokker block.
I want my 37-edo Lattice to show the mappings to the primes
above 5, which the TM-basis does not succeed in doing.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Apr 5, 2012 at 3:41 AM, joemonz <joemonz@...> wrote:
> >
> > . the best mapping of 7:4 is to 30 degrees of 37,
> > but the TM-basis instead maps 30deg37 to the 16:9 ratio;
>
> I'm confused - when you say you're looking for a basis, do you mean
> that you're looking for a JI Fokker block that corresponds to 37-EDO?
> Otherwise I'm not sure I understand your question - since 64/63 is
> tempered out, 16/9 and 7/4 are the same thing.
>
> -Mike
>

I wanted to add that it's usually fairly easy to simply
uncheck the "TM-basis" box in Tonescape, and then you get
the whole list of possible unison-vectors within the
arbitrary Lattice size limits you choose. Then you can
select unison-vectors more-or-less randomly and see
what types of Fokker blocks result.

However, with the Lattice showing a 3,5,7,11,13 prime-space,
there are so many pitches included at first that each time
a unison-vector is selected, it takes my computer much
too long to trim out the pitches which disappear, and to
show the huge resulting Lattice of pitches which remain.
(And this is on a c.2008 machine with a 64-bit processor
and 4 gb of RAM.)

Of course the incomplete "Fokker block" gets smaller
with each new unison-vector which is selected ... but
with so many dimensions it takes an uncomfortably long time.
I'm sure there must be some algorithm which points out
the desired unison-vectors.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Yes, exactly: a Fokker block which corresponds to 37-edo.
> I need to choose unison-vectors which leave certain pitches
> inside the block, in particular, 7:4, 8:7, 11:8, 16:11, 13:8,
> 16:13, etc.
>
> When Tonescape creates an equal-temperament, the Lattice
> only shows the JI pitches which fall inside the Fokker block.
> I want my 37-edo Lattice to show the mappings to the primes
> above 5, which the TM-basis does not succeed in doing.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Yes, exactly: a Fokker block which corresponds to 37-edo.
> I need to choose unison-vectors which leave certain pitches
> inside the block, in particular, 7:4, 8:7, 11:8, 16:11, 13:8,
> 16:13, etc.

Give 3584/3575, 65536/65065, 9295/9216, 4225/4224, 385/384 a try. Am I right in assuming you use symmetrical blocks when the et is odd?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "monz" <joemonz@> wrote:
> >
> > Yes, exactly: a Fokker block which corresponds to 37-edo.
> > I need to choose unison-vectors which leave certain pitches
> > inside the block, in particular, 7:4, 8:7, 11:8, 16:11, 13:8,
> > 16:13, etc.
>
> Give 3584/3575, 65536/65065, 9295/9216, 4225/4224, 385/384 a try. Am I right in assuming you use symmetrical blocks when the et is odd?

Here's something less extreme which still should work: 169/168, 176/175, 91/90, 55/54, 64/63.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning-math@yahoogroups.com, "monz" <joemonz@> wrote:
> > >
> > > Yes, exactly: a Fokker block which corresponds to 37-edo.
> > > I need to choose unison-vectors which leave certain pitches
> > > inside the block, in particular, 7:4, 8:7, 11:8, 16:11, 13:8,
> > > 16:13, etc.
> >
> > Give 3584/3575, 65536/65065, 9295/9216, 4225/4224, 385/384 a try. Am I right in assuming you use symmetrical blocks when the et is odd?
>
> Here's something less extreme which still should work: 169/168, 176/175, 91/90, 55/54, 64/63.
>

Sorry, nice block but it misses 11/8 and 16/11.