Quick Review of TM-reduced basis

Okay, this is pretty bad. Could someone give me a quick review
of a TM-reduced basis. I've read the definition on Monz's Encyclopedia,
I think I get the general idea, but if someone could show how
128/125 & 81/80 meet the criteria for TM-reduced basis, perhaps
other things will come back to me.

Paul :(

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> Okay, this is pretty bad. Could someone give me a quick review
> of a TM-reduced basis. I've read the definition on Monz's Encyclopedia,
> I think I get the general idea, but if someone could show how
> 128/125 & 81/80 meet the criteria for TM-reduced basis, perhaps
> other things will come back to me.

(1) The reduced val which sends both 128/125 and 81/80 to zero is
<12 19 28|. These two together are in fact a basis for the kernel of
<12 19 28|.

(2) If we take all products p/q = (128/125)^i (81/80)^j such that p/q>1,
and gcd(p,q)=1, then all such products not equal to 128/125 or 81/80
have p*q > 128*125 = 16000.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
> >
> > Okay, this is pretty bad. Could someone give me a quick review
> > of a TM-reduced basis. I've read the definition on Monz's
Encyclopedia,
> > I think I get the general idea, but if someone could show how
> > 128/125 & 81/80 meet the criteria for TM-reduced basis, perhaps
> > other things will come back to me.
>
> (1) The reduced val which sends both 128/125 and 81/80 to zero is
> <12 19 28|. These two together are in fact a basis for the kernel
of
> <12 19 28|.
>
> (2) If we take all products p/q = (128/125)^i (81/80)^j such that
p/q>1,
> and gcd(p,q)=1, then all such products not equal to 128/125 or 81/80
> have p*q > 128*125 = 16000.
>
Thanks. I get (1), no problem. (2) presents me with the challenge of
always having a gcd(p,q)>=2^4 unless of course i and j are of
opposite signs.

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> > (2) If we take all products p/q = (128/125)^i (81/80)^j such that
> p/q>1,
> > and gcd(p,q)=1, then all such products not equal to 128/125 or 81/80
> > have p*q > 128*125 = 16000.
> >
> Thanks. I get (1), no problem. (2) presents me with the challenge of
> always having a gcd(p,q)>=2^4 unless of course i and j are of
> opposite signs.

Sorry, I don't understand what you are saying.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
> >
> > Okay, this is pretty bad. Could someone give me a quick review
> > of a TM-reduced basis. I've read the definition on Monz's
Encyclopedia,
> > I think I get the general idea, but if someone could show how
> > 128/125 & 81/80 meet the criteria for TM-reduced basis, perhaps
> > other things will come back to me.
>
> (1) The reduced val which sends both 128/125 and 81/80 to zero is
> <12 19 28|. These two together are in fact a basis for the kernel
of
> <12 19 28|.
>
> (2) If we take all products p/q = (128/125)^i (81/80)^j such that
p/q>1,
> and gcd(p,q)=1, then all such products not equal to 128/125 or 81/80
> have p*q > 128*125 = 16000.

Finally, this is all making sense (almost). Taking the kernel space,
a=[-19,12,0; 7,0,-3; 0, 28, -19] I see that null(a)=(12,19,28) Now
(-4,4,-1) & (7, 0, -3) wedge to (12,19,28). Question: How to find
(-4,4 -1). I can get the middle 4, but don't see how to get the outer
values. Is it just solving -3x-7y=19? y=19+3x/-7... Trying to find a
way beyond trial and error. Thanks

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:

> Finally, this is all making sense (almost). Taking the kernel
space,
> a=[-19,12,0; 7,0,-3; 0, 28, -19] I see that null(a)=(12,19,28) Now
> (-4,4,-1) & (7, 0, -3) wedge to (12,19,28). Question: How to find
> (-4,4 -1). I can get the middle 4, but don't see how to get the
outer
> values. Is it just solving -3x-7y=19? y=19+3x/-7... Trying to find
a
> way beyond trial and error. Thanks

I looked this up, one can just solve the Diophantine equation using
the Generalized Euclidean Algorithm. Using the three two-prime commas
one gets three different kernel bases for (12,19,28)

(19,-12,0) & (-4,4,-1) (Pythagorean comma and syntonic comma)
(7,0,-3) & (-4,4,-1) (Diesis and syntonic comma)
(0,28,-19) & (1,-8,5) (What and What?)

Hate to be such a straggler(okay exteme straggler), not expecting a
response unless someone wants to give names to the last kernel basis
I have here! I think it's cool that the first two kernel bases
involve the three simplest commas in 12t-ET. (Is the diesis a comma?)

Paul

Hi Paul,

I haven't really been following discussions here much lately,
but thought that i would direct you to my "bingo-card" page
if you don't already know it, as it might shed some light
on this business:

http://tonalsoft.com/enc/b/bingo.aspx

At the bottom of the page is a "5-limit bingo card viewer",
where you can click on some selected EDO cardinalities to
see a 5-limit bingo-card lattice of that EDO, with the
"vanishing commas" clearly visible as cells which are
outlined in black.

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

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Finally, this is all making sense (almost). Taking the kernel
> space,
> > a=[-19,12,0; 7,0,-3; 0, 28, -19] I see that null(a)=(12,19,28) Now
> > (-4,4,-1) & (7, 0, -3) wedge to (12,19,28). Question: How to find
> > (-4,4 -1). I can get the middle 4, but don't see how to get the
> outer
> > values. Is it just solving -3x-7y=19? y=19+3x/-7... Trying to find
> a
> > way beyond trial and error. Thanks
>
> I looked this up, one can just solve the Diophantine equation using
> the Generalized Euclidean Algorithm. Using the three two-prime commas
> one gets three different kernel bases for (12,19,28)
>
> (19,-12,0) & (-4,4,-1) (Pythagorean comma and syntonic comma)
> (7,0,-3) & (-4,4,-1) (Diesis and syntonic comma)
> (0,28,-19) & (1,-8,5) (What and What?)
>
> Hate to be such a straggler(okay exteme straggler), not expecting a
> response unless someone wants to give names to the last kernel basis
> I have here! I think it's cool that the first two kernel bases
> involve the three simplest commas in 12t-ET. (Is the diesis a comma?)
>
> Paul
>

On 05/01/07, Paul G Hjelmstad <paul_hjelmstad@allianzlife.com> wrote:

> I looked this up, one can just solve the Diophantine equation using
> the Generalized Euclidean Algorithm. Using the three two-prime commas
> one gets three different kernel bases for (12,19,28)

What's the algorithm for multiple commas? This is still an unsolved
problem for me. According to Mathword, "Generalized Euclidean
Algorithm" isn't specific.

Graham

On 06/01/07, Graham Breed <gbreed@gmail.com> wrote:
> On 05/01/07, Paul G Hjelmstad <paul_hjelmstad@allianzlife.com> wrote:
>
> > I looked this up, one can just solve the Diophantine equation using
> > the Generalized Euclidean Algorithm. Using the three two-prime commas
> > one gets three different kernel bases for (12,19,28)
>
> What's the algorithm for multiple commas? This is still an unsolved
> problem for me. According to Mathword, "Generalized Euclidean
> Algorithm" isn't specific.

Hang on, Mathworld lists LLL reduction as a suitable algorithm. That
doesn't solve the difficult problem because it only *reduces* the
lattice. It doesn't find a simpler lattice without torsion. The
difficult problem here is finding a set of commas that don't have
torsion.

Use mystery to check your algorithm. As a printout, it's

1/3

41.362 cents period
15.934 cents generator

mapping by period and generator:
[29, 46, 67, 81, 100, 107]
[0, 0, 1, 1, 1, 1]

mapping by steps:
(29, 46, 67, 81, 100, 107)
(58, 92, 135, 163, 201, 215)

complexity measure: 12.490
RMS weighted error: 0.512 cents/octave
max weighted error: 0.906 cents/octave

and the vectorized wedgie is

[0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7]

If you can get sensible unison vectors for that you've probably got a
solid algorithm.

Graham

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> Hang on, Mathworld lists LLL reduction as a suitable algorithm.
That
> doesn't solve the difficult problem because it only *reduces* the
> lattice. It doesn't find a simpler lattice without torsion. The
> difficult problem here is finding a set of commas that don't have
> torsion.

Triprime commas will do that for you. My personal algorithm is
Hermite reduction of the triprime commas, followed by LLL. This in
practice always gives something close to the TM basis.

> Use mystery to check your algorithm. As a printout, it's

> [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7]
>
> If you can get sensible unison vectors for that you've probably got
a
> solid algorithm.

For comparison purposes, I got {364/363, 847/845, 441/440, 1575/1573}.
This isn't TM reduced, but the commas are sensible and will get you
there.

On 06/01/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

> Triprime commas will do that for you. My personal algorithm is
> Hermite reduction of the triprime commas, followed by LLL. This in
> practice always gives something close to the TM basis.

Oh, yes, I remember this. What's Hermite reduction? As I recall,
triprime commas do *not* do it on their own.

Graham

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> On 06/01/07, Graham Breed <gbreed@...> wrote:
> > On 05/01/07, Paul G Hjelmstad <paul_hjelmstad@...> wrote:
> >
> > > I looked this up, one can just solve the Diophantine equation
using
> > > the Generalized Euclidean Algorithm. Using the three two-prime
commas
> > > one gets three different kernel bases for (12,19,28)
> >
> > What's the algorithm for multiple commas? This is still an
unsolved
> > problem for me. According to Mathword, "Generalized Euclidean
> > Algorithm" isn't specific.
>
> Hang on, Mathworld lists LLL reduction as a suitable algorithm.
That
> doesn't solve the difficult problem because it only *reduces* the
> lattice. It doesn't find a simpler lattice without torsion. The
> difficult problem here is finding a set of commas that don't have
> torsion.
>
> Use mystery to check your algorithm. As a printout, it's
>
> 1/3
>
> 41.362 cents period
> 15.934 cents generator
>
> mapping by period and generator:
> [29, 46, 67, 81, 100, 107]
> [0, 0, 1, 1, 1, 1]
>
> mapping by steps:
> (29, 46, 67, 81, 100, 107)
> (58, 92, 135, 163, 201, 215)
>
> complexity measure: 12.490
> RMS weighted error: 0.512 cents/octave
> max weighted error: 0.906 cents/octave
>
> and the vectorized wedgie is
>
> [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7]
>
> If you can get sensible unison vectors for that you've probably got
a
> solid algorithm.
>
>
> Graham

Thanks Graham. I need to figure out your post. (Thanks for including
mapping by steps which is another thing (of yours) I have been
working on). Question: What exactly is the "vectorized wedgie"?
Why is "mapping by steps (29, 58) for the octave?

Anyway, this was all that I used:

http://www.alpertron.com.ar/QUAD.HTM

You get -4 + 19t, 4+12t which produces x, y for the first part of
this kernel basis: (-4,4,-1)&(19,-12,0). So 4, -1 on the bingo
card lattice (thanx monz) work out nicely, and so does the next
solution 16, -1 of course! (Throwing out x and just using y, z)

Still not done with your paper. Nephews were making too many baby
noises:)

Harmonic Series Question - Calc Formula?

Hi, I'm trying to understand a weird sort ot math,
music related problem. Wondering if anyone out there
may be able to assist with ideas, possible formulas
for computing, deriving values and feedback;

A range of 22 values exist. Conceptually the values
comprise a sort of harmonic series, or possibly an
irregular scale or octave.

We only know two values in the range of 22.

From these two values we want to derive the other
possible remaining twenty values;

1.842030768
3.2

We also want to determine what the relationship is
between the two values interval wise;

For instance, if we say 1.842030768 is the 1/1 of the
scale, what interval would we associate 3.2 with?

Or alternatively from a different perspective, if we
have only these two values, and we know they belong to
a scale of twenty two irregular notes but we do not
know the values of the other notes or intervals how do
we compute them based on these values, would we say
that 1.842030768 is closest to a minor 7th and 3.2
a minor 6th an octave up?

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

On 08/01/07, Paul G Hjelmstad <paul_hjelmstad@allianzlife.com> wrote:
> --- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> > Use mystery to check your algorithm. As a printout, it's
> >
> > 1/3
> >
> > 41.362 cents period
> > 15.934 cents generator
> >
> > mapping by period and generator:
> > [29, 46, 67, 81, 100, 107]
> > [0, 0, 1, 1, 1, 1]
> >
> > mapping by steps:
> > (29, 46, 67, 81, 100, 107)
> > (58, 92, 135, 163, 201, 215)
> >
> > complexity measure: 12.490
> > RMS weighted error: 0.512 cents/octave
> > max weighted error: 0.906 cents/octave
> >
> > and the vectorized wedgie is
> >
> > [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7]
> >
> > If you can get sensible unison vectors for that you've probably got
> a
> > solid algorithm.

> Thanks Graham. I need to figure out your post. (Thanks for including
> mapping by steps which is another thing (of yours) I have been
> working on). Question: What exactly is the "vectorized wedgie"?
> Why is "mapping by steps (29, 58) for the octave?

The vectorized wedgie is the wedgie in vector form. That is, the
wedge product of the two mappings flattened in a way that's standard
in these parts.

It's bases on equal temperaments of 29 and 58 notes to the octave.

> Anyway, this was all that I used:
>
> http://www.alpertron.com.ar/QUAD.HTM
>
> You get -4 + 19t, 4+12t which produces x, y for the first part of
> this kernel basis: (-4,4,-1)&(19,-12,0). So 4, -1 on the bingo
> card lattice (thanx monz) work out nicely, and so does the next
> solution 16, -1 of course! (Throwing out x and just using y, z)

Is that (-4, 4, -1) the same as 81:80? It shouldn't be because this
isn't a meantone variant. Gene got a correct kernel but I still don't
know how.

> Still not done with your paper. Nephews were making too many baby
> noises:)

That's okay, I was wondering where you were with it.

Graham

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:
>
> On 08/01/07, Paul G Hjelmstad <paul_hjelmstad@...> wrote:
> > --- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@>
wrote:
>
> > > Use mystery to check your algorithm. As a printout, it's
> > >
> > > 1/3
> > >
> > > 41.362 cents period
> > > 15.934 cents generator
> > >
> > > mapping by period and generator:
> > > [29, 46, 67, 81, 100, 107]
> > > [0, 0, 1, 1, 1, 1]
> > >
> > > mapping by steps:
> > > (29, 46, 67, 81, 100, 107)
> > > (58, 92, 135, 163, 201, 215)
> > >
> > > complexity measure: 12.490
> > > RMS weighted error: 0.512 cents/octave
> > > max weighted error: 0.906 cents/octave
> > >
> > > and the vectorized wedgie is
> > >
> > > [0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7]
> > >
> > > If you can get sensible unison vectors for that you've probably
got
> > a
> > > solid algorithm.
>
> > Thanks Graham. I need to figure out your post. (Thanks for
including
> > mapping by steps which is another thing (of yours) I have been
> > working on). Question: What exactly is the "vectorized wedgie"?
> > Why is "mapping by steps (29, 58) for the octave?
>
> The vectorized wedgie is the wedgie in vector form. That is, the
> wedge product of the two mappings flattened in a way that's standard
> in these parts.

Yes I see it is the wedge of the two mapping by steps vectors, which
have been tinkered with a little. I'll figure it out, no need to
explain!
>
> It's bases on equal temperaments of 29 and 58 notes to the octave.

I see. This also gives me some depth on your "mapping by steps" thing

> > Anyway, this was all that I used:
> >
> > http://www.alpertron.com.ar/QUAD.HTM
> >
> > You get -4 + 19t, 4+12t which produces x, y for the first part of
> > this kernel basis: (-4,4,-1)&(19,-12,0). So 4, -1 on the bingo
> > card lattice (thanx monz) work out nicely, and so does the next
> > solution 16, -1 of course! (Throwing out x and just using y, z)
>
> Is that (-4, 4, -1) the same as 81:80? It shouldn't be because this
> isn't a meantone variant. Gene got a correct kernel but I still
don't know how.

Yes. Well, meantone or not, the cross product of (-4, 4, -1) x (19, -
12, 0) is (12,19, 28) even though the signs are reversed from what I
thought was normal (that is, I get -28, 19, -12 instead of 28, -19,
12) Unfortunately, little things like that really slow me down.

> > Still not done with your paper. Nephews were making too many baby
> > noises:)
>
> That's okay, I was wondering where you were with it.

I'm on page 9. Excellent so far! I peeked ahead a little. Will finish
soon. If I can get it, you've done great :)

Paul Hj

--- In tuning-math@yahoogroups.com, "Graham Breed" <gbreed@...> wrote:

> Is that (-4, 4, -1) the same as 81:80? It shouldn't be because this
> isn't a meantone variant. Gene got a correct kernel but I still don't
> know how.

Integral Gaussian reduction in place of Hermote reduction (which is a
standard form) should work. I don't know what algorithms you have on
call with your Python stuff.