how to find the generator?

hello all,

i need a solution to this problem:

given:
1) the set of prime-factors
2) an EDO cardinality
3) an identity-vector

how do i find the generator?

can i get a unique generator from this given data?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hello all,
>
>
> i need a solution to this problem:
>
> given:
> 1) the set of prime-factors
> 2) an EDO cardinality
> 3) an identity-vector
>
> how do i find the generator?

What's an "identity vector"? A comma? Is "set of prime factors" the
same as prime limit? Are you asking for rational generators or not?

If you have one comma, you get a codimension one temperament, meaning
you only get an equal temperament in the 3-limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hello all,
> >
> >
> > i need a solution to this problem:
> >
> > given:
> > 1) the set of prime-factors
> > 2) an EDO cardinality
> > 3) an identity-vector
> >
> > how do i find the generator?
>
> What's an "identity vector"? A comma?

same as "equivalence interval". almost always it's a 2:1,
but it can be anything else.

> Is "set of prime factors" the same as prime limit?

not exactly. "prime-limit" is simply the largest
prime in the set, and if you skip any primes, then
"prime-limit" doesn't tell you that. (i.e., if the
user wants to create a tuning which uses only 2, 3,
and 7, then technically it's 7-limit, but there is
no 5.)

so all of the primes used in the prime-space have
to be specified.

> Are you asking for rational generators or not?

not. this is for calculating equal-temperaments.

> If you have one comma, you get a codimension one
> temperament, meaning you only get an equal temperament
> in the 3-limit.

hmm ... well, thanks very much for saying that, because
it reminded me that i left out the very important first
part of this question !!!

i also need a method for calculating what the promos
(projective monzos) are for any given EDO, given the
same data as above. we want to be able to find these
mathematically, without having to actually do a search
of all the ratios.

once we have the set of promos, that would be the #4
item to go with the other three i listed above.
then we can find the periodicity-block that goes
with the EDO, which is ultimately what we're after.

-monz

oops ... my bad. i really messed up this question,
so i'm starting all over.

we want to create an ET, and draw a lattice of
how it emulates some rational tuning system.

we know:

1) the equivalence interval (usually 2:1, but it
can be anything), which is divided equally into
a certain number of ET steps

2) the ET cardinality, which is that number of ET steps

3) the set of prime-factors which generate the
rational tuning system to be emulated

we know that the ET emulates the rational tuning system
by tempering out a lattice of vapros.

( / vanishing promos / vanishing commas /
vanishing unison-vectors ... whatever you
want to call them -- i find "vapro" short
and precise)

each vapro is represented by one monzo,
whose multiples (or powers if you prefer
to think in terms of ratios) form that vapro.

how do we find the lattice of vapros?

??????

the result should have a way of showing that
the block is torsional if that is the case.

(as, for example, with these two vapros:
skhisma [-15 8, 1> and diesis [7 0, -3> .)

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> how do we find the lattice of vapros?
>
> ??????
>
>
> the result should have a way of showing that
> the block is torsional if that is the case.
>
> (as, for example, with these two vapros:
> skhisma [-15 8, 1> and diesis [7 0, -3> .)

this torsional stuff does confuse things.

for this example we would already know beforehand
that we have a cardinality of 12 (i should have
specified that in the original post and didn't,
sorry), and the determinant of the matrix of these
two monzos is 24.

so i suppose these two vapros would not pop out
of the formula in the first place?

-monz

>we want to create an ET,

That's easy. :)

>and draw a lattice of
>how it emulates some rational tuning system.

Can you give an example of what you want this to
look like?

>we know:
>
>1) the equivalence interval (usually 2:1, but it
>can be anything), which is divided equally into
>a certain number of ET steps
>
>2) the ET cardinality, which is that number of ET steps

Ok.

>3) the set of prime-factors which generate the
>rational tuning system to be emulated

That's called a "basis".

>we know that the ET emulates the rational tuning system
>by tempering out a lattice of vapros.

What's a lattice of vapros?

>(as, for example, with these two vapros:
>skhisma [-15 8, 1> and diesis [7 0, -3> .)

I wonder if you're looking for a val that tempers
these out?

In that case, you have to make sure that the ET (which
is a given according to the above) supports such a val.
You can do that by checking if the ET approximates the
given commas as zero steps. If the ET is consistent
wrt the given basis, you can do this by simply rounding
the commas in cents to the nearest ET step. Otherwise
you have to compute the lattice paths for the commas.

If the ET is consistent wrt to given basis and tempers
out the given commas, you can just use the standard
val for that ET. Otherwise...

I think the procedure you want is wedging the vanishing
monzos together, and then taking the complement to get
some kind of val. If the monzos you started with define
an ET, I think you get a single val out. Otherwise you
get a bunch of vals (map). But hopefully Gene or
someone will chime in here.

-Carl

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> we want to create an ET, and draw a lattice of
> how it emulates some rational tuning system.

Good!

> we know:
>
> 1) the equivalence interval (usually 2:1, but it
> can be anything), which is divided equally into
> a certain number of ET steps
>
> 2) the ET cardinality, which is that number of ET steps
>
> 3) the set of prime-factors which generate the
> rational tuning system to be emulated

This isn't enough. If the et cardinality is n, you need an et *val*
for that n and your set of primes. This could be eg the standard val,
defined as what you get by rounding n*log2(q) to the nearest integer
for the primes q in your set; or the TOP val, where you do the same
but for the TOP tuning.

I have Maple code which does what you want, but it relies on built-in
Maple functions for Hermite reduction and LLL lattice basis reduction.
However, if you have something available for Gaussian reduction, which
is a lot more likely, you could use that.

The basic idea I used was to find the commas using the bicommas of the
val, where a bicomma is c = q^a/r^b, q, and r in the prime set, a and
b relatively prime integers, and c>1. I then feed that to the Hermite
reduction, which is really just a standardized Gaussian reduction, and
take the result and feed that to LLL lattice basis reduction. This
gives me something close enough to the reduced TM basis that I then
can go ahead an TM reduce it to that.

> how do we find the lattice of vapros?

If all you want is the lattice, Gaussian reduction of the bicommas
will give you a lattice basis.

> the result should have a way of showing that
> the block is torsional if that is the case.

Since you start from a val using my method, you know right off the bat
if the val is torsional.

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> I think the procedure you want is wedging the vanishing
> monzos together, and then taking the complement to get
> some kind of val.

I thought he wanted to start from the val and get the monzos; going
the other way is a lot easier, as you point out.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
>
> > I think the procedure you want is wedging the vanishing
> > monzos together, and then taking the complement to get
> > some kind of val.
>
> I thought he wanted to start from the val and get the monzos;
> going the other way is a lot easier, as you point out.

yes, that's right, that's what we need to do.

we already have the ET, and we therefore have the
standard val and can see a few other possibilities
for other vals.

we need to find the monzos which vanish.

using 3-dimensional (primes 2,3,5) examples,
i know how to find the common vapro (vanishing "comma")
shared by two different ETs ... the problem we need
solved is to find the *two* independent vapros which
define *one* ET.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> [Gene]
> I have Maple code which does what you want, but it relies
> on built-in Maple functions for Hermite reduction and LLL
> lattice basis reduction. However, if you have something
> available for Gaussian reduction, which is a lot more likely,
> you could use that.

as long as we understand how to do this stuff, we can write
plenty of code ourselves. thanks for giving us pointers.

> The basic idea I used was to find the commas using the
> bicommas of the val, where a bicomma is c = q^a/r^b,
> q, and r in the prime set, a and b relatively prime integers,
> and c>1. I then feed that to the Hermite reduction,
> which is really just a standardized Gaussian reduction,
> and take the result and feed that to LLL lattice basis
> reduction. This gives me something close enough to the
> reduced TM basis that I then can go ahead an TM reduce
> it to that.

yes, i'm pretty sure that this is what we need. thanks!!

> > how do we find the lattice of vapros?
>
> If all you want is the lattice, Gaussian reduction of the bicommas
> will give you a lattice basis.
>
> > the result should have a way of showing that
> > the block is torsional if that is the case.
>
> Since you start from a val using my method, you know right
> off the bat if the val is torsional.

that's what i thought.

my thinking is that we can't just throw away
torsional-blocks. there are tunings such as
Helmholtz-24 and Groven-36 schismic which work
exactly this way.

so we want to be able to allow the user to create
torsional-blocks, but we want to keep it obvious what
is a regular periodicity-block, and what is torsional.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> The basic idea I used was to find the commas using the
> bicommas of the val,

in the language i'm using (which i thought was what
you were using), you're talking about a bimonzo here,
right?

but we need this to be more general, so that we
can also work with trimonzos, tetramonzos, etc., if
the dimensionality of the prime-space is that high.

> where a bicomma is c = q^a/r^b,
> q, and r in the prime set,
> a and b relatively prime integers,
> and c>1.

can you spell this out with a few examples?

also, i know the easy rule of flipping the order of
numbers and the middle sign of the val, to find the
bimonzo, in the 3-dimensional case ... but can you
show the general method for converting
multival => multimonzo for any dimensionality?

-monz

I then feed that to the Hermite
> reduction, which is really just a standardized Gaussian reduction,
and
> take the result and feed that to LLL lattice basis reduction. This
> gives me something close enough to the reduced TM basis that I then
> can go ahead an TM reduce it to that.
>
> > how do we find the lattice of vapros?
>
> If all you want is the lattice, Gaussian reduction of the bicommas
> will give you a lattice basis.
>
> > the result should have a way of showing that
> > the block is torsional if that is the case.
>
> Since you start from a val using my method, you know right off the
bat
> if the val is torsional.

>> I thought he wanted to start from the val and get the monzos;
>> going the other way is a lot easier, as you point out.
>
>yes, that's right, that's what we need to do.

Ah, as I re-read your original message now that's clear.
I read it the first time at work. :(

>we already have the ET, and we therefore have the
>standard val and can see a few other possibilities
>for other vals.
>
>we need to find the monzos which vanish.

Gene's method is what you want, then.

>using 3-dimensional (primes 2,3,5) examples,
>i know how to find the common vapro (vanishing "comma")
>shared by two different ETs ... the problem we need
>solved is to find the *two* independent vapros which
>define *one* ET.

There are lots of solutions for any ET. Gene's
method using LLL and TM reduction seem to be the way
to go.

-Carl

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > The basic idea I used was to find the commas using the
> > bicommas of the val,
>
>
> in the language i'm using (which i thought was what
> you were using), you're talking about a bimonzo here,
> right?

No, I'm talking about a two-prime comma, so I should have said biprime
comma.

> but we need this to be more general, so that we
> can also work with trimonzos, tetramonzos, etc., if
> the dimensionality of the prime-space is that high.
>
>
>
> > where a bicomma is c = q^a/r^b,
> > q, and r in the prime set,
> > a and b relatively prime integers,
> > and c>1.
>
>
> can you spell this out with a few examples?

In the case of <12 19 28|, 12 and 19 are relatively prime, leading to
the comma 3^12/2^19. 12 and 28 have a common factor of 4; taking that
out gives 3 and 7, leading to the comma 2^7/5^3. 19 and 28 are again
relatively prime, and this gives us 3^28/5^19.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s..
.>
> > wrote:
> >
> > > The basic idea I used was to find the commas using the
> > > bicommas of the val,
> >
> >
> > in the language i'm using (which i thought was what
> > you were using), you're talking about a bimonzo here,
> > right?
>
> No, I'm talking about a two-prime comma, so I should
> have said biprime comma.

that's still not clearing up my confusion.

so you're really not talking about a bimonzo?

> > but we need this to be more general, so that we
> > can also work with trimonzos, tetramonzos, etc., if
> > the dimensionality of the prime-space is that high.
> >
> >
> >
> > > where a bicomma is c = q^a/r^b,
> > > q, and r in the prime set,
> > > a and b relatively prime integers,
> > > and c>1.
> >
> >
> > can you spell this out with a few examples?
>
> In the case of <12 19 28|, 12 and 19 are relatively prime,
> leading to the comma 3^12/2^19. 12 and 28 have a common
> factor of 4; taking that out gives 3 and 7, leading to the
> comma 2^7/5^3. 19 and 28 are again relatively prime, and
> this gives us 3^28/5^19.

is there a rule for knowing which primes go with which
elements of the val? i don't quite see a pattern.

also, can you give two more examples in higher dimensionality,
one 7-limit and one 11-limit? and explain the rules for
associating primes with val elements?

thanks.

-monz

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > > where a bicomma is c = q^a/r^b,
> > > q, and r in the prime set,
> > > a and b relatively prime integers,
> > > and c>1.
> >
> >
> > can you spell this out with a few examples?
>
> In the case of <12 19 28|, 12 and 19 are relatively prime,
> leading to the comma 3^12/2^19. 12 and 28 have a common
> factor of 4; taking that out gives 3 and 7, leading to
> the comma 2^7/5^3. 19 and 28 are again relatively prime,
> and this gives us 3^28/5^19.

ok, it's obvious that [-19 12, 0> (= 3^12/2^19) and
[7 0, -3> (= 2^7/5^3) are the 2,3,5-monzos for the
simplest pair of vapros which define 12edo,
assuming [1 0, 0> (= 2^1) as the equivalence-interval.

so what's the [0 28, -19> (= 3^28/5^19) ?

that's ratio 8158105:6801806, ~ 314.7794608 cents ...
within one cent of 6:5. what's its significance?

... and i'm still looking forward to the
higher-dimension examples.

-monz

hi Gene,

still waiting for a reply to this ...

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:

> > In the case of <12 19 28|, 12 and 19 are relatively prime,
> > leading to the comma 3^12/2^19. 12 and 28 have a common
> > factor of 4; taking that out gives 3 and 7, leading to the
> > comma 2^7/5^3. 19 and 28 are again relatively prime, and
> > this gives us 3^28/5^19.
>
>
>
> is there a rule for knowing which primes go with which
> elements of the val? i don't quite see a pattern.
>
> also, can you give two more examples in higher dimensionality,
> one 7-limit and one 11-limit? and explain the rules for
> associating primes with val elements?
>
> thanks.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene,
>
> still waiting for a reply to this ...

Hey, I'm still awaiting that lattice diagram. Since I didn't seem to
be getting through I was waiting to see if someone else stepped in,
but usually that means Paul.

> > > In the case of <12 19 28|, 12 and 19 are relatively prime,
> > > leading to the comma 3^12/2^19. 12 and 28 have a common
> > > factor of 4; taking that out gives 3 and 7, leading to the
> > > comma 2^7/5^3. 19 and 28 are again relatively prime, and
> > > this gives us 3^28/5^19.

> > is there a rule for knowing which primes go with which
> > elements of the val? i don't quite see a pattern.

If you have primes q, r and a val v, then q^v(r)/r^v(q) is a comma.
You can take the gcd out of v(q) and v(r), and then normalize so that
the comma is greater than 1, and you get what I was calling a biprime
comma for the val.

> > also, can you give two more examples in higher dimensionality,
> > one 7-limit and one 11-limit? and explain the rules for
> > associating primes with val elements?

In the 7-limit you get 6, and in the 11-limit 10 commas, so this gets
a little out of hand. If the val is <12 19 28 34| you get the three
commas for the 5-limit, plus 2^17/7^6, 3^34/7^19, and 5^17/7^14.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> > [Gene:]
> > In the case of <12 19 28|, 12 and 19 are relatively prime,
> > leading to the comma 3^12/2^19. 12 and 28 have a common
> > factor of 4; taking that out gives 3 and 7, leading to
> > the comma 2^7/5^3. 19 and 28 are again relatively prime,
> > and this gives us 3^28/5^19.
>
>
> ok, it's obvious that [-19 12, 0> (= 3^12/2^19) and
> [7 0, -3> (= 2^7/5^3) are the 2,3,5-monzos for the
> simplest pair of vapros which define 12edo,
> assuming [1 0, 0> (= 2^1) as the equivalence-interval.
>
> so what's the [0 28, -19> (= 3^28/5^19) ?
>
> that's ratio 8158105:6801806, ~ 314.7794608 cents ...
> within one cent of 6:5. what's its significance?

oops ... i rounded that ratio incorrectly ... it's
actually 22876792454961:19073486328125.

> ... and i'm still looking forward to the
> higher-dimension examples.

... and still waiting ...

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> > ... and i'm still looking forward to the
> > higher-dimension examples.
>
>
> ... and still waiting ...

It looks like no one is going to help me out here. I wish someone
would; I gave a higher limit example and Monz is still waiting for one
anyway. I am out of clues for how to proceed.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > ... and i'm still looking forward to the
> > > higher-dimension examples.
> >
> >
> > ... and still waiting ...
>
> It looks like no one is going to help me out here. I wish
> someone would; I gave a higher limit example and Monz is
> still waiting for one anyway. I am out of clues for how
> to proceed.

i think our schedules just got crossed ... i didn't
see your example until late last night. this one:

>> In the 7-limit you get 6, and in the 11-limit 10 commas,
>> so this gets little out of hand. If the val is <12 19 28 34|
>> you get the three commas for the 5-limit, plus 2^17/7^6,
>> 3^34/7^19, and 5^17/7^14.

that's fine -- now i can see what's going on.

yes, i can also see that it gets "out of hand" rapidly
as the dimensions increase ... but our software right now
allows the user to create a tuning with up to 7 dimensions,
so we have to be able to do this.

i forgotten now ... what lattice are you waiting on from me?

-monz

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> In the 7-limit you get 6, and in the 11-limit 10 commas,
> so this gets a little out of hand. If the val is <12 19 28 34|
> you get the three commas for the 5-limit, plus 2^17/7^6,
> 3^34/7^19, and 5^17/7^14.

just to be sure i'm getting this, can you check my work?

for 11-limit, if the val is <12 19 28 34 42|,
i get all the commas for 5- and 7-limit, plus these:
2^7/11^2, 3^42/11^19, 5^3/11^2, and 7^21/11^17.

right?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i forgotten now ... what lattice are you waiting on from me?

/tuning/topicId_55583.html#55623

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i forgotten now ... what lattice are you waiting on from me?
>
> /tuning/topicId_55583.html#55623

ok, thanks. wow, that really got left behind in the dust.
i'm working on too many things at once.

(including earning my rent money, alas ... the music-lesson
business slows down every summer, but picks up again in a
big way around the end of August.)

i did start that, and will try to finish it as soon
as i can.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> just to be sure i'm getting this, can you check my work?
>
> for 11-limit, if the val is <12 19 28 34 42|,
> i get all the commas for 5- and 7-limit, plus these:
> 2^7/11^2, 3^42/11^19, 5^3/11^2, and 7^21/11^17.

Right. From that, you could eventually find the TM basis, which
is {36/35, 45/44, 50/49, 56/55}.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > just to be sure i'm getting this, can you check my work?
> >
> > for 11-limit, if the val is <12 19 28 34 42|,
> > i get all the commas for 5- and 7-limit, plus these:
> > 2^7/11^2, 3^42/11^19, 5^3/11^2, and 7^21/11^17.
>
> Right. From that, you could eventually find the TM basis,
> which is {36/35, 45/44, 50/49, 56/55}.

nice. thanks.

... i'll probably have some more questions for you
when i'm ready to figure out how to find the TM basis.

;-)

-monz