Finding Generators to Primes etc

Imagine a triangle representing

1. Generators to Primes
2. Commas
3. Temperaments (such as 12&19)

I am solid in my understanding of the leg between 2 and 1.(Even
though I understand that going from 1 to 2 is more difficult because
of contorsion). I have some understanding of the leg between 2 and 3
(by mapping Linear Temperaments as lines on the Zoom diagrams, these
also represent commas, even though I am not sure how to extract them)
Lastly, even though it should be the easiest, I really have no idea
how to get generators to primes from Temperaments (like a Linear
Temperament like 12&19). I know Hermite Normal Form was mentioned
but I am not sure how to implement it. (Obviously, if I understood
how to get from 3 to 1, then I would also be solid going between 3
and 2...) Sorry if I am lagging behind here and once again even
partial information is appreciated

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Imagine a triangle representing
>
> 1. Generators to Primes
> 2. Commas
> 3. Temperaments (such as 12&19)
>
> I am solid in my understanding of the leg between 2 and 1.(Even
> though I understand that going from 1 to 2 is more difficult
because
> of contorsion). I have some understanding of the leg between 2 and 3
> (by mapping Linear Temperaments as lines on the Zoom diagrams, these
> also represent commas, even though I am not sure how to extract
them)

The leg between 2 and 3 is actually the easiest, it seems to me. Our
recent discussion on wedge products, with the particular example of
cross products, should be helpful to you here.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > Imagine a triangle representing
> >
> > 1. Generators to Primes
> > 2. Commas
> > 3. Temperaments (such as 12&19)
> >
> > I am solid in my understanding of the leg between 2 and 1.(Even
> > though I understand that going from 1 to 2 is more difficult
> because
> > of contorsion). I have some understanding of the leg between 2
and 3
> > (by mapping Linear Temperaments as lines on the Zoom diagrams,
these
> > also represent commas, even though I am not sure how to extract
> them)
>
> The leg between 2 and 3 is actually the easiest, it seems to me.
Our
> recent discussion on wedge products, with the particular example of
> cross products, should be helpful to you here.

It would be cool if you or someone could give an example of the
number crunching used to, say, get 81/80 from 12&19 Temperaments.
Can this be done using matrices? I know the wedge product of the
comma is equal to the wedge product of the val.. but still don't see
how you get from 12&19 TO 81/80...

>It would be cool if you or someone could give an example of the
>number crunching used to, say, get 81/80 from 12&19 Temperaments.
>Can this be done using matrices? I know the wedge product of the
>comma is equal to the wedge product of the val.. but still don't see
>how you get from 12&19 TO 81/80...

The other Paul demonstrated this recently -- you take the cross
product of two vals. So

< 12 19 28 |

is h12 and

< 19 30 44 |

is h19. Except there's something about using the transpose of
one of them to get it into a form where the cross product will
give you a monzo. Which in this case is

| -4 4 -1 > = 81/80

Do I have that right, guys?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >It would be cool if you or someone could give an example of the
> >number crunching used to, say, get 81/80 from 12&19 Temperaments.
> >Can this be done using matrices? I know the wedge product of the
> >comma is equal to the wedge product of the val.. but still don't
see
> >how you get from 12&19 TO 81/80...
>
> The other Paul demonstrated this recently -- you take the cross
> product of two vals. So
>
> < 12 19 28 |
>
> is h12 and
>
> < 19 30 44 |
>
> is h19. Except there's something about using the transpose of
> one of them to get it into a form where the cross product will
> give you a monzo. Which in this case is
>
> | -4 4 -1 > = 81/80
>
> Do I have that right, guys?
>
> -Carl

Thanx!!

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

> < 12 19 28 |
>
> is h12 and
>
> < 19 30 44 |
>
> is h19. Except there's something about using the transpose of
> one of them to get it into a form where the cross product will
> give you a monzo. Which in this case is
>
> | -4 4 -1 > = 81/80
>
> Do I have that right, guys?

~(<12 19 28| ^ <19 30 44|) = |-4 4 -1>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > Imagine a triangle representing
> > >
> > > 1. Generators to Primes
> > > 2. Commas
> > > 3. Temperaments (such as 12&19)
> > >
> I am now solid (Thanks to Carl) on the leg between 3 and 2. Since
I know how to go from 2 to 1, (Thanks to Graham) it is merely
transitive to go from 3 to 1. The hard thing is to reverse direction!
(because of torsion, I take it...) Going from 1 to 2 for example.
I imagine you just reverse all the matricial (sp?) calculations.
Well I'll play with this this weekend. I also am a little hazy when
the period is not just an octave, for example, a tritone.(.5). I am
doing a lot of learning by comparison...

I am thinking of buying both Mathematica and Maple, even though Maple
is fairly expensive, I imagine. It's really neat how much one can do
in Excel! Also Python is pretty cool. Now on to getting complexity
rms error and badness...

Paul

At 03:11 PM 11/21/2003, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
>> < 12 19 28 |
>>
>> is h12 and
>>
>> < 19 30 44 |
>>
>> is h19. Except there's something about using the transpose of
>> one of them to get it into a form where the cross product will
>> give you a monzo. Which in this case is
>>
>> | -4 4 -1 > = 81/80
>>
>> Do I have that right, guys?
>
>~(<12 19 28| ^ <19 30 44|) = |-4 4 -1>

^ is the wedge product. ~ is ? Complement? So the wedging with
a complement is the same as crossing? Please answer each question,
I'm just guessing.

I still don't know a simple procedure to calculate a wedge product.

Oh, and can anyone explain in one sentence why I should care about
bi- and tri- things?

-Carl

>I am thinking of buying both Mathematica and Maple, even though Maple
>is fairly expensive, I imagine.

Both are very expensive, unless you're a student at a university.

Mathematica makes Maple look like a joke, if you ask me.

>It's really neat how much one can do in Excel!

Have you seen/heard Dave Keenan's tumbling dekany? (!)

http://www.uq.net.au/~zzdkeena/Music/StereoDekany.htm

-Carl

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > Imagine a triangle representing
> > >
> > > 1. Generators to Primes
> > > 2. Commas
> > > 3. Temperaments (such as 12&19)
> > >
> > > I am solid in my understanding of the leg between 2 and 1.(Even
> > > though I understand that going from 1 to 2 is more difficult
> > because
> > > of contorsion). I have some understanding of the leg between 2
> and 3
> > > (by mapping Linear Temperaments as lines on the Zoom diagrams,
> these
> > > also represent commas, even though I am not sure how to extract
> > them)
> >
> > The leg between 2 and 3 is actually the easiest, it seems to me.
> Our
> > recent discussion on wedge products, with the particular example
of
> > cross products, should be helpful to you here.
>
> It would be cool if you or someone could give an example of the
> number crunching used to, say, get 81/80 from 12&19 Temperaments.
> Can this be done using matrices? I know the wedge product of the
> comma is equal to the wedge product of the val.. but still don't
see
> how you get from 12&19 TO 81/80...

write down the representations of the primes {2,3,5} in 12:

|12 19 28>

and in 19:

|19 30 44>

now take the usual cross-product between these two and you get:

<-4 4 -1|

these is the "monzo" or prime-exponent-vector for 81/80, as you can
see by computing

2^(-4) * 3^4 * 5^(-1).

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >It would be cool if you or someone could give an example of the
> >number crunching used to, say, get 81/80 from 12&19 Temperaments.
> >Can this be done using matrices? I know the wedge product of the
> >comma is equal to the wedge product of the val.. but still don't
see
> >how you get from 12&19 TO 81/80...
>
> The other Paul demonstrated this recently -- you take the cross
> product of two vals. So
>
> < 12 19 28 |
>
> is h12 and
>
> < 19 30 44 |
>
> is h19. Except there's something about using the transpose of
> one of them to get it into a form where the cross product will
> give you a monzo. Which in this case is
>
> | -4 4 -1 > = 81/80
>
> Do I have that right, guys?

The numbers are right, but you don't use the transpose of one of them.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > Imagine a triangle representing
> > > >
> > > > 1. Generators to Primes
> > > > 2. Commas
> > > > 3. Temperaments (such as 12&19)
> > > >
> > > > I am solid in my understanding of the leg between 2 and 1.
(Even
> > > > though I understand that going from 1 to 2 is more difficult
> > > because
> > > > of contorsion). I have some understanding of the leg between
2
> > and 3
> > > > (by mapping Linear Temperaments as lines on the Zoom
diagrams,
> > these
> > > > also represent commas, even though I am not sure how to
extract
> > > them)
> > >
> > > The leg between 2 and 3 is actually the easiest, it seems to
me.
> > Our
> > > recent discussion on wedge products, with the particular
example
> of
> > > cross products, should be helpful to you here.
> >
> > It would be cool if you or someone could give an example of the
> > number crunching used to, say, get 81/80 from 12&19 Temperaments.
> > Can this be done using matrices? I know the wedge product of the
> > comma is equal to the wedge product of the val.. but still don't
> see
> > how you get from 12&19 TO 81/80...
>
> write down the representations of the primes {2,3,5} in 12:
>
> |12 19 28>
>
> and in 19:
>
> |19 30 44>
>
> now take the usual cross-product between these two and you get:
>
> <-4 4 -1|
>
> these is the "monzo" or prime-exponent-vector for 81/80, as you can
> see by computing
>
> 2^(-4) * 3^4 * 5^(-1).

sorry, i had the left-pointing and right-pointing notation backwards.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 03:11 PM 11/21/2003, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> >> < 12 19 28 |
> >>
> >> is h12 and
> >>
> >> < 19 30 44 |
> >>
> >> is h19. Except there's something about using the transpose of
> >> one of them to get it into a form where the cross product will
> >> give you a monzo. Which in this case is
> >>
> >> | -4 4 -1 > = 81/80
> >>
> >> Do I have that right, guys?
> >
> >~(<12 19 28| ^ <19 30 44|) = |-4 4 -1>
>
> ^ is the wedge product.

yes.

> ~ is ? Complement?

yes.

> So the wedging with
> a complement is the same as crossing?

no, look at the parentheses. the complement of the wedge product is
the cross product (when you're dealing with a 3 dimensional problem).

> Please answer each question,
> I'm just guessing.
>
> I still don't know a simple procedure to calculate a wedge product.

Graham just explained that.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I am thinking of buying both Mathematica and Maple, even though
Maple
> >is fairly expensive, I imagine.
>
> Both are very expensive, unless you're a student at a university.
>
> Mathematica makes Maple look like a joke, if you ask me.

Geometers and analysts tend to prefer Mathematica, and algebraists
and number theorists Maple. Set theorists don't give a good goddamn.
Maple certainly isn't a joke.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> no, look at the parentheses. the complement of the wedge product is
> the cross product (when you're dealing with a 3 dimensional
problem).

Not exactly. A cross product takes vectors to vectors (or
pseudovectors, if you are a physicist) and in fact a three
dimensional real vector space with cross product is the real Lie
algebra o(3). The complement of a wedge product of bra vectors is a
ket vector, and conversely.

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

> Geometers and analysts tend to prefer Mathematica, and algebraists
> and number theorists Maple. Set theorists don't give a good
goddamn.
> Maple certainly isn't a joke.

Full disclosure--the Maple code for finding Galois groups is partly
based on stuff I did.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > no, look at the parentheses. the complement of the wedge product is
> > the cross product (when you're dealing with a 3 dimensional
> problem).
>
> Not exactly. A cross product takes vectors to vectors (or
> pseudovectors, if you are a physicist) and in fact a three
> dimensional real vector space with cross product is the real Lie
> algebra o(3). The complement of a wedge product of bra vectors is a
> ket vector, and conversely.

This is hilarious. I couldn't parody this any better than you're doing
it yourself. :-)

I thought Paul's statement was perfectly clear and correct.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> This is hilarious. I couldn't parody this any better than you're
doing
> it yourself. :-)

Your hilarity is noted. There is a way to relate the wedge product of
two n-dimensional vectors to an n by n antisymmetric matrix in a
natural way, and in concrete terms the Lie algebra product for o(n)
can be thought of as the commutator for two such matricies. This
gives us a product for pairs of linear temperaments in any prime
limit, which would be nice if anyone could explain what the product
meant and what use it was. If I find out, I'll be sure to tell you.

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

> Your hilarity is noted. There is a way to relate the wedge product of
> two n-dimensional vectors to an n by n antisymmetric matrix in a
> natural way, and in concrete terms the Lie algebra product for o(n)
> can be thought of as the commutator for two such matricies. This
> gives us a product for pairs of linear temperaments in any prime
> limit, which would be nice if anyone could explain what the product
> meant and what use it was. If I find out, I'll be sure to tell you.

No doubt inevitably, I started fiddling with this and it seems there
actually might be some uses. If you take the Lie bracket of two
bivals, you get a bival which is orthogonal, using the usual inner
product, to both. These means it really makes more sense for most
purposes to consider it a bimonzo. This bracket seems to allow us to
do in a more general way what I was doing in the 7-limit with
pfaffians a while back--seeing whether two temperaments were related
or not, and giving a measure of by how much.

Taking wedgies of some important 7-limit temperaments for an example,
if mean = <<1 4 10 4 13 12]] is the meantone wedgie,
mir = <<6 -7 -2 -25 -20 15]] is the miracle wedgie,
paj = <<2 -4 -4 -11 -12 2]] is the pajara wedgie, and
orw = <<7 -3 8 -21 -7 27]] is the orwell wedgie, and if [a,b] is a
modified Lie bracket, where the product is interpreted to be a
bimonzo, then

[mean,mir] = [mean,orw] = [orw,mir] = [[246 -223 46 -4 422 -307>>

On the other hand, the brackets of mean, orw and mir with paj are all
distinct.

If we take the complement of these bimonzos, we get a bival again,
which we can regard as a wedgie. What connection, if any, the
resulting temperament has with the ones we started from I don't know.
In the case of [mean,mir], the negative of the complement gives us

<<307 422 404 -46 -223 -246]]

This is certainly a wedgie for an honest temperament, with rms error
(3.7 cents) in the vicinity of where we started from, but a much
higher complexity, and so a high badness. An et which belongs to it is
31, and in 31 it is the same as the supermajor seconds temperament,
with generator 6/31. We might note that 31 belongs to meantone,
miracle and orwell, but not pajara.

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

> This is certainly a wedgie for an honest temperament, with rms error
> (3.7 cents) in the vicinity of where we started from, but a much
> higher complexity, and so a high badness.

The MT basis is [[4 26 -17 -2>, [11 10 9 -17>]

two not very distinguished commas. The generator mapping is, using a
notation I'd like comments on,

[<1 61 84 81], <0 -307 -422 -404]]

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> At 03:11 PM 11/21/2003, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> >> < 12 19 28 |
> >>
> >> is h12 and
> >>
> >> < 19 30 44 |
> >>
> >> is h19. Except there's something about using the transpose of
> >> one of them to get it into a form where the cross product will
> >> give you a monzo. Which in this case is
> >>
> >> | -4 4 -1 > = 81/80
> >>
> >> Do I have that right, guys?

Pretty much. Right answer anyway. Although I think most of us agree we
should use the square brackets instead of the vertical bar, except if
we write a "dot product" as < ... | ... >. But I have to say I can't
see a lot of point in writing them that way.

Forget about transpose here. You take the wedge product of the two ET
mappings to get a "bi-mapping" (bi-val) << -1 -4 -4 ]] and then you
take the complement of that to get the exponent vector for the comma
that vanishes. In 3D only (i.e. 5-limit), that whole sequence happens
to be the same as simply taking the cross-product.

> ^ is the wedge product.

Yes.

For 5-limit this is

<a1 a2 a3] ^ <b1 b2 b3]

= << a1*b2-a2*b1 a1*b3-a3*b1 a2*b3-a3*b2 ]]

How to calculate it in general is described in the thread starting here
/tuning-math/message/7854

> ~ is ?

The complement.

For 5-limit bi-mappings it is simply

~<<c12 c13 c23]]

= [c23 -c13 c12> (reverse the order and negate the middle one)

How to calculate it in general, is described in the thread starting here
/tuning-math/message/7845

> So the wedging with
> a complement is the same as crossing? Please answer each question,
> I'm just guessing.

Not quite. The complement of the wedging is the same as crossing (but
crossing is only defined for 3D (5-limit)).

Sorry if I'm repeating stuff other people have already told you.

> Oh, and can anyone explain in one sentence why I should care about
> bi- and tri- things?

Because bi- and tri- mappings (vals) represent linear and planar
temperaments respectively, in a way that is independent of any
particular choice of generators or vanishing commas.

>Not quite. The complement of the wedging is the same as crossing (but
>crossing is only defined for 3D (5-limit)).

Is the cross product really only defined, for anything, for 3-item
vectors?

>Sorry if I'm repeating stuff other people have already told you.

Not at all; thanks for the references.

>> Oh, and can anyone explain in one sentence why I should care about
>> bi- and tri- things?
>
>Because bi- and tri- mappings (vals) represent linear and planar
>temperaments respectively, in a way that is independent of any
>particular choice of generators or vanishing commas.

And you're the first to answer this.

-C.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > no, look at the parentheses. the complement of the wedge product
is
> > the cross product (when you're dealing with a 3 dimensional
> problem).
>
> Not exactly. A cross product takes vectors to vectors (or
> pseudovectors, if you are a physicist) and in fact a three
> dimensional real vector space with cross product is the real Lie
> algebra o(3). The complement of a wedge product of bra vectors is a
> ket vector, and conversely.

*hands thrown up in air*

so why "Not exactly"???

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Not quite. The complement of the wedging is the same as crossing
(but
> >crossing is only defined for 3D (5-limit)).
>
> Is the cross product really only defined, for anything, for 3-item
> vectors?

yes. see

http://mathworld.wolfram.com/CrossProduct.html

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Not quite. The complement of the wedging is the same as crossing
(but
> >crossing is only defined for 3D (5-limit)).
>
> Is the cross product really only defined, for anything, for 3-item
> vectors?

The orthogonal Lie algebras o(n), which concretely can be represented
as commutator brackets for nxn antisymmetric matricies, is a
generalization. My new thread was on how one could try to apply this
to linear temperaments.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> *hands thrown up in air*
>
> so why "Not exactly"???

The cross product of two bra vectors would be a bra vector, not a ket
vector, if we define things as usual and make the cross products of
two vectors be a vector in the same vector space. This is how it
manages to be an *algebra*.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > Imagine a triangle representing
> > > >
> > > > 1. Generators to Primes
> > > > 2. Commas
> > > > 3. Temperaments (such as 12&19)
> > > >
> > I am now solid (Thanks to Carl) on the leg between 3 and 2. Since
> I know how to go from 2 to 1, (Thanks to Graham) it is merely
> transitive to go from 3 to 1. The hard thing is to reverse
direction!

Another question - is it possible to go from 3 to 1, without going
through 2? Also, do you always need 2 commas (in the 5-limit) to get
Generators to Primes?

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

> Another question - is it possible to go from 3 to 1, without going
> through 2?

Yes -- in the two-ET case, the period will be 1/n octaves where n is
the greatest common divisor of the two ET cardinalities, and the
generator will be in the vicinity of the best-matching other interval
between the two ETs. For example, for 12&19, the period is 1 octave,
while the best other match is between 5/12 oct. and 8/19 oct., so the
generator is near these values. For 12&22, the period is 1/2 octave,
and the best other match is between 1/12 oct. and 2/22 oct., so the
generator is near these values. The mapping from *this* period and
generator to primes should be identical for both ETs, and this
carries over to the '&'.

> Also, do you always need 2 commas (in the 5-limit) to get
> Generators to Primes?

No, with one comma you get a so-called 'linear temperament' (or 2D
temperament -- one period and one generator), just as you get from
12&19 or 12&22 above.

I know you've seen this before, but on each row in this table:

/tuning/database?
method=reportRows&tbl=10&sortBy=3

The single comma is described in the first five columns, the period
and optimal generator are shown in the eighth column, and the mapping
from period and generator to primes is shown in the ninth column.

81:80 corresponds to the 12&19 case above, and 2048:2025 corresponds
to the 12&22 case above.

The comma is more unique, since 12&19 might be written 19&31, etc.,
12&22 might be written 22&34, etc. . . .

Paul G Hjelmstad wrote:

> Another question - is it possible to go from 3 to 1, without going > through 2? Also, do you always need 2 commas (in the 5-limit) to get
> Generators to Primes?

It's possible to do anything you like without commas, unless it explicitly involves commas. If you're using commas, you need 2 of them to get a 5-limit equal temperament.

What's 3 to 1, a linear temperament to a pair of equal temperaments? The usual mapping of primes to generators already gives you this. They may be bizarre equal temperments, with 1 or 0 notes to the octave. But you can add them together to get a different 1 note equal temperment. And then add the two 1s to get a 2 note temperament. And so on, as far as you like.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> The usual mapping of primes to generators

generators to primes?

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> may be bizarre equal temperments, with 1 or 0 notes to the octave.

A good reason to call them vals.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
>
> > The usual mapping of primes to generators
>
> generators to primes?

Graham's way of saying it really makes more sense. We have a mapping
h, for which h(2) gives the number of generator steps to 2, h(3) to
3, and so forth.

Graham:
...
> > may be bizarre equal temperments, with 1 or 0 notes to the octave.

Gene:
> A good reason to call them vals.

Or single-generator mappings.

But whether he called them mappings or vals he would still have wanted
to say that they can be considered as mappings or vals _for_ equal
temperaments, albeit bizarre ones. And that linear combinations of
them are all that is needed to give you mappings for more reasonable ETs.

I had not realised that. Thanks Graham.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> But whether he called them mappings or vals he would still have
wanted
> to say that they can be considered as mappings or vals _for_ equal
> temperaments, albeit bizarre ones.

I don't believe in the 0-equal temperament.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > Another question - is it possible to go from 3 to 1, without
going
> > through 2?
>
> Yes -- in the two-ET case, the period will be 1/n octaves where n
is
> the greatest common divisor of the two ET cardinalities, and the
> generator will be in the vicinity of the best-matching other
interval
> between the two ETs. For example, for 12&19, the period is 1
octave,
> while the best other match is between 5/12 oct. and 8/19 oct., so
the
> generator is near these values. For 12&22, the period is 1/2
octave,
> and the best other match is between 1/12 oct. and 2/22 oct., so the
> generator is near these values. The mapping from *this* period and
> generator to primes should be identical for both ETs, and this
> carries over to the '&'.
>
Interesting. So is rms or minimax applied, to say, 5/12 and 8/19
for 12&19 together?

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > The mapping from *this* period and
> > generator to primes should be identical for both ETs, and this
> > carries over to the '&'.
> >
> Interesting. So is rms or minimax applied, to say, 5/12 and 8/19
> for 12&19 together?
>
> Paul

Well, what I was thinking was that, once you've found the period and
generator, you can determine the mapping between primes and
generators using either ET. But that's not quite right, you need
both. For example, prime 3 is -1 generator but also 11 generators in
12-equal, and prime 3 is -1 generator but also 18 generators in 19-
equal. So you need to pick the one that's in common between the two
ETs. Once you've done that, you can drop any reference to the ETs
themselves, and just optimize, rms or minimaxm, using the mapping
itself.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > The mapping from *this* period and
> > > generator to primes should be identical for both ETs, and this
> > > carries over to the '&'.
> > >
> > Interesting. So is rms or minimax applied, to say, 5/12 and 8/19
> > for 12&19 together?
> >
> > Paul
>
> Well, what I was thinking was that, once you've found the period
and
> generator, you can determine the mapping between primes and
> generators using either ET. But that's not quite right, you need
> both. For example, prime 3 is -1 generator but also 11 generators
in
> 12-equal, and prime 3 is -1 generator but also 18 generators in 19-
> equal. So you need to pick the one that's in common between the two
> ETs. Once you've done that, you can drop any reference to the ETs
> themselves, and just optimize, rms or minimaxm, using the mapping
> itself.

Thanks. Could you give an example?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > > wrote:
> > > > The mapping from *this* period and
> > > > generator to primes should be identical for both ETs, and
this
> > > > carries over to the '&'.
> > > >
> > > Interesting. So is rms or minimax applied, to say, 5/12 and 8/19
> > > for 12&19 together?
> > >
> > > Paul
> >
> > Well, what I was thinking was that, once you've found the period
> and
> > generator, you can determine the mapping between primes and
> > generators using either ET. But that's not quite right, you need
> > both. For example, prime 3 is -1 generator but also 11 generators
> in
> > 12-equal, and prime 3 is -1 generator but also 18 generators in
19-
> > equal. So you need to pick the one that's in common between the
two
> > ETs. Once you've done that, you can drop any reference to the ETs
> > themselves, and just optimize, rms or minimaxm, using the mapping
> > itself.
>
> Thanks. Could you give an example?

Hmm, I thought this *was* an example . . . you already know how to do
the optimization using the mapping between primes and generators,
right?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
> <perlich@a...>
> > > > wrote:
> > > > > The mapping from *this* period and
> > > > > generator to primes should be identical for both ETs, and
> this
> > > > > carries over to the '&'.
> > > > >
> > > > Interesting. So is rms or minimax applied, to say, 5/12 and
8/19
> > > > for 12&19 together?
> > > >
> > > > Paul
> > >
> > > Well, what I was thinking was that, once you've found the
period
> > and
> > > generator, you can determine the mapping between primes and
> > > generators using either ET. But that's not quite right, you
need
> > > both. For example, prime 3 is -1 generator but also 11
generators
> > in
> > > 12-equal, and prime 3 is -1 generator but also 18 generators in
> 19-
> > > equal. So you need to pick the one that's in common between the
> two
> > > ETs. Once you've done that, you can drop any reference to the
ETs
> > > themselves, and just optimize, rms or minimaxm, using the
mapping
> > > itself.
> >
> > Thanks. Could you give an example?
>
> Hmm, I thought this *was* an example . . . you already know how to
do
> the optimization using the mapping between primes and generators,
> right?

Yes. I guess what I meant was, could you give me the pre-optimized
generator in this case. Thanx!

>Yes. I guess what I meant was, could you give me the pre-optimized
>generator in this case. Thanx!

Many examples are given here in the linear temperament database.

/tuning/database/

-Carl

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

> > Hmm, I thought this *was* an example . . . you already know how
to
> do
> > the optimization using the mapping between primes and generators,
> > right?
>
> Yes. I guess what I meant was, could you give me the pre-optimized
> generator in this case. Thanx!

I don't know what you mean by 'pre-optimized' generator! Sorry, I
really do want to help . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > > Hmm, I thought this *was* an example . . . you already know how
> to
> > do
> > > the optimization using the mapping between primes and
generators,
> > > right?
> >
> > Yes. I guess what I meant was, could you give me the pre-
optimized
> > generator in this case. Thanx!
>
> I don't know what you mean by 'pre-optimized' generator! Sorry, I
> really do want to help . . .

Hmm. Sorry, let me approach it this way. in 12&19, you have 5/12 for
the one and 8/19 for the other. How do you come up with one raw
generator-to-prime mapping. Is it x/31? And then is rms applied after
that?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > > Hmm, I thought this *was* an example . . . you already know
how
> > to
> > > do
> > > > the optimization using the mapping between primes and
> generators,
> > > > right?
> > >
> > > Yes. I guess what I meant was, could you give me the pre-
> optimized
> > > generator in this case. Thanx!
> >
> > I don't know what you mean by 'pre-optimized' generator! Sorry, I
> > really do want to help . . .
>
> Hmm. Sorry, let me approach it this way. in 12&19, you have 5/12 for
> the one and 8/19 for the other. How do you come up with one raw
> generator-to-prime mapping.

As I was saying, for each prime, use the mapping common to both 5/12
and 8/19, which is

prime 2 = 1 period
prime 3 = 2 periods - 1 generator
prime 5 = 4 periods - 4 generators

> And then is rms applied after
> that?

yes, you want to 'solve' the above system of equations for the
generator, to minimize your desired error function.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> As I was saying, for each prime, use the mapping common to both 5/12
> and 8/19, which is
>
> prime 2 = 1 period
> prime 3 = 2 periods - 1 generator
> prime 5 = 4 periods - 4 generators
>
> > And then is rms applied after
> > that?
>
> yes, you want to 'solve' the above system of equations for the
> generator, to minimize your desired error function.

Paul H,

I hope you've installed the optional Solver Add-in for Excel.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> >
> > Hmm. Sorry, let me approach it this way. in 12&19,
> > you have 5/12 for the one and 8/19 for the other.
> > How do you come up with one raw generator-to-prime mapping.
>
> As I was saying, for each prime, use the mapping
> common to both 5/12 and 8/19, which is
>
> prime 2 = 1 period
> prime 3 = 2 periods - 1 generator
> prime 5 = 4 periods - 4 generators

i'm not understanding a lot of this ... but i am curious
about this: why are you overshooting the prime with the
periods and then subtracting generators (instead of
coming as close under the prime as you can with the periods,
then adding generators)?

the latter is the way i've always thought of prime-mapping.
is there some special reason to do it "backwards" like this?

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > >
> > > > > Hmm, I thought this *was* an example . . . you already know
> how
> > > to
> > > > do
> > > > > the optimization using the mapping between primes and
> > generators,
> > > > > right?
> > > >
> > > > Yes. I guess what I meant was, could you give me the pre-
> > optimized
> > > > generator in this case. Thanx!
> > >
> > > I don't know what you mean by 'pre-optimized' generator! Sorry,
I
> > > really do want to help . . .
> >
> > Hmm. Sorry, let me approach it this way. in 12&19, you have 5/12
for
> > the one and 8/19 for the other. How do you come up with one raw
> > generator-to-prime mapping.
>
> As I was saying, for each prime, use the mapping common to both
5/12
> and 8/19, which is
>
> prime 2 = 1 period
> prime 3 = 2 periods - 1 generator
> prime 5 = 4 periods - 4 generators
>
> > And then is rms applied after
> > that?
>
> yes, you want to 'solve' the above system of equations for the
> generator, to minimize your desired error function.

I see, finally. Now my triangle is complete, Generators - Commas -
Temperaments. Thanks

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > >
> > > Hmm. Sorry, let me approach it this way. in 12&19,
> > > you have 5/12 for the one and 8/19 for the other.
> > > How do you come up with one raw generator-to-prime mapping.
> >
> > As I was saying, for each prime, use the mapping
> > common to both 5/12 and 8/19, which is
> >
> > prime 2 = 1 period
> > prime 3 = 2 periods - 1 generator
> > prime 5 = 4 periods - 4 generators
>
>
> i'm not understanding a lot of this ...

This should be very familiar territory for you -- it's just meantone
we're talking about here.

> but i am curious
> about this: why are you overshooting the prime with the
> periods and then subtracting generators (instead of
> coming as close under the prime as you can with the periods,
> then adding generators)?

Because that's how meantone approximates the primes.

> the latter is the way i've always thought of prime-mapping.
> is there some special reason to do it "backwards" like this?

Monz, sometimes you have to use minus signs. This is one of those
cases. You might think that all you have to do is restate the
generator to be the fifth (7/12 oct. or 11/19 oct.) instead of the
fourth, and then you won't have any minus signs. But that wouldn't
help in other cases, for example schismic:

If we state the generator of schismic as the fourth (~498.3 cents),
we have

prime 2 = 1 period
prime 3 = 2 periods - 1 generator
prime 5 = -1 period + 8 generators

If we state the generator of schismic as the fifth (~701.7 cents), we
have

prime 2 = 1 period
prime 3 = 1 periods + 1 generator
prime 5 = 7 periods - 8 generators

So either way, we have minus signs to contend with. In general, they
will be needed, and one should not be afraid of them.

It's fairly conventional around here to state the generator in
smallest possible (cents) terms.

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

> I see, finally. Now my triangle is complete, Generators - Commas -
> Temperaments. Thanks

Now *I* need to figure out how to get generators from commas!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Now *I* need to figure out how to get generators from commas!

My personal approach is to turn everything into wedgies, and then
derive everything *from* wedgies.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Now *I* need to figure out how to get generators from commas!
>
> My personal approach is to turn everything into wedgies, and then
> derive everything *from* wedgies.

if there's only one comma, is that the wedgie?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > I see, finally. Now my triangle is complete, Generators - Commas -
> > Temperaments. Thanks
>
> Now *I* need to figure out how to get generators from commas!

Graham Breed has a website that shows how to do this with matrices.
(I also want to learnt the wedgie way...) It's pretty cool ...

http://microtonal.co.uk/lintemp.htm

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > I see, finally. Now my triangle is complete, Generators -
Commas -
> > > Temperaments. Thanks
> >
> > Now *I* need to figure out how to get generators from commas!
>
> Graham Breed has a website that shows how to do this with matrices.
> (I also want to learnt the wedgie way...) It's pretty cool ...
>
> http://microtonal.co.uk/lintemp.htm

Isn't he *assuming* that the fifth is the generator here? Sorry, i'm
having trouble following his reasoning . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > >
> > > > I see, finally. Now my triangle is complete, Generators -
> Commas -
> > > > Temperaments. Thanks
> > >
> > > Now *I* need to figure out how to get generators from commas!
> >
> > Graham Breed has a website that shows how to do this with
matrices.
> > (I also want to learnt the wedgie way...) It's pretty cool ...
> >
> > http://microtonal.co.uk/lintemp.htm
>
> Isn't he *assuming* that the fifth is the generator here? Sorry,
i'm
> having trouble following his reasoning . . .

But what I really want is a way of getting it through geometrical
understanding, yet without assuming a metric . . .

Paul Erlich wrote:

> Isn't he *assuming* that the fifth is the generator here? Sorry, i'm > having trouble following his reasoning . . .

Yes, but that page is out of date, and wrong anyway for octave equivalent vectors. But it's the best that's currently "published". I've explained the modern way several times on this list, but as you've obviously forgotten I'll try again.

You form a matrix with the octave (1 0 0 ...) at the top, then a chromatic unison vector (it doesn't matter which) and below them the commas. Take the adjoint (the inverse multiplied by the determinant). The left hand column is an equal temperament mapping -- describing the periodicity block corresponding to the chromatic UV. The gcd of the next column is the number of periods to the octave, and when you divide through by that GCD you get the generator mapping.

The wedge product version I pretty much showed in a recent bra/ket post.

Graham

> if there's only one comma, is that the wedgie?

Yes.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
>
> > Isn't he *assuming* that the fifth is the generator here? Sorry,
i'm
> > having trouble following his reasoning . . .
>
> Yes, but that page is out of date, and wrong anyway for octave
> equivalent vectors. But it's the best that's
currently "published".
> I've explained the modern way several times on this list, but as
you've
> obviously forgotten I'll try again.

If I don't have a direct understanding of how something works, I
won't retain it. Sorry.

> You form a matrix with the octave (1 0 0 ...) at the top, then a
> chromatic unison vector (it doesn't matter which)

Is this one of those cases where you're saying chromatic unison
vector but don't really mean it?

Anyway, thanks, and I hope you'll update your pages.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Now *I* need to figure out how to get generators from commas!
>
> My personal approach is to turn everything into wedgies, and then
> derive everything *from* wedgies.

OK. Now it's time for me to wrestle with the term "wedgie". It this a
synonym for "multivector" (both contravariant and covariant)?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > Now *I* need to figure out how to get generators from commas!
> >
> > My personal approach is to turn everything into wedgies, and then
> > derive everything *from* wedgies.
>
> if there's only one comma, is that the wedgie?

It is by my usage; if you were going to treat the 5-limit the same as
all the other limits you'd need to take the complement and make the
first nonzero coefficent positive. I see no point in that excess of
consistency.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> OK. Now it's time for me to wrestle with the term "wedgie". It this
a
> synonym for "multivector" (both contravariant and covariant)?

No; these correspond to regular temperaments. An (m-1)-D regular
temperament is determined by m vals. Take the wedge product of the
vals and divide through by the gcd of the coefficients. If the first
nonzero coefficient is negative, change signs. The result is the
wedgie. You may also start from (n-m) commas, which you then wedge
and take the complement, followed by the above procedure.

The wedgie for a regular temperament is unique, and even if their are
torsion problems in the vals or commas you start from there are none
in the wedgie they define.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:

> You form a matrix with the octave (1 0 0 ...) at the top, then a
> chromatic unison vector (it doesn't matter which) and below them
the
> commas. Take the adjoint (the inverse multiplied by the
determinant).

This proceedure only works if you have a linear temperament.
Something else you might try is finding a basis for the nullspace of
the matrix formed from the commas alone, without your additions, and
using this to obtain a reduced set of vals (which could involve some
extra work.) From there, one can put the vals in the form you like; I
am partial to Hermite reduction unless we are dealing with linear
temperaments, in which case we do period-generator and make the the
generator as small, greater than one, as possible, to get a standard
reduced form.

Sometimes it suffices to simply find all standard vals which make all
of the commas zero and use this to start with. Finding the wedgie
from the commas, and the matrix from the wedgie, will also work; that
is how I would do this but I use Maple's Hermite reduction function
for it.

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> You form a matrix with the octave (1 0 0 ...) at the top, then a
> chromatic unison vector (it doesn't matter which) and below them
the
> commas. Take the adjoint (the inverse multiplied by the
determinant).
> The left hand column is an equal temperament mapping -- describing
the
> periodicity block corresponding to the chromatic UV. The gcd of
the
> next column is the number of periods to the octave, and when you
divide
> through by that GCD you get the generator mapping.
>
> The wedge product version I pretty much showed in a recent bra/ket
post.
>
>
> Graham

I've been fiddling with this. It's great for finding the equal
temperament mapping, number of periods per octave and generator
mapping. What's missing is the period mapping. For example, I got
a matrix with two periods per octave and 2, -4, -16, 22 in the second
column. Dividing by the gcd=2 results in 1, -2, -8, 11. This requires
a period mapping of (3,5,7,5). Is there anyway to calculate this,
or do you just have to do it by hand?
Thanks

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <graham@m...>
wrote:
> > You form a matrix with the octave (1 0 0 ...) at the top, then a
> > chromatic unison vector (it doesn't matter which) and below them
> the
> > commas. Take the adjoint (the inverse multiplied by the
> determinant).
> > The left hand column is an equal temperament mapping --
describing
> the
> > periodicity block corresponding to the chromatic UV. The gcd of
> the
> > next column is the number of periods to the octave, and when you
> divide
> > through by that GCD you get the generator mapping.
> >
> > The wedge product version I pretty much showed in a recent
bra/ket
> post.
> >
> >
> > Graham
>
> I've been fiddling with this. It's great for finding the equal
> temperament mapping, number of periods per octave and generator
> mapping. What's missing is the period mapping. For example, I got
> a matrix with two periods per octave and 2, -4, -16, 22 in the
second
> column. Dividing by the gcd=2 results in 1, -2, -8, 11. This
requires
> a period mapping of (3,5,7,5). Is there anyway to calculate this,
> or do you just have to do it by hand?
> Thanks
Also I should add that I can only get this to work right if I skip
the "chromatic unison vector" and go straight to the commas
i.e. 1 0 0 0 0 and then 4 commas or 1 0 0 0 and 3 commas

Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > At 03:11 PM 11/21/2003, you wrote:
> > >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >
> > >> < 12 19 28 |
> > >>
> > >> is h12 and
> > >>
> > >> < 19 30 44 |
> > >>
> > >> is h19. Except there's something about using the transpose of
> > >> one of them to get it into a form where the cross product will
> > >> give you a monzo. Which in this case is
> > >>
> > >> | -4 4 -1 > = 81/80
> > >>
> > >> Do I have that right, guys?
> > >
> > >~(<12 19 28| ^ <19 30 44|) = |-4 4 -1>
> >
> > ^ is the wedge product.
>
> yes.
>
> > ~ is ? Complement?
>
> yes.
>
> > So the wedging with
> > a complement is the same as crossing?
>
> no, look at the parentheses. the complement of the wedge product is
> the cross product (when you're dealing with a 3 dimensional
problem).
>
> > Please answer each question,
> > I'm just guessing.
> >
> > I still don't know a simple procedure to calculate a wedge
product.
>
> Graham just explained that.

The cross product appears more like the wedge product of the
complement! (As opposed to the complement of the wedge product.
Could someone kindly show an example?

Sorry, never mind. Got my mental wires crossed (again)

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > At 03:11 PM 11/21/2003, you wrote:
> > > >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...>
wrote:
> > > >
> > > >> < 12 19 28 |
> > > >>
> > > >> is h12 and
> > > >>
> > > >> < 19 30 44 |
> > > >>
> > > >> is h19. Except there's something about using the transpose
of
> > > >> one of them to get it into a form where the cross product
will
> > > >> give you a monzo. Which in this case is
> > > >>
> > > >> | -4 4 -1 > = 81/80
> > > >>
> > > >> Do I have that right, guys?
> > > >
> > > >~(<12 19 28| ^ <19 30 44|) = |-4 4 -1>
> > >
> > > ^ is the wedge product.
> >
> > yes.
> >
> > > ~ is ? Complement?
> >
> > yes.
> >
> > > So the wedging with
> > > a complement is the same as crossing?
> >
> > no, look at the parentheses. the complement of the wedge product
is
> > the cross product (when you're dealing with a 3 dimensional
> problem).
> >
> > > Please answer each question,
> > > I'm just guessing.
> > >
> > > I still don't know a simple procedure to calculate a wedge
> product.
> >
> > Graham just explained that.
>
> The cross product appears more like the wedge product of the
> complement! (As opposed to the complement of the wedge product.
> Could someone kindly show an example?