Questions for Carl

Carl,

I was wondering if you could bring me up to speed regarding some
gaps I have in my understanding of things that are posted on this
list.

1. I notice that (sometimes)generators-to-primes "line up" with
temperament values (like 12 19 28) when you invert a matrix of
commas, and then multiply by the determinant. Is there a rule
for this?

2. I still am not clear on how period values are calculated. (Using
matrices). Using wedge products I see how they are calculated
(from the wedgie) but I still am having trouble convincing myself
why wedging period ^ generators leads to the same wedgie as you
would obtain from monzo ^ monzo or value ^ value. (I know that
monzo wedgie is backwards from value wedgie, etc)

3. I get how periods may be part of an octave, when gcd(values)
is not 1. Once again, is there a rule where these generators
and periods "line up" with temperament values?

4. Geometry. I've got some questions... I'll discuss in the
relevant post

Thanks! Anyone else, feel free to chime in...

Paul Hj

>I was wondering if you could bring me up to speed regarding some
>gaps I have in my understanding of things that are posted on this
>list.

You may have me confused with someone who understands the things
posted on this list. :)

>1. I notice that (sometimes)generators-to-primes "line up" with
>temperament values (like 12 19 28) when you invert a matrix of
>commas, and then multiply by the determinant. Is there a rule
>for this?

This sounds vaguely like Paul E.'s procedure to find the number
of notes in a periodicity block, which would correspond to the
number of pitches in an ET based on those commas (barring torsion).

In general, there are recurrence relations that give the number
of pitches in 'good' temperaments. Some of them can be picked
off the Stern-Brocot tree. Gene knows more about this.

>2. I still am not clear on how period values are calculated. (Using
>matrices). Using wedge products I see how they are calculated
>(from the wedgie) but I still am having trouble convincing myself
>why wedging period ^ generators leads to the same wedgie as you
>would obtain from monzo ^ monzo or value ^ value. (I know that
>monzo wedgie is backwards from value wedgie, etc)

I have yet to understand the techniques based on wedge products.
Maybe you can help explain them to me once you've mastered them!

>3. I get how periods may be part of an octave, when gcd(values)
>is not 1. Once again, is there a rule where these generators
>and periods "line up" with temperament values?

This sounds like your first question. What exactly do you mean
by "temperament values"? And "line up"?

-Carl

Paul G Hjelmstad wrote:

> 1. I notice that (sometimes)generators-to-primes "line up" with
> temperament values (like 12 19 28) when you invert a matrix of
> commas, and then multiply by the determinant. Is there a rule
> for this? "Invert a matrix ... and then multiply by the determinant" gives the adjoint. Once you know the word, it's easier to use it, because the inverse as such isn't that interesting.

Yes, the column of the adjoint corresponding to the row of the original matrix that represents the octave gives you a representative ET mapping/val/constant structure. If you want to temper out all the commas, that's the equal temperament you were looking for.

If you don't temper out any columns, the constant structure refers to the periodicity block. Fokker only worked with octave-equivalent matrices, so his determinant only told him how many notes there were. The octave-specific method gives you the mappings for the other primes, and also helps you find and remove torsion.

If you temper out some but not all commas, this is one equal temperament that's a special case of whatever dimensioned temperament you end up with.

> 2. I still am not clear on how period values are calculated. (Using
> matrices). Using wedge products I see how they are calculated
> (from the wedgie) but I still am having trouble convincing myself
> why wedging period ^ generators leads to the same wedgie as you
> would obtain from monzo ^ monzo or value ^ value. (I know that
> monzo wedgie is backwards from value wedgie, etc)

Do you mean the whole column mapping intervals to periods? It's a fiddly calculation, and I'd need to check with my source code, which you have anyway. But do it once and you can forget about it.

It happens that the period and generator mappings are both vals. The generator mapping is special (and unique) in that it refers to an imaginary equal temperament with zero notes to the octave. It must be a val, because you get it from the adjoint of the matrix of commas. It's what you get when one of the "commas" you temper out is an octave. You could get it by repeatedly subtracting more sensible vals, because the difference between vals is always a val (even if it isn't an equal temperament).

For temperaments like mystery that divide the octave into a middling number of steps, it's more obvious the period mapping is a val as well.

> 3. I get how periods may be part of an octave, when gcd(values)
> is not 1. Once again, is there a rule where these generators
> and periods "line up" with temperament values?

I don't see what you mean by this.

> 4. Geometry. I've got some questions... I'll discuss in the > relevant post
> > Thanks! Anyone else, feel free to chime in...

Yes, well, I've done so. I don't know if you've given up on me yet.

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >I was wondering if you could bring me up to speed regarding some
> >gaps I have in my understanding of things that are posted on this
> >list.
>
> You may have me confused with someone who understands the things
> posted on this list. :)
>
> >1. I notice that (sometimes)generators-to-primes "line up" with
> >temperament values (like 12 19 28) when you invert a matrix of
> >commas, and then multiply by the determinant. Is there a rule
> >for this?
>
> This sounds vaguely like Paul E.'s procedure to find the number
> of notes in a periodicity block, which would correspond to the
> number of pitches in an ET based on those commas (barring torsion).
>
> In general, there are recurrence relations that give the number
> of pitches in 'good' temperaments. Some of them can be picked
> off the Stern-Brocot tree. Gene knows more about this.
>
> >2. I still am not clear on how period values are calculated. (Using
> >matrices). Using wedge products I see how they are calculated
> >(from the wedgie) but I still am having trouble convincing myself
> >why wedging period ^ generators leads to the same wedgie as you
> >would obtain from monzo ^ monzo or value ^ value. (I know that
> >monzo wedgie is backwards from value wedgie, etc)
>
> I have yet to understand the techniques based on wedge products.
> Maybe you can help explain them to me once you've mastered them!
>
> >3. I get how periods may be part of an octave, when gcd(values)
> >is not 1. Once again, is there a rule where these generators
> >and periods "line up" with temperament values?
>
> This sounds like your first question. What exactly do you mean
> by "temperament values"? And "line up"?
>
> -Carl

Thanks. I didn't word these questions that well. What I meant about
generators "lining up with temperament values" is the case where
a generator IS a step in the given temperament. (Perhaps 6 steps
out of 31-et, for example). I'll look into the Stern-Brocot tree.

- Paul Hjelmstad

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Paul G Hjelmstad wrote:
>
> > 1. I notice that (sometimes)generators-to-primes "line up" with
> > temperament values (like 12 19 28) when you invert a matrix of
> > commas, and then multiply by the determinant. Is there a rule
> > for this?
>
> "Invert a matrix ... and then multiply by the determinant" gives
the
> adjoint. Once you know the word, it's easier to use it, because
the
> inverse as such isn't that interesting.
>
> Yes, the column of the adjoint corresponding to the row of the
original
> matrix that represents the octave gives you a representative ET
> mapping/val/constant structure. If you want to temper out all the
> commas, that's the equal temperament you were looking for.
>
> If you don't temper out any columns, the constant structure refers
to
> the periodicity block. Fokker only worked with octave-equivalent
> matrices, so his determinant only told him how many notes there
were.
> The octave-specific method gives you the mappings for the other
primes,
> and also helps you find and remove torsion.
>
> If you temper out some but not all commas, this is one equal
temperament
> that's a special case of whatever dimensioned temperament you end
up with.
>
> > 2. I still am not clear on how period values are calculated.
(Using
> > matrices). Using wedge products I see how they are calculated
> > (from the wedgie) but I still am having trouble convincing myself
> > why wedging period ^ generators leads to the same wedgie as you
> > would obtain from monzo ^ monzo or value ^ value. (I know that
> > monzo wedgie is backwards from value wedgie, etc)
>
> Do you mean the whole column mapping intervals to periods? It's a
> fiddly calculation, and I'd need to check with my source code,
which you
> have anyway. But do it once and you can forget about it.
>
> It happens that the period and generator mappings are both vals.
The
> generator mapping is special (and unique) in that it refers to an
> imaginary equal temperament with zero notes to the octave. It must
be a
> val, because you get it from the adjoint of the matrix of commas.
It's
> what you get when one of the "commas" you temper out is an octave.
You
> could get it by repeatedly subtracting more sensible vals, because
the
> difference between vals is always a val (even if it isn't an equal
> temperament).
>
> For temperaments like mystery that divide the octave into a
middling
> number of steps, it's more obvious the period mapping is a val as
well.
>
> > 3. I get how periods may be part of an octave, when gcd(values)
> > is not 1. Once again, is there a rule where these generators
> > and periods "line up" with temperament values?
>
> I don't see what you mean by this.
>
> > 4. Geometry. I've got some questions... I'll discuss in the
> > relevant post
> >
> > Thanks! Anyone else, feel free to chime in...
>
> Yes, well, I've done so. I don't know if you've given up on me yet.
>
>
> Graham

Thanks. (Forget about 3. Either generators and periods line up, or
they don't I guess. Anyway, once you RMS or minimax them, generators
don't line up any more)

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

> Yes, the column of the adjoint corresponding to the row of the original
> matrix that represents the octave gives you a representative ET
> mapping/val/constant structure. If you want to temper out all the
> commas, that's the equal temperament you were looking for.

If the monzo matrix is unimodular, ie has determinant +-1, then all of
the columns correspond to vals; if the monzos are comma monzos, then
the vals will be +- equal temperament vals.

>Thanks. I didn't word these questions that well. What I meant about
>generators "lining up with temperament values" is the case where
>a generator IS a step in the given temperament. (Perhaps 6 steps
>out of 31-et, for example). I'll look into the Stern-Brocot tree.

Generator representations aren't unique. In 31-et, any interval
will generate the tuning since 31 is prime.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Thanks. I didn't word these questions that well. What I meant about
> >generators "lining up with temperament values" is the case where
> >a generator IS a step in the given temperament. (Perhaps 6 steps
> >out of 31-et, for example). I'll look into the Stern-Brocot tree.
>
> Generator representations aren't unique. In 31-et, any interval
> will generate the tuning since 31 is prime.
>
> -Carl

Only if you assume that everything is cyclic around the octave.
Gene's math never assumes that, so the only generator for 31-equal is
(plus or minus) 1/31 of an octave. Possibly 1/31 of a tempered octave.

But I don't think this is pertinent to the passage you were quoting.

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

> Only if you assume that everything is cyclic around the octave.
> Gene's math never assumes that, so the only generator for 31-equal is
> (plus or minus) 1/31 of an octave. Possibly 1/31 of a tempered octave.
>
> But I don't think this is pertinent to the passage you were quoting.

To be even more impertinent, this made me consider the [1,1/31] linear
temperament systems. In the five limit, |46 -10 -13>, in the seven
limit <13 -10 6 -46 -27 42| with TM commas 225/224 and 589824/588245.
TOP generators 1200.45 and 38.44 in the five limit, and 1200.34 and
38.43 in the seven limit. In the eleven limit, two systems have some
plausibilty, but clearly the one to choose is the one with mapping
[<1 2 2 3 4|, <0 -13 10 -6 -17|] and TM commas {225/224, 385/384,
1331/1323}, giving 1200.38 and 38.39 as generators. This is more
interesting that one might suppose, since the improvement in the
tuning over 31 equal or TOP 31 is considerable.