The same page

Just to be sure we are on it, in terms of defintions of compexity and
error, here is my page.

5-limit, comma = n/d

Complexity is log2(n*d), so log(complexity) is loglog(n*d). Error is
distance from the TOP tuning to the JIP, or in other words the max of
the absolute values of the errors for 2, 3 and 5 in TOP tuning,
divided by log2(2), log2(3) and log2(5) respectively. Log(error) is
the log of this. Loglog plots compare loglog(n*d) with log(error).

7-limit planar, comma = n/d

Complexity is log2(n*d) (this is always the TOP complexity in
codimension one) and error is again the distance from the JIP for the
7-limit planar TOP tuning for n/d. Log(error) is log of this error.

7-limit linear

Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6|| is
the wedgie, then complexity is
|a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|. Log complexity
is log of this. Error is the distance from the JIP of the 7-limit TOP
tuning for the temperament; log(complexity) and log(error) are logs
of complexity and error, so defined.

> Complexity is the Erlich magic L1 norm; if
> <<a1 a2 a3 a4 a5 a6|| is the wedgie, then complexity is
> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|.
> Log complexity is log of this. Error is the distance from
> the JIP of the 7-limit TOP tuning for the temperament; log
> (complexity) and log(error) are logs of complexity and
> error, so defined.

What are p3, p5, etc.?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Just to be sure we are on it, in terms of defintions of compexity
and
> error, here is my page.

Where are ETs?

> 5-limit, comma = n/d
>
> Complexity is log2(n*d),

Yes, but this can also be expressed in other ways, for example if

<<a1 a2 a3||

is the val-wedgie (dual to the comma), it can be expressed as

|a1/p3| + |a2/p5| + |a3/p2p5|

> so log(complexity) is loglog(n*d). Error is
> distance from the TOP tuning to the JIP, or in other words the max
of
> the absolute values of the errors for 2, 3 and 5 in TOP tuning,
> divided by log2(2), log2(3) and log2(5) respectively.

It also can be expressed as log(n/d)/log(n*d) (*1200).

> Log(error) is
> the log of this. Loglog plots compare loglog(n*d) with log(error).

i.e., log(log(n*d)) with log(log(n/d)/log(n*d)).
>
> 7-limit linear
>
> Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6|| is
> the wedgie,

val-wedgie, yes.

> then complexity is
> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|.

Note the similarity. It really should be one formula for all cases.

> Error is the distance from the JIP of the 7-limit TOP
> tuning for the temperament;

Or same as 5-limit linear error but with an additional term for 7

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > Complexity is the Erlich magic L1 norm; if
> > <<a1 a2 a3 a4 a5 a6|| is the wedgie, then complexity is
> > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|.
> > Log complexity is log of this. Error is the distance from
> > the JIP of the 7-limit TOP tuning for the temperament; log
> > (complexity) and log(error) are logs of complexity and
> > error, so defined.
>
> What are p3, p5, etc.?
>
> -Carl

log2(3), log2(5), etc.

Sorry Gene, I thought I was responding to Carl, otherwise I wouldn't
have posted all that math that must be obvious to you.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > Just to be sure we are on it, in terms of defintions of compexity
> and
> > error, here is my page.
>
> Where are ETs?
>
> > 5-limit, comma = n/d
> >
> > Complexity is log2(n*d),
>
> Yes, but this can also be expressed in other ways, for example if
>
> <<a1 a2 a3||
>
> is the val-wedgie (dual to the comma), it can be expressed as
>
> |a1/p3| + |a2/p5| + |a3/p2p5|
>
> > so log(complexity) is loglog(n*d). Error is
> > distance from the TOP tuning to the JIP, or in other words the
max
> of
> > the absolute values of the errors for 2, 3 and 5 in TOP tuning,
> > divided by log2(2), log2(3) and log2(5) respectively.
>
> It also can be expressed as log(n/d)/log(n*d) (*1200).
>
> > Log(error) is
> > the log of this. Loglog plots compare loglog(n*d) with log(error).
>
> i.e., log(log(n*d)) with log(log(n/d)/log(n*d)).
> >
> > 7-limit linear
> >
> > Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6||
is
> > the wedgie,
>
> val-wedgie, yes.
>
> > then complexity is
> > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|.
>
> Note the similarity. It really should be one formula for all cases.
>
> > Error is the distance from the JIP of the 7-limit TOP
> > tuning for the temperament;
>
> Or same as 5-limit linear error but with an additional term for 7

>log2(3), log2(5), etc.

Thanks. I think that's the same as Gene was using before, then.

-Carl

>> > 5-limit, comma = n/d
>> >
>> > Complexity is log2(n*d),
>>
>> Yes, but this can also be expressed in other ways, for example if
>>
>> <<a1 a2 a3||
>>
>> is the val-wedgie (dual to the comma),

I thought val ^ val -> comma, so val ^ val must not be a val-wedgie.
What's a val-wedgie?

Anybody have a handy asci 'units' table for popular wedge products
in ket notation? ie,

[ val > ^ [ val > -> [[ wedgie >>
< monzo ] ^ < monzo ] -> ?

...etc.

>> > Error is the distance from the JIP of the 7-limit TOP
>> > tuning for the temperament;
>>
>> Or same as 5-limit linear error but with an additional term for 7.

What's linear error?

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > Just to be sure we are on it, in terms of defintions of compexity
> and
> > error, here is my page.
>
> Where are ETs?

Forgot 'em, but you seem to have them figured out. Modulo some slight
fiddling if you must fiddle, complexity is n for the n-et, so log
complexity is log(n).

> > so log(complexity) is loglog(n*d). Error is
> > distance from the TOP tuning to the JIP, or in other words the
max
> of
> > the absolute values of the errors for 2, 3 and 5 in TOP tuning,
> > divided by log2(2), log2(3) and log2(5) respectively.
>
> It also can be expressed as log(n/d)/log(n*d) (*1200).

How can either log(n*d) or loglog(n*d) also be expressed as
epimericity, which this is very close to being?

>
> > Log(error) is
> > the log of this. Loglog plots compare loglog(n*d) with log(error).
>
> i.e., log(log(n*d)) with log(log(n/d)/log(n*d)).
> >
> > 7-limit linear
> >
> > Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5 a6||
is
> > the wedgie,
>
> val-wedgie, yes.

That's how "wedgie" is defined.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> > 5-limit, comma = n/d
> >> >
> >> > Complexity is log2(n*d),
> >>
> >> Yes, but this can also be expressed in other ways, for example if
> >>
> >> <<a1 a2 a3||
> >>
> >> is the val-wedgie (dual to the comma),
>
> I thought val ^ val -> comma,

No, it's dual to the comma.

> so val ^ val must not be a val-wedgie.

Yes, I meant val ^ val.

> What's a val-wedgie?
>
> Anybody have a handy asci 'units' table for popular wedge products
> in ket notation? ie,
>
> [ val > ^ [ val > -> [[ wedgie >>
> < monzo ] ^ < monzo ] -> ?

<val] ^ <val] -> <<bival||
[monzo> ^ [monzo> -> ||bimonzo>>

In 3D (e.g., 5-limit), for linear temperaments the bival is dual to
the monzo, and for equal temperaments the bimonzo is dual to the val.

In 4D (e.g., 7-limit), for linear temperaments the bival is dual to
the bimonzo, and both are referred to as the "wedgie" (though Gene
uses the bival form).

> ...etc.
>
> >> > Error is the distance from the JIP of the 7-limit TOP
> >> > tuning for the temperament;
> >>
> >> Or same as 5-limit linear error but with an additional term for
7.
>
> What's linear error?

No -- (5-limit linear) error. See the original message.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > Just to be sure we are on it, in terms of defintions of
compexity
> > and
> > > error, here is my page.
> >
> > Where are ETs?
>
> Forgot 'em, but you seem to have them figured out. Modulo some
slight
> fiddling if you must fiddle,

I'd like to understand this slight fiddling, and apply this
understanding to the 7-limit linear case (and elsewhere).

> complexity is n for the n-et, so log
> complexity is log(n).
>
> > > so log(complexity) is loglog(n*d). Error is
> > > distance from the TOP tuning to the JIP, or in other words the
> max
> > of
> > > the absolute values of the errors for 2, 3 and 5 in TOP tuning,
> > > divided by log2(2), log2(3) and log2(5) respectively.
> >
> > It also can be expressed as log(n/d)/log(n*d) (*1200).
>
> How can either log(n*d) or loglog(n*d) also be expressed as
> epimericity, which this is very close to being?

Not following.

> > > Log(error) is
> > > the log of this. Loglog plots compare loglog(n*d) with log
(error).
> >
> > i.e., log(log(n*d)) with log(log(n/d)/log(n*d)).
> > >
> > > 7-limit linear
> > >
> > > Complexity is the Erlich magic L1 norm; if <<a1 a2 a3 a4 a5
a6||
> is
> > > the wedgie,
> >
> > val-wedgie, yes.
>
> That's how "wedgie" is defined.

They're merely duals of one another, but why this definition? I can
understand taking the wedge product of monzos much better than I
understand the wedge product of vals.

>> Anybody have a handy asci 'units' table for popular wedge products
>> in ket notation? ie,
>>
>> [ val > ^ [ val > -> [[ wedgie >>
>> < monzo ] ^ < monzo ] -> ?
>
><val] ^ <val] -> <<bival||
>[monzo> ^ [monzo> -> ||bimonzo>>

Great, so what happens when the monzos are commas being
tempered out?

A chart running over comma useful things would help our
endeavor tremendously.

>In 3D (e.g., 5-limit), for linear temperaments the bival is dual to
>the monzo, and for equal temperaments the bimonzo is dual to the val.
>
>In 4D (e.g., 7-limit), for linear temperaments the bival is dual to
>the bimonzo, and both are referred to as the "wedgie" (though Gene
>uses the bival form).

Ok great. But what's all about this algebraic dual? Is this
something I can do to matrices, like complement and transpose?
Again, a chart of these things would be awesome.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Anybody have a handy asci 'units' table for popular wedge
products
> >> in ket notation? ie,
> >>
> >> [ val > ^ [ val > -> [[ wedgie >>
> >> < monzo ] ^ < monzo ] -> ?
> >
> ><val] ^ <val] -> <<bival||
> >[monzo> ^ [monzo> -> ||bimonzo>>
>
> Great, so what happens when the monzos are commas being
> tempered out?

That's what they always represent here.

> A chart running over comma useful things would help our
> endeavor tremendously.

What would you like to see?

> >In 3D (e.g., 5-limit), for linear temperaments the bival is dual
to
> >the monzo, and for equal temperaments the bimonzo is dual to the
val.
> >
> >In 4D (e.g., 7-limit), for linear temperaments the bival is dual
to
> >the bimonzo, and both are referred to as the "wedgie" (though Gene
> >uses the bival form).
>
> Ok great. But what's all about this algebraic dual? Is this
> something I can do to matrices, like complement and transpose?

The wedgie is a sort of vector, not a matrix. The dual involves
reversing the order of the entries, and flipping some of the signs.
There was an extensive thread here recently explaining this.

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

> > Forgot 'em, but you seem to have them figured out. Modulo some
> slight
> > fiddling if you must fiddle,
>
> I'd like to understand this slight fiddling, and apply this
> understanding to the 7-limit linear case (and elsewhere).

You can take the val and simply choose the first number in it, the
number of steps in an octave. Or, you can normalize it by
1/log2(prime), and take the maximum. Or, you can TOP tune it,
normalize that, and take the maximum.

> > That's how "wedgie" is defined.
>
> They're merely duals of one another, but why this definition?

It's nicer than the monzo version, as the first part of it gives you
the period and the mapping to primes of the generator, and the signs
of the mapping are not screwed around with.

I can
> understand taking the wedge product of monzos much better than I
> understand the wedge product of vals.

See above. Vals is clearly the correct choice.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > Forgot 'em, but you seem to have them figured out. Modulo some
> > slight
> > > fiddling if you must fiddle,
> >
> > I'd like to understand this slight fiddling, and apply this
> > understanding to the 7-limit linear case (and elsewhere).
>
> You can take the val and simply choose the first number in it, the
> number of steps in an octave. Or, you can normalize it by
> 1/log2(prime), and take the maximum. Or, you can TOP tune it,
> normalize that, and take the maximum.

Or you can take the sum. What I'm trying to get at is what these
*mean*, beginning with the 3-limit cases in the "Attn: Gene 2" post.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Anybody have a handy asci 'units' table for popular wedge products
> >> in ket notation? ie,
> >>
> >> [ val > ^ [ val > -> [[ wedgie >>

<val| ^ <val| = || wedgie >>

> >In 4D (e.g., 7-limit), for linear temperaments the bival is dual to
> >the bimonzo, and both are referred to as the "wedgie" (though Gene
> >uses the bival form).

Both are referred to as the "wedgie" by whom?

> Ok great. But what's all about this algebraic dual? Is this
> something I can do to matrices, like complement and transpose?

It's the complement.

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

> > >In 4D (e.g., 7-limit), for linear temperaments the bival is dual
to
> > >the bimonzo, and both are referred to as the "wedgie" (though
Gene
> > >uses the bival form).
>
> Both are referred to as the "wedgie" by whom?

For example, in the original post to Paul Hj. explaining Pascal's
triangle. Clearly there, when there's only one val involved, the
wedgie can only be a multimonzo, not a multival.

> > Ok great. But what's all about this algebraic dual? Is this
> > something I can do to matrices, like complement and transpose?
>
> It's the complement.

Oh yeah -- sorry!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > > >In 4D (e.g., 7-limit), for linear temperaments the bival is dual
> to
> > > >the bimonzo, and both are referred to as the "wedgie" (though
> Gene
> > > >uses the bival form).
> >
> > Both are referred to as the "wedgie" by whom?
>
> For example, in the original post to Paul Hj. explaining Pascal's
> triangle. Clearly there, when there's only one val involved, the
> wedgie can only be a multimonzo, not a multival.

With one val, the wedgie by definition is that val. The only special
case I know is 5-limit linear temperaments, where using the comma as a
wedgie seems a better plan than sticking with the definition.

>> >> Anybody have a handy asci 'units' table for popular wedge
>> >> products in ket notation? ie,
>> >>
>> >> [ val > ^ [ val > -> [[ wedgie >>
>> >> < monzo ] ^ < monzo ] -> ?
>> >
>> ><val] ^ <val] -> <<bival||
>> >[monzo> ^ [monzo> -> ||bimonzo>>
>>
>> Great, so what happens when the monzos are commas being
>> tempered out?
>
>That's what they always represent here.

Yes of course, but in that case, what does the bimonzo give
us? Anything musical?

>> A chart running over comma useful things would help our
>> endeavor tremendously.
>
>What would you like to see?

A dummy chart for what I need to wedge in order to get what
I care about about temperaments.

>> >In 3D (e.g., 5-limit), for linear temperaments the bival is dual
>> >to the monzo, and for equal temperaments the bimonzo is dual to
>> >the val.
>> >
>> >In 4D (e.g., 7-limit), for linear temperaments the bival is dual
>> >to the bimonzo, and both are referred to as the "wedgie" (though
>> >Gene uses the bival form).
>>
>> Ok great. But what's all about this algebraic dual? Is this
>> something I can do to matrices, like complement and transpose?
>
>The wedgie is a sort of vector, not a matrix. The dual involves
>reversing the order of the entries, and flipping some of the signs.
>There was an extensive thread here recently explaining this.

Ok, so the form for that should be on the chart.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > > >In 4D (e.g., 7-limit), for linear temperaments the bival is
dual
> > to
> > > > >the bimonzo, and both are referred to as the "wedgie"
(though
> > Gene
> > > > >uses the bival form).
> > >
> > > Both are referred to as the "wedgie" by whom?
> >
> > For example, in the original post to Paul Hj. explaining Pascal's
> > triangle. Clearly there, when there's only one val involved, the
> > wedgie can only be a multimonzo, not a multival.
>
> With one val, the wedgie by definition is that val. The only special
> case I know is 5-limit linear temperaments, where using the comma
as a
> wedgie seems a better plan than sticking with the definition.

Why not admit both versions of the wedgie in all instances? They're
so similar it's hard to see why one would make a big deal out of it.
And personally, I have a far better intuitive grasp of monzo-wedgies
than val-wedgies, but since we only care about the absolute values,
converting from one to the other is trivial.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> Anybody have a handy asci 'units' table for popular wedge
> >> >> products in ket notation? ie,
> >> >>
> >> >> [ val > ^ [ val > -> [[ wedgie >>
> >> >> < monzo ] ^ < monzo ] -> ?
> >> >
> >> ><val] ^ <val] -> <<bival||
> >> >[monzo> ^ [monzo> -> ||bimonzo>>
> >>
> >> Great, so what happens when the monzos are commas being
> >> tempered out?
> >
> >That's what they always represent here.
>
> Yes of course, but in that case, what does the bimonzo give
> us? Anything musical?

Sure; in the 5-limit it gives the periodicity block, and so on.

> >> A chart running over comma useful things would help our
> >> endeavor tremendously.
> >
> >What would you like to see?
>
> A dummy chart for what I need to wedge in order to get what
> I care about about temperaments.

Can I see an example of what you have in mind?

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

> Why not admit both versions of the wedgie in all instances?

The wedgie then no longer corresponds 1-1 with temperaments, as there
are two of them.

>> >> >> Anybody have a handy asci 'units' table for popular wedge
>> >> >> products in ket notation? ie,
>> >> >>
>> >> >> [ val > ^ [ val > -> [[ wedgie >>
>> >> >> < monzo ] ^ < monzo ] -> ?
>> >> >
>> >> ><val] ^ <val] -> <<bival||
>> >> >[monzo> ^ [monzo> -> ||bimonzo>>
>> >>
>> >> Great, so what happens when the monzos are commas being
>> >> tempered out?
>> >
>> >That's what they always represent here.
>>
>> Yes of course, but in that case, what does the bimonzo give
>> us? Anything musical?
>
>Sure; in the 5-limit it gives the periodicity block, and so on.
>
>> >> A chart running over comma useful things would help our
>> >> endeavor tremendously.
>> >
>> >What would you like to see?
>>
>> A dummy chart for what I need to wedge in order to get what
>> I care about about temperaments.
>
>Can I see an example of what you have in mind?

Above! For all operations one would want to do. With templates
for dual and every other damn thing that can be done to a vector by
flipping signs, rearranging elements, and other trivial operations.
If I could do any better than this I'd make the thing myself!

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Why not admit both versions of the wedgie in all instances?
>
> The wedgie then no longer corresponds 1-1 with temperaments, as
there
> are two of them.

So the correspondence is 1-1-1. Why is that a problem?

--- 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:
> >
> > > Why not admit both versions of the wedgie in all instances?
> >
> > The wedgie then no longer corresponds 1-1 with temperaments, as
> there
> > are two of them.
>
> So the correspondence is 1-1-1. Why is that a problem?

If I say "here is a wedgie for a 7-limit temperament" you no longer
know which temperament.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> Anybody have a handy asci 'units' table for popular wedge
> >> >> >> products in ket notation? ie,
> >> >> >>
> >> >> >> [ val > ^ [ val > -> [[ wedgie >>
> >> >> >> < monzo ] ^ < monzo ] -> ?
> >> >> >
> >> >> ><val] ^ <val] -> <<bival||
> >> >> >[monzo> ^ [monzo> -> ||bimonzo>>
> >> >>
> >> >> Great, so what happens when the monzos are commas being
> >> >> tempered out?
> >> >
> >> >That's what they always represent here.
> >>
> >> Yes of course, but in that case, what does the bimonzo give
> >> us? Anything musical?
> >
> >Sure; in the 5-limit it gives the periodicity block, and so on.
> >
> >> >> A chart running over comma useful things would help our
> >> >> endeavor tremendously.
> >> >
> >> >What would you like to see?
> >>
> >> A dummy chart for what I need to wedge in order to get what
> >> I care about about temperaments.
> >
> >Can I see an example of what you have in mind?
>
> Above! For all operations one would want to do. With templates
> for dual and every other damn thing that can be done to a vector by
> flipping signs, rearranging elements, and other trivial operations.
> If I could do any better than this I'd make the thing myself!

Is this a start? ~= will mean "equal when one side is complemented".

2 primes:

<val] ~= [monzo>

3 primes:

()ET:

[monzo> /\ [monzo> ~= <val]

()LT:

[monzo> ~= <val] /\ <val]

4 primes:

()ET:

[monzo> /\ [monzo> /\ [monzo> ~= <val]

()LT:

[monzo> /\ [monzo> ~= <val] /\ <val]

()PT:

[monzo> ~= <val} /\ <val] /\ <val]

Hopefully the pattern is clear.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- 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:
> > >
> > > > Why not admit both versions of the wedgie in all instances?
> > >
> > > The wedgie then no longer corresponds 1-1 with temperaments, as
> > there
> > > are two of them.
> >
> > So the correspondence is 1-1-1. Why is that a problem?
>
> If I say "here is a wedgie for a 7-limit temperament" you no longer
> know which temperament.

Yes you do, since we're using bra-ket notation . . .

>>> >> >> Anybody have a handy asci 'units' table for popular wedge
>>> >> >> products in ket notation? ie,
>>> >> >>
>>> >> >> [ val > ^ [ val > -> [[ wedgie >>
>>> >> >> < monzo ] ^ < monzo ] -> ?
>>> >> >
>>> >> ><val] ^ <val] -> <<bival||
>>> >> >[monzo> ^ [monzo> -> ||bimonzo>>
>>> >>
>>> >> Great, so what happens when the monzos are commas being
>>> >> tempered out?
>>> >
>>> >That's what they always represent here.
>>>
>>> Yes of course, but in that case, what does the bimonzo give
>>> us? Anything musical?
>>
>>Sure; in the 5-limit it gives the periodicity block, and so on.
>>
>>> >> A chart running over comma useful things would help our
>>> >> endeavor tremendously.
>>> >
>>> >What would you like to see?
>>>
>>> A dummy chart for what I need to wedge in order to get what
>>> I care about about temperaments.
>>
>>Can I see an example of what you have in mind?
>
>Above! For all operations one would want to do. With templates
>for dual and every other damn thing that can be done to a vector by
>flipping signs, rearranging elements, and other trivial operations.
>If I could do any better than this I'd make the thing myself!

Sorry to be so hasty here. Like any true addict, I was making
myself late for my interview. I managed to arrive on time, thanks
to favorable traffic conditions, and it went well.

I meant to add that scenarios for different dimensionalities
should be included. For example, do I need to wedge three vals
to get a 7-limit codimenision 1 temperament, or...? etc.

-Carl

>Is this a start?

Yes, great!!

> ~= will mean "equal when one side is complemented".
>
>2 primes:
>
><val] ~= [monzo>
>
>3 primes:
>
>()ET:
>[monzo> /\ [monzo> ~= <val]
>()LT:
>[monzo> ~= <val] /\ <val]
>
>4 primes:
>
>()ET:
>[monzo> /\ [monzo> /\ [monzo> ~= <val]
>()LT:
>[monzo> /\ [monzo> ~= <val] /\ <val]
>()PT:
>[monzo> ~= <val} /\ <val] /\ <val]
>
>Hopefully the pattern is clear.

I'm missing wedgies here. And maps. And dual/complement.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Is this a start?
>
> Yes, great!!
>
> > ~= will mean "equal when one side is complemented".
> >
> >2 primes:
> >
> ><val] ~= [monzo>
> >
> >3 primes:
> >
> >()ET:
> >[monzo> /\ [monzo> ~= <val]
> >()LT:
> >[monzo> ~= <val] /\ <val]
> >
> >4 primes:
> >
> >()ET:
> >[monzo> /\ [monzo> /\ [monzo> ~= <val]
> >()LT:
> >[monzo> /\ [monzo> ~= <val] /\ <val]
> >()PT:
> >[monzo> ~= <val} /\ <val] /\ <val]
> >
> >Hopefully the pattern is clear.
>
> I'm missing wedgies here. And maps. And dual/complement.

/\ is the wedgie, and ~= is the dual/complement.

>> > ~= will mean "equal when one side is complemented".
>> >
>> >2 primes:
>> >
>> ><val] ~= [monzo>
>> >
>> >3 primes:
>> >
>> >()ET:
>> >[monzo> /\ [monzo> ~= <val]
>> >()LT:
>> >[monzo> ~= <val] /\ <val]
>> >
>> >4 primes:
>> >
>> >()ET:
>> >[monzo> /\ [monzo> /\ [monzo> ~= <val]
>> >()LT:
>> >[monzo> /\ [monzo> ~= <val] /\ <val]
>> >()PT:
>> >[monzo> ~= <val} /\ <val] /\ <val]
>> >
>> >Hopefully the pattern is clear.
>>
>> I'm missing wedgies here. And maps. And dual/complement.
>
>/\ is the wedgie,

/\ is the wedge product. I mean, you're not showing calculations
where the inputs involve wedgies.

>and ~= is the dual/complement.

So sorry, I read "when one side is completed", or something.
That leaves me:

() What is the form form complement?

() Does dual = complement?

-Carl

>() What is the form form complement?

Form *for*, sorry. :) -C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> > ~= will mean "equal when one side is complemented".
> >> >
> >> >2 primes:
> >> >
> >> ><val] ~= [monzo>
> >> >
> >> >3 primes:
> >> >
> >> >()ET:
> >> >[monzo> /\ [monzo> ~= <val]
> >> >()LT:
> >> >[monzo> ~= <val] /\ <val]
> >> >
> >> >4 primes:
> >> >
> >> >()ET:
> >> >[monzo> /\ [monzo> /\ [monzo> ~= <val]
> >> >()LT:
> >> >[monzo> /\ [monzo> ~= <val] /\ <val]
> >> >()PT:
> >> >[monzo> ~= <val} /\ <val] /\ <val]
> >> >
> >> >Hopefully the pattern is clear.
> >>
> >> I'm missing wedgies here. And maps. And dual/complement.
> >
> >/\ is the wedgie,
>
> /\ is the wedge product. I mean, you're not showing calculations
> where the inputs involve wedgies.
>
> >and ~= is the dual/complement.
>
> So sorry, I read "when one side is completed", or something.
> That leaves me:
>
> () What is the form form complement?

Form-form complement? Never heard of it.

> () Does dual = complement?

I think so.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >() What is the form form complement?
>
> Form *for*, sorry. :) -C.

The complement reverses the order of the entries and changes some of
the signs. Dave Keenan and others made extensive posts here detailing
this. I haven't yet had to worry about the change of signs, as the
measures I've looked at so far take the absolute values anyway.

>> >() What is the form form complement?
>>
>> Form *for*, sorry. :) -C.
>
>The complement reverses the order of the entries and changes some of
>the signs. Dave Keenan and others made extensive posts here detailing
>this. I haven't yet had to worry about the change of signs, as the
>measures I've looked at so far take the absolute values anyway.

How am I ever going to find these posts of Dave's to get to

| a b c > ~= | -b c a >

or whatever?

And I'm still missing things to do with wedgies. For example,
your L1 complexity. Gene gives

>Erlich magic L1 norm; if
>
> <<a1 a2 a3 a4 a5 a6||
>
>is the wedgie, then complexity is
>
> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|

Where wedgie is val-wedgie. But apparently there's a monzo-wedgie
formualation...

>the L1 norm of the *monzo-wedgie* in the Tenney lattice.
>In other words, it's the 'taxicab area' of the (nontorsional)
>vanishing bivector, something which seems to give three times
>the number of notes in the 5-limit ET case.

C'mon guys, help me publish this stuff in one place. I'm
begging you.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >() What is the form form complement?
> >>
> >> Form *for*, sorry. :) -C.
> >
> >The complement reverses the order of the entries and changes some
of
> >the signs. Dave Keenan and others made extensive posts here
detailing
> >this. I haven't yet had to worry about the change of signs, as the
> >measures I've looked at so far take the absolute values anyway.
>
> How am I ever going to find these posts of Dave's to get to
>
> | a b c > ~= | -b c a >
>
> or whatever?

Shouldn't be too hard, it was a fairly recent discussion here.

> And I'm still missing things to do with wedgies. For example,
> your L1 complexity. Gene gives
>
> >Erlich magic L1 norm; if
> >
> > <<a1 a2 a3 a4 a5 a6||
> >
> >is the wedgie, then complexity is
> >
> > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|
>
> Where wedgie is val-wedgie. But apparently there's a monzo-wedgie
> formualation...

Simply reverse the order of the entries.

> >the L1 norm of the *monzo-wedgie* in the Tenney lattice.
> >In other words, it's the 'taxicab area' of the (nontorsional)
> >vanishing bivector, something which seems to give three times
> >the number of notes in the 5-limit ET case.
>
> C'mon guys, help me publish this stuff in one place. I'm
> begging you.

We should be thanking you. The 5-limit ET case (12 was used as the
example) was referred to a number of times; I'll see if I can locate
that post . . .

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

> We should be thanking you. The 5-limit ET case (12 was used as the
> example) was referred to a number of times; I'll see if I can
locate
> that post . . .

/tuning-math/message/9052

Apparently these weren't the kinds of questions Gene wants to try
answering . . .

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

> /tuning-math/message/9052
>
> Apparently these weren't the kinds of questions Gene wants to try
> answering . . .

I was planning on getting around to it. Is it urgent for some reason?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > /tuning-math/message/9052
> >
> > Apparently these weren't the kinds of questions Gene wants to try
> > answering . . .
>
> I was planning on getting around to it.

You were? Oh, sorry . . .

>Is it urgent for some
>reason?

Yes, I wanted know why the correspondence breaks down beyond the 3-
limit (I suppose that's clear enough mathematically, but . . .) and
what implications that has on how we should consider defining
complexity (and perhaps even error) beyond the 3-limit. Clearly the
exact choice here isn't going to have much impact on how our ET
graphs look and what we choose from them. But in general, my
understanding is not completely crystallized, so every little bit
helps . . .

>> >Erlich magic L1 norm; if
>> >
>> > <<a1 a2 a3 a4 a5 a6||
>> >
>> >is the wedgie, then complexity is
>> >
>> > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|
>>
>> Where wedgie is val-wedgie. But apparently there's a monzo-wedgie
>> formualation...
>
>Simply reverse the order of the entries.

Not sure what you're saying.

monzo-wedgie = reverse(val-wedgie)

or? Can you give the form?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Erlich magic L1 norm; if
> >> >
> >> > <<a1 a2 a3 a4 a5 a6||
> >> >
> >> >is the wedgie, then complexity is
> >> >
> >> > |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|
> >>
> >> Where wedgie is val-wedgie. But apparently there's a monzo-
wedgie
> >> formualation...
> >
> >Simply reverse the order of the entries.
>
> Not sure what you're saying.
>
> monzo-wedgie = reverse(val-wedgie)

Up to some of the signs, yes. Since the above expression for
complexity takes the absolute values anyway, you don't have to worry
about the signs. The point is that the complexity you end up
calculating is the same. We could write

If ||a6 a5 a4 a3 a2 a1>> is the monzo-wedgie, then the complexity is
|a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|

and we'd be getting the same answer as above.

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

> > I was planning on getting around to it.
>
> You were? Oh, sorry . . .

I guess I was according it about the same urgency you give to my
requests. :)

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> How am I ever going to find these posts of Dave's to get to
>
> | a b c > ~= | -b c a >
>
> or whatever?

Try:
/tuning-math/message/7852

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > How am I ever going to find these posts of Dave's to get to
> >
> > | a b c > ~= | -b c a >
> >
> > or whatever?
>
> Try:
> /tuning-math/message/7852

Thanks Dave, for looking in even after you've gone away!