Gene's mail server

Hi Gene,

Your mail server rejected my e-mail response to you as "probable
SPAM".

Assuming you disagree with your server's assessment, let me know how
I should proceed.

-Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> Hi Gene,
>
> Your mail server rejected my e-mail response to you as "probable
> SPAM".
>
> Assuming you disagree with your server's assessment, let me know how
> I should proceed.

Considering the huge quantity of spam which makes it through, this is
kind of depressing. Maybe upload it, or post it if it is postable. I
suppose I should get an alternative email address.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > Hi Gene,
> >
> > Your mail server rejected my e-mail response to you as "probable
> > SPAM".
> >
> > Assuming you disagree with your server's assessment, let me know
how
> > I should proceed.
>
> Considering the huge quantity of spam which makes it through, this
is
> kind of depressing. Maybe upload it, or post it if it is postable. I
> suppose I should get an alternative email address.

Anyway, here we go:

>> >"Temperament, when implemented in a regular manner, reduces the
often
>> >bewildering variety of interval sizes in a scale or tuning system
to a
>> >manageable few."
>> >
>> >It only produces a finite number in the case equal temperaments;
>> > otherwise, the intervals are dense in the real continuum of
pitches. Why
>> > not say it makes intervals more managable?
>>
>> I should indeed say that in a few places. But here, I'm not
talking about
>> an *infinite* tuning system, let alone an infinite scale. So isn't
my
>> assertion true?

>If you have an equal temperament, it is theoretically infinite, but
finite in
>practice since only a finite number of tones are audible. If you
have
>something with more than one generator, whether JI or temperament,
you get
>density. Meantone is finite only in the same way and for the same
reason that
>5-limit JI is finite; you can't distinguish an infinite number of
tones.

Are you completely ignoring what I'm saying here, or what? I
changed "tuning system" to "finite tuning system". Is my assertion
now true, or not?

>> >"A fancier way of writing this _expression is
>> ><1200 1901.96 2786.31|-4 4 -1> = 21.5;
>> >this is an example of an inner product operation between two
vectors."
>> >
>> >Technically, it would be better to call this an angle bracket.
Most of the
>> >time, an inner product means between two vectors in the same
space.
>
>> Is it an interior product?

>No, though there is a relationship. It's the linear mapping of an
element of
>the dual space V* acting on an element of the original vector space
V.

>Here's what Math World has on dual vector spaces:

>http://mathworld.wolfram.com/DualVectorSpace.html

>http://mathworld.wolfram.com/LinearFunctional.html

Can't I just refer to it as some kind of product? I really don't want
to burden the reader with abstruse mathematics, and I think 99% of
them would thank me for this.

>> >and
>> >the mapping to primes can be read off the bival more readily.
>
>> Maybe the generator part of the mapping is a tiny bit more direct.
But the
>> period part of the mapping, without which the other part does
little good,
>> doesn't.follow directly from either. The table tells you both, so
who
>> cares?

>I care; I find the bival form very much more convenient. If you
don't care,
>why do you insist on not using it?

Because it would add significantly to the length and complexity of
the paper.

>> >Of course
>> >switiching from one to the other is not difficult, but why saddle
us with
>> > a wholly unnecessary headache?
>
>> What is this headache, exactly? I think introducing vals would be a
>> headache.

>But there they are, in your table, in bimonzo form.

The 'bimonzos' are there, yes. Geometrically they represent the
periodic unit of the lattice, when the 'commas' are applied as
equivalence relations. Relating this to vals is outside the scope of
the paper, as beautiful as they may be mathematically. However, I
will be explaining the meaning and role of the "ket vector" a little
more fully in the 3-limit case, so an opportunity might open up for
you to provide a very informative footnote.

>>There's a footnote about Graham Breed's "melodic" approach being
>> the algebraic dual to this one.

>Graham, can you explain (maybe on tuning-math) what the melodic
approach is?

Take a look at Graham's webpages on meantone, schismic,
diaschismic . . . It seems to be closely wedded to the val approach,
which is why I suggested we use the term "breeds" to refer to vals
(as long as we're using "monzos" to refer to lattice vectors).

>> How do you know which is 5-limit and which is 7-limit?

>If they have the same tuning,

They don't. Different tunings, different horagrams. Take another
look. These systems do *not* have identical tunings in the 5-limit
and 7-limit (at least according to the 7-limit results you
furnished). However, for meantone, porcupine, superpyth, and magic,
I'm already doing this, because the tunings are indeed the same. Did
you really not see this?

>I guess I *could*
> redo these as long as whatever you propose stays in the same place
> alphabetically . . .

Please think about this again. Otherwise you'll be "saddled" with the
ugly names

>> >Why orson?
>
>> Son of Orwell, or ORson WELLs, or whatever . . .

>Why not just orwell?

Different tuning, different horagrams. "Mork to Orson!" Hello?

:)

BTW, quantum mechanics texts do refer to

<x|y>

as an "inner product". Are you telling me that I need to be *more*
mathematical than a quantum mechanics text?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> BTW, quantum mechanics texts do refer to
>
> <x|y>
>
> as an "inner product". Are you telling me that I need to be *more*
> mathematical than a quantum mechanics text?

<x|y> is indeed sometimes used to notate a inner product; this is
where x and y are vectors in the *same* vector space. It is *also*
used to notate the effect of applying a linear fuctional <x| to a
vector |y>. Students often find this confusing, I fear, and I would
not mention it in your paper, but I would avoid "inner product"
because it could be very confusing to those knowing enough math to be
confused. If you want a "product" name for it, I suggest "bracket
product", which is used for both the inner product and the linear
functional and vector thing so long as the bra-ket notation, or the
bracket notation for inner product, is being used.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> >I care; I find the bival form very much more convenient. If you
> don't care,
> >why do you insist on not using it?
>
> Because it would add significantly to the length and complexity of
> the paper.

Bivals and bimonzos are equally complex.

> >> >Of course
> >> >switiching from one to the other is not difficult, but why
saddle
> us with
> >> > a wholly unnecessary headache?
> >
> >> What is this headache, exactly? I think introducing vals would
be a
> >> headache.

But you go ahead and use bimonzos!

> >But there they are, in your table, in bimonzo form.
>
> The 'bimonzos' are there, yes. Geometrically they represent the
> periodic unit of the lattice, when the 'commas' are applied as
> equivalence relations. Relating this to vals is outside the scope
of
> the paper, as beautiful as they may be mathematically.

Are you expecting people to read the comma values off of the bimonzo?
> >> How do you know which is 5-limit and which is 7-limit?
>
> >If they have the same tuning,
>
> They don't. Different tunings, different horagrams.

Ah! The light dawns--you are proposing, in effect, a naming system
where two temperaments at different limits carry the same name if,
and only if, they have the same TOP generators. This would indeed be
simple and logical, and the only problem would arise when they have
*almost* the same TOP generators, where you will be drawing a fine
distinction which may not mean much in practice.

I we adopt this plan, I suggest that we do link the names of related
systems, as in augmented-august-augene and orson-orwell. I'd be
interested in what other people think of a sweeping revision of
temperament nomenclature along these lines; I've been trying to work
it in this direction, but not systematically.

> Different tuning, different horagrams. "Mork to Orson!" Hello?

7-limit orson, according to the TOP approach, would be the +53
adjustment to orwell. It would have the orson/semicomma comma
together with 4375/4374. It would also be only marginally
interesting, but it would get the name anyway.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > >I care; I find the bival form very much more convenient. If you
> > don't care,
> > >why do you insist on not using it?
> >
> > Because it would add significantly to the length and complexity
of
> > the paper.
>
> Bivals and bimonzos are equally complex.

Not when you've already introduced the "monzos" concept but haven't
introduced the "vals" concept.

> > >> >Of course
> > >> >switiching from one to the other is not difficult, but why
> saddle
> > us with
> > >> > a wholly unnecessary headache?
> > >
> > >> What is this headache, exactly? I think introducing vals would
> be a
> > >> headache.
>
> But you go ahead and use bimonzos!

Yup -- and I just said why, too.

> > >But there they are, in your table, in bimonzo form.
> >
> > The 'bimonzos' are there, yes. Geometrically they represent the
> > periodic unit of the lattice, when the 'commas' are applied as
> > equivalence relations. Relating this to vals is outside the scope
> of
> > the paper, as beautiful as they may be mathematically.
>
> Are you expecting people to read the comma values off of the
bimonzo?

No. But as long as we're on the subject here, it might be worth
reviewing here for list memmbers how you do that. Not in the paper.

Anyway, since these names are so ugly, does *anyone* have suggestions
for renaming them (Dimipent, Dimisept, Negripent, Negrisept,
Sensipent, Sensisept) that preserves their approximate alphabetical
location?

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

> Not when you've already introduced the "monzos" concept but haven't
> introduced the "vals" concept.

You introduce both, more or less, in your paper, at least to the
extent of an implictly defined vector space togher with its dual. That
is how you ended up with a bracket product despite the fact that no
inner product space is being discussed, or would make any sense in the
context.

> Yup -- and I just said why, too.

And your reason is not correct.

> Anyway, since these names are so ugly, does *anyone* have suggestions
> for renaming them (Dimipent, Dimisept, Negripent, Negrisept,
> Sensipent, Sensisept) that preserves their approximate alphabetical
> location?

Just shortening them would help--dimip, dimis, negrip, negris, sensip,
sensis. Possibly needs work to keep people from reading negris as
Negress (a word many people find offensive) and confusing sensis with
census.

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

> And your reason is not correct.

It *isn't* the periodic unit in the tempered lattice? Where did I go
wrong?

> Just shortening them would help--dimip, dimis, negrip, negris,
sensip,
> sensis.

Those don't strike me as better names, but thanks.

hi Paul and Gene,

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

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

> <snip>
>
> > Graham, can you explain (maybe on tuning-math) what
> > the melodic approach is?
>
> Take a look at Graham's webpages on meantone, schismic,
> diaschismic . . . It seems to be closely wedded to the
> val approach, which is why I suggested we use the term
> "breeds" to refer to vals (as long as we're using "monzos"
> to refer to lattice vectors).

i like that !! ;-)

i recall some posts a while back which complained of
using "val", instead of a new term.

>> > Why orson?
>
>> Son of Orwell, or ORson WELLs, or whatever . . .
>
>Why not just orwell?
>
> Different tuning, different horagrams. "Mork to Orson!" Hello?
>
> :)

... hmm ... monz predicts that Gene's next temperament
family will be named "Mork"

... that will be one zany tuning !!!

... wonder what properties it will have?

i'm sure Gene will give a full description of it,
including all the monzos and breeds. :)

(seems like this cyber-community is becoming
a nice cozy family)

what tuning concept will we ever call a "smith"?

;-) ;-)

-monz

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

> what tuning concept will we ever call a "smith"?
>
> ;-) ;-)

Aaron J. suggested that the "wedgie" be renamed the "genie" . . .

hi Gene and Paul

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

> <snip>
>
> Ah! The light dawns--you are proposing, in effect,
> a naming system where two temperaments at different limits
> carry the same name if, and only if, they have the same
> TOP generators. This would indeed be simple and logical,
> and the only problem would arise when they have *almost*
> the same TOP generators, where you will be drawing a fine
> distinction which may not mean much in practice.

can you explain why TOP generators are so important?

> I we adopt this plan, I suggest that we do link the names
> of related systems, as in augmented-august-augene and
> orson-orwell. I'd be interested in what other people think
> of a sweeping revision of temperament nomenclature along
> these lines; I've been trying to work it in this direction,
> but not systematically.

i support that idea whole-heartedly !!!

-monz

hi again, Gene and Paul,

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

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

> <snip>
>
> > Are you expecting people to read the comma values
> > off of the bimonzo?
>
> No. But as long as we're on the subject here, it might
> be worth reviewing here for list memmbers how you do that.
> Not in the paper.

yes, please do review it!

>
> Anyway, since these names are so ugly, does *anyone*
> have suggestions for renaming them (Dimipent, Dimisept,
> Negripent, Negrisept, Sensipent, Sensisept) that preserves
> their approximate alphabetical location?

my suggestion:

i think that the past decade has seen a great expansion
of the study of tuning, thanks largely to the internet.

now that we have so much broader a view of large numbers
of tunings, we should subject the whole "tuning universe"
to deep review, and come up with a really good and logical
system of classification and naming.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene and Paul
>
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
>
> > <snip>
> >
> > Ah! The light dawns--you are proposing, in effect,
> > a naming system where two temperaments at different limits
> > carry the same name if, and only if, they have the same
> > TOP generators. This would indeed be simple and logical,
> > and the only problem would arise when they have *almost*
> > the same TOP generators, where you will be drawing a fine
> > distinction which may not mean much in practice.
>
>
>
> can you explain why TOP generators are so important?

I'm not actually proposing what Gene said I'm proposing above. TOP is
great, but I wouldn't make it so all-important. I'm just naming
horagrams in my paper, that's all.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi again, Gene and Paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
>
> > <snip>
> >
> > > Are you expecting people to read the comma values
> > > off of the bimonzo?
> >
> > No. But as long as we're on the subject here, it might
> > be worth reviewing here for list memmbers how you do that.
> > Not in the paper.
>
>
> yes, please do review it!
>
>
>
> >
> > Anyway, since these names are so ugly, does *anyone*
> > have suggestions for renaming them (Dimipent, Dimisept,
> > Negripent, Negrisept, Sensipent, Sensisept) that preserves
> > their approximate alphabetical location?
>
>
> my suggestion:
>
> i think that the past decade has seen a great expansion
> of the study of tuning, thanks largely to the internet.
>
> now that we have so much broader a view of large numbers
> of tunings, we should subject the whole "tuning universe"
> to deep review, and come up with a really good and logical
> system of classification and naming.

OK, but this was asked in the context of my paper, which has to be
submitted very soon. Did you see the draft?

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

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

>
> > Not when you've already introduced the
> > "monzos" concept but haven't introduced
> > the "vals" concept.
>
> You introduce both, more or less, in your paper,
> at least to the extent of an implictly defined
> vector space togher with its dual.

is that "together with its dual"?

what is the dual? if you could explain it along
the lines of my "prime-space" definition, that
would help me.

http://tonalsoft.com/enc/primespace.htm

> That is how you ended up with a bracket product
> despite the fact that no inner product space is
> being discussed, or would make any sense in the context.

what's an "inner product space"?

is "vector space" here synonymous with my
definition of "prime-space"?

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > what tuning concept will we ever call a "smith"?
> >
> > ;-) ;-)
>
> Aaron J. suggested that the "wedgie" be renamed the "genie" . . .

ok, so my encylopaedia needs TV themes for these
shows, *in* the tunings named after them, for
this new category of tuning names!

Herman, what do you say? :)

i write the definitions, you make the webpages full
of MIDI files, and we link to each other.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Not when you've already introduced the "monzos" concept but
haven't
> > introduced the "vals" concept.
>
> You introduce both, more or less, in your paper, at least to the
> extent of an implictly defined vector space togher with its dual.

Something "implicit" isn't a concept that I introduced in the paper.
Is it possible for you to think like a non-mathematician for a
moment? Look, the next version of the paper will show, a little more
explicitly, what you call the "JIP val" -- and what I would
call "pitch altitude" or something -- in the 3-limit lattice. At that
point, maybe you could meet me halfway and try to give specific
suggestions that I can implement readily, rather than overarching
criticisms which would require, essentially, a new paper . . .

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

>Here's what Math World has on dual vector spaces:
>
>http://mathworld.wolfram.com/DualVectorSpace.html
>
>http://mathworld.wolfram.com/LinearFunctional.html

Did you notice that this second link was in the footnotes of my paper?

And here's what it says on Angle Bracket:

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

It says that the bra is a covariant 1-vector, and the ket is a
contravariant one-form. It also says that the combination of the two,
as in <v|w>, is an inner product.

hi Paul,

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

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

> > now that we have so much broader a view of large numbers
> > of tunings, we should subject the whole "tuning universe"
> > to deep review, and come up with a really good and logical
> > system of classification and naming.
>
> OK, but this was asked in the context of my paper, which
> has to be submitted very soon. Did you see the draft?

yes, but i only had time to skim it quickly. actually,
i'm printing it out right now so that i can give it a
proper reading, which i'll do later this morning.

but it does seem to me that the time may be right for
a drastic reconsideration of existing nomenclature ...
considering especially the advances made in tuning theory
since Gene joined the tuning cyber-community, and the
fact that sagittal notation and my own little contribution
(Tonalsoft software) are about to be born to the world.

-monz

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

> can you explain why TOP generators are so important?

They have an intrinsic definition in terms of bounding the error
relative to Tenney height, which in turn seems the best way to define
interval complexity. Hence, they define a tuning standard which is
less arbitary than the alternatives, and so consequently a seemingly
better choice on which to base a naming system.

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

> It says that the bra is a covariant 1-vector, and the ket is a
> contravariant one-form. It also says that the combination of the two,
> as in <v|w>, is an inner product.

I know people talk that way. I also know it is very confusing to
people trying to learn this stuff. But probably not worth worrying
about in connection with your paper. I'd also flush all of that stuff
about 1-forms from your brain immediately.

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

> Something "implicit" isn't a concept that I introduced in the paper.
> Is it possible for you to think like a non-mathematician for a
> moment? Look, the next version of the paper will show, a little more
> explicitly, what you call the "JIP val" -- and what I would
> call "pitch altitude" or something -- in the 3-limit lattice. At that
> point, maybe you could meet me halfway and try to give specific
> suggestions that I can implement readily, rather than overarching
> criticisms which would require, essentially, a new paper . . .

You've lost me. I've made no overarching criticisms, only suggestions
for minute changes (such as "angle bracket" for "inner product") which
would be trivial to implement.

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

> what is the dual? if you could explain it along
> the lines of my "prime-space" definition, that
> would help me.

Paul and I have been tossing this back and forth at each other:

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

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

> http://tonalsoft.com/enc/primespace.htm
>
>
> > That is how you ended up with a bracket product
> > despite the fact that no inner product space is
> > being discussed, or would make any sense in the context.
>
>
> what's an "inner product space"?

http://mathworld.wolfram.com/VectorSpace.html
http://en.wikipedia.org/wiki/Vector_space

http://mathworld.wolfram.com/InnerProductSpace.html
http://en.wikipedia.org/wiki/Inner_product_space

However, since we aren't using inner product spaces it is only vector
spaces which need concern us. Other relevant encyclopedia pages are
for abelian group

http://en.wikipedia.org/wiki/Abelian_group

You could also look up bra-ket notation, but because it assumes we are
in an inner product space and we are not, it may not be that great.
This is my problem with Paul wanting to use "inner product" for the
bracket--we don't actually have an inner product, whereas in quantum
mechanics we do. This isn't QM, thank heavens.

> is "vector space" here synonymous with my
> definition of "prime-space"?

I'm afraid your definition was too far from the way mathematicians
define things to let me have any assurance I understand what you are
saying. I define Tenney space here

http://66.98.148.43/~xenharmo/top.htm

Inside of Tenney space, monzos or musical intervals live as a lattice,
which in my way of thinking must include a means of measuring the
distance between the lattice points.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > It says that the bra is a covariant 1-vector, and the ket is a
> > contravariant one-form. It also says that the combination of the
two,
> > as in <v|w>, is an inner product.
>
> I know people talk that way. I also know it is very confusing to
> people trying to learn this stuff. But probably not worth worrying
> about in connection with your paper. I'd also flush all of that
stuff
> about 1-forms from your brain immediately.

Well, I remember trying to understand differential forms when
cramming for a big freshman math final, but really there's nothing to
flush.

I just looked at Robert Griffiths' book "Consistent Quantum
Mechanics", and it introduces kets explitictly as linear functionals,
and then the bracket product as the "inner product". Luckily (or
perhaps unluckily?), the music theory doesn't have to deal with
taking complex conjugates.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > what is the dual? if you could explain it along
> > the lines of my "prime-space" definition, that
> > would help me.
>
> Paul and I have been tossing this back and forth at each other:
>
> http://mathworld.wolfram.com/DualVectorSpace.html
>
> http://mathworld.wolfram.com/LinearFunction.html
>
> > http://tonalsoft.com/enc/primespace.htm
> >
> >
> > > That is how you ended up with a bracket product
> > > despite the fact that no inner product space is
> > > being discussed, or would make any sense in the context.
> >
> >
> > what's an "inner product space"?
>
> http://mathworld.wolfram.com/VectorSpace.html
> http://en.wikipedia.org/wiki/Vector_space
>
> http://mathworld.wolfram.com/InnerProductSpace.html
> http://en.wikipedia.org/wiki/Inner_product_space
>
> However, since we aren't using inner product spaces it is only
vector
> spaces which need concern us. Other relevant encyclopedia pages are
> for abelian group
>
> http://en.wikipedia.org/wiki/Abelian_group
>
> You could also look up bra-ket notation, but because it assumes we
are
> in an inner product space and we are not, it may not be that great.
> This is my problem with Paul wanting to use "inner product" for the
> bracket--we don't actually have an inner product, whereas in quantum
> mechanics we do. This isn't QM, thank heavens.

What do you mean? In quantum mechanics, the bracket or "inner
product" is simply the application of a linear functional acting on a
vector.

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

> What do you mean? In quantum mechanics, the bracket or "inner
> product" is simply the application of a linear functional acting on a
> vector.

In QM state vectors are unit vectors in a complex Hilbert space,
meaning there is a Hermitian inner product on the space. An
eigenvector for a Hermitian linear operator (ie, an "observable") with
discrete spectrum will *also* be a state vector if normalized to a
unit vector. They live in the same space, so the eigenvector will be a
wave function like the state vector, if those are wave functions. If
you take the absolute value of the inner product and square it, you
get the probability of a measurement coming up with with corresponding
eigenvalue as a result of the measurement of the observable (=
Hermitian operator.) If you don't have a discrete spectrum you need to
resort to spectral theory, but it's basically similar. The upshot is
that the eigenvectors are bounded linear functionals on the states,
but since we are in an inner product space we can identify these with
states. It's like identifying a row vector with a column vector.

Applications to music? I dunno; but the discrete spectrum business is
intriging. Someone should tune up a hydrogen atom when taken down
enough octaves, I guess.

hi Gene,

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

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > What do you mean? In quantum mechanics, the bracket
> > or "inner product" is simply the application of a
> > linear functional acting on a vector.
>
> In QM state vectors are unit vectors in a complex
> Hilbert space, meaning there is a Hermitian inner product
> on the space. An eigenvector for a Hermitian linear operator
> (ie, an "observable") with discrete spectrum will *also*
> be a state vector if normalized to a unit vector. They
> live in the same space, so the eigenvector will be a
> wave function like the state vector, if those are wave
> functions. If you take the absolute value of the inner
> product and square it, you get the probability of a
> measurement coming up with with corresponding eigenvalue
> as a result of the measurement of the observable
> (= Hermitian operator.) If you don't have a discrete spectrum
> you need to resort to spectral theory, but it's basically
> similar. The upshot is that the eigenvectors are bounded
> linear functionals on the states, but since we are in an
> inner product space we can identify these with states.
> It's like identifying a row vector with a column vector.
>
> Applications to music? I dunno; but the discrete spectrum
> business is intriging. Someone should tune up a hydrogen atom
> when taken down enough octaves, I guess.

i understand very little of what you wrote here, but ever
since i came up with the idea of finity and bridging in 1998 ...

http://tonalsoft.com/enc/finity.htm
http://tonalsoft.com/enc/bridging.htm

... what i *do* understand about QM has had me believing
that it might have some application to music. i.e.,
there's a sort of "uncertainty principle" with regard
to our perception and comprehension of pitch / tuning / harmony.

-monz

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

> > > Are you expecting people to read the comma values
> > > off of the bimonzo?
> >
> > No. But as long as we're on the subject here, it might
> > be worth reviewing here for list memmbers how you do that.
> > Not in the paper.

> yes, please do review it!

If you have a bimonzo ||a1 a2 a3 a4 a5 a6>> then you can read off the
odd comma of the temperament (the comma which is a ratio of odd
integers) by taking out the common factor if needed of a1, a2, a3 to
get b1, b2, b3, and then the comma is 3^b1 5^b2 7^b3, or its
reciprocal if you need to make it bigger than 1.

In general, however, reading the commas from a bimonzo is no easier
than reading them from a bival, and in fact probably harder, and I
don't think this makes much of a reason for using bimonzos. I really
would like to know why Paul insists on this so stubbornly.

If you have a bival <<a1 a2 a3 a4 a5 a6||, then

2^a4 3^(-a2) 5^a1 gives the 5-limit comma. That's 2 to the power of
the (3,5) coefficient, 3 to minus the power of the (2,5) coefficent,
and 5 to the power of the (2,3) coefficient. You can figure out the
signs by putting the primes in the circular order, 235; then 35, 52
and 23 have the same sign, so we give 25 a sign opposite to 35 and 23,
and we have the comma. Of course the prime we attach the exponent to
is the prime not in the (a,b) of "(a,b) coefficient", and the
coefficients are ordered <<(2,3), (2,5), (2,7), (3,5), (3,7), (5,7)||.

The rest of these work similarly.

2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power
of the (3,7) coefficient, 3 to minus the power of the (2,7)
coefficient, and 7 to the power of the (2,3) coefficient; 237 in order
tells you that (2,3) and (3,7) have a sign opposite from (2, 7).

2^a6 5^(-a3) 7^a2 gives the {2,5,7}-comma

3^a6 5^(-a5) 7^a4 gives the {3,5,7}-comma (the odd comma.)

You get similar stuff in higher limits, but with a lot more commas.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > > Are you expecting people to read the comma values
> > > > off of the bimonzo?
> > >
> > > No. But as long as we're on the subject here, it might
> > > be worth reviewing here for list memmbers how you do that.
> > > Not in the paper.
>
> > yes, please do review it!
>
> If you have a bimonzo ||a1 a2 a3 a4 a5 a6>> then you can read off
the
> odd comma of the temperament (the comma which is a ratio of odd
> integers) by taking out the common factor if needed of a1, a2, a3 to
> get b1, b2, b3, and then the comma is 3^b1 5^b2 7^b3, or its
> reciprocal if you need to make it bigger than 1.
>
> In general, however, reading the commas from a bimonzo is no easier
> than reading them from a bival, and in fact probably harder,

Why harder? Can you show this?

> and I
> don't think this makes much of a reason for using bimonzos. I really
> would like to know why Paul insists on this so stubbornly.

I welcome constructive suggestions for making the paper go bival,
without adding to its math-heaviness.

> If you have a bival <<a1 a2 a3 a4 a5 a6||, then
>
> 2^a4 3^(-a2) 5^a1 gives the 5-limit comma.

You don't need to remove common factors?

> 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power
> of the (3,7) coefficient, 3 to minus the power of the (2,7)
> coefficient, and 7 to the power of the (2,3) coefficient;

How is this easier than the bimonzo case?

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

> Why harder? Can you show this?

From my explanation, you can see it is easy to read off the prime
factors in a frontward way and figure out the signs in a bival. With
the 7-limit bimonzo, you don't have the same regular pattern where two
signs are the same, and one opposite; sometimes they all have the same
sign, and sometimes they don't. I think bivals are clearly easier.
Moreover, for higher limit linear temperaments, they are still bivals
and you can use the same rule, whereas trimonzos are what you get in
the 11-limit, etc. A big fat mess by comparison.

> > and I
> > don't think this makes much of a reason for using bimonzos. I really
> > would like to know why Paul insists on this so stubbornly.
>
> I welcome constructive suggestions for making the paper go bival,
> without adding to its math-heaviness.

It's really, really, really easy. Simply replace the bimonzos you list
with the corresponding bivals, and you are done. You need explain
nothing, nor define anything.

> > If you have a bival <<a1 a2 a3 a4 a5 a6||, then
> >
> > 2^a4 3^(-a2) 5^a1 gives the 5-limit comma.
>
> You don't need to remove common factors?

I said "gives the comma", not "is the comma".

> > 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the power
> > of the (3,7) coefficient, 3 to minus the power of the (2,7)
> > coefficient, and 7 to the power of the (2,3) coefficient;
>
> How is this easier than the bimonzo case?

Because there is a simple rule I can explain which works for any prime
limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Why harder? Can you show this?
>
> From my explanation, you can see it is easy to read off the prime
> factors in a frontward way and figure out the signs in a bival. With
> the 7-limit bimonzo, you don't have the same regular pattern where
two
> signs are the same, and one opposite; sometimes they all have the
same
> sign, and sometimes they don't.

When don't they? I must have missed something.

> I think bivals are clearly easier.
> Moreover, for higher limit linear temperaments, they are still
bivals
> and you can use the same rule, whereas trimonzos are what you get in
> the 11-limit, etc. A big fat mess by comparison.

I already answered this, so I won't repeat myself.

> > > and I
> > > don't think this makes much of a reason for using bimonzos. I
really
> > > would like to know why Paul insists on this so stubbornly.
> >
> > I welcome constructive suggestions for making the paper go bival,
> > without adding to its math-heaviness.
>
> It's really, really, really easy. Simply replace the bimonzos you
list
> with the corresponding bivals, and you are done.

You've got to be kidding me.

> You need explain
> nothing, nor define anything.

I'd like to do better by my readers.

> > > If you have a bival <<a1 a2 a3 a4 a5 a6||, then
> > >
> > > 2^a4 3^(-a2) 5^a1 gives the 5-limit comma.
> >
> > You don't need to remove common factors?
>
> I said "gives the comma", not "is the comma".

But when you were talking about the bimonzo, you made it seem a whole
lot more complicated by introducing b1, b2, b3.

> > > 2^(a5) 3^(-a3) 7^(a1) gives the {2,3,7}-comma; that's 2 to the
power
> > > of the (3,7) coefficient, 3 to minus the power of the (2,7)
> > > coefficient, and 7 to the power of the (2,3) coefficient;
> >
> > How is this easier than the bimonzo case?
>
> Because there is a simple rule I can explain which works for any
prime
> limit.

Show me how a bimonzo gets so bad in a higher prime limit.

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

I'm a little late replying, because I've got a new computer now. I'm
taking a break from things like config files.

> > It's really, really, really easy. Simply replace the bimonzos you
> list
> > with the corresponding bivals, and you are done.
>
> You've got to be kidding me.

Why? You said you were not introducing any multilinar algebra, I
thought. Certainly you cannot do so and stay suitably nonmathematical.

> > You need explain
> > nothing, nor define anything.
>
> I'd like to do better by my readers.

How does simply tossing a bimonzo in their face do better by them?

> Show me how a bimonzo gets so bad in a higher prime limit.

The bimonzo isn't even what you would use in a higher prime limit,
for starters. To get a linear temperament, you need a monzo in the 5-
limit, a bimonzo in the 7-limit, a trimonzo in the 11-limit, and so
forth. It's bi*val*, not bimonzo, which gives linear temperaments
always.

(1) In any limit, the first pi(p)-1 entries of the bival give us the
period, and the generator part of the period-generator map. For any
limit above 5, the advantage goes on this point to bivals.

(2) In all limits, we can read off the three-prime commas of the
temperament by taking two of the exponents to have one sign, and the
other to have another. This makes the rule for these commas an easier
one in all limits above 5.

For example, in the 7 limit, suppose we have the bival

<<a23 a25 a27 a35 a37 a57||

Then the commas can be found from

2^a35 3^-a25 5^a23

2^a37 3^-a27 7^a23

2^a57 5^-a27 7^a25

3^a57 5^-a37 7^a35

All of the commas have two exponents of one sign and one of the
opposite sign in terms of the components of the bival. If we take the
complement of this, and so use the bimonzo signs, two of the commas
(the odd comma, {3,5,7}, and the 5-limit comma, {2,3,5}) have all the
exponents the same sign, and the other two have two the same and one
the opposite.

In the 11 and higher limits, it is similar. The bival commas are all
of the ++- form, but the complementary multimonzo does not give all
of them the same form. In the 11-limit, the {2,5,11}, {3,7,11} and
{5,7,11} commas have the +++ form, and the rest the --- form. Again,
the bival form on this basis is clearly superior.

I would really like to know why you are so intransigent on this
score; there seems to be no advantage in using multimonzos beyond the
5-limit as a canonical form, and you don't seem to have any argument
beyond a dislike of vals for not using bivals. But this isn't a good
argument, because vals are important and should be introduced before
multilinear algebra is even mentioned.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> I'm a little late replying, because I've got a new computer now.
I'm
> taking a break from things like config files.
>
> > > It's really, really, really easy. Simply replace the bimonzos
you
> > list
> > > with the corresponding bivals, and you are done.
> >
> > You've got to be kidding me.
>
> Why? You said you were not introducing any multilinar algebra, I
> thought. Certainly you cannot do so and stay suitably
nonmathematical.

Cannot do so? Cannot do what?

> > > You need explain
> > > nothing, nor define anything.
> >
> > I'd like to do better by my readers.
>
> How does simply tossing a bimonzo in their face do better by them?

It won't be tossed in their face, it'll be clearly illustrated.

> > Show me how a bimonzo gets so bad in a higher prime limit.
>
> The bimonzo isn't even what you would use in a higher prime limit,
> for starters. To get a linear temperament,

Stop right there. You're changing the subject. We were only talking
about bimonzos. And we weren't only talking about linear
temperaments. You keep telling me this stuff I know, but you're not
looking at the context of the paper.

> (1) In any limit, the first pi(p)-1 entries of the bival give us
the
> period, and the generator part of the period-generator map. For any
> limit above 5, the advantage goes on this point to bivals.

You're setting up an unfair comparison, bivals vs. multimonzos. What
if I set it up as bimonzos vs. multivals? That's still a middle path,
just one that rides closer to the JI edge than to the ET edge.

> All of the commas have two exponents of one sign and one of the
> opposite sign in terms of the components of the bival. If we take
the
> complement of this, and so use the bimonzo signs, two of the commas
> (the odd comma, {3,5,7}, and the 5-limit comma, {2,3,5}) have all
the
> exponents the same sign,

Oh. I was only testing the 5-limit comma, {2,3,5), and I liked the
fact that you didn't have to selectively mess with the exponents, the
way you do with the bival.

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

> > Why? You said you were not introducing any multilinar algebra, I
> > thought. Certainly you cannot do so and stay suitably
> nonmathematical.
>
> Cannot do so? Cannot do what?

You can be not very mathematical, or you can introduce wedge
products, but you can't do both.

> > How does simply tossing a bimonzo in their face do better by them?
>
> It won't be tossed in their face, it'll be clearly illustrated.

Is this why you want bimonzos--to show a wedge product of two commas
pictorially?

> > > Show me how a bimonzo gets so bad in a higher prime limit.
> >
> > The bimonzo isn't even what you would use in a higher prime
limit,
> > for starters. To get a linear temperament,
>
> Stop right there. You're changing the subject. We were only talking
> about bimonzos.

Your paper does not introduce bimonzos for 11-limit planar
temperaments. The only time they get used, they are used for linear
temperaments. Hence, this is what we are discussing.

And we weren't only talking about linear
> temperaments.

I saw nothing about planar temperaments.

> Oh. I was only testing the 5-limit comma, {2,3,5), and I liked the
> fact that you didn't have to selectively mess with the exponents,
the
> way you do with the bival.

Commas are exellent for 5-limit temperaments, or for codimension one
in general. However, most of the time that isn't what we are looking
at. In any case, the bival for a 5-limit linear temperament does, in
fact, give the mapping; <1 4 4| gets to the mapping for meantone more
directly than |-4 4 -1> does.

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

> > > How does simply tossing a bimonzo in their face do better by
them?
> >
> > It won't be tossed in their face, it'll be clearly illustrated.
>
> Is this why you want bimonzos--to show a wedge product of two
commas
> pictorially?

Something like that. Did you not see the post I wrote to you
yesterday in which I mentioned that the 12-equal case will be
presented, similarly to what's in the gentle introduciton to
periodicity blocks?

> > > > Show me how a bimonzo gets so bad in a higher prime limit.
> > >
> > > The bimonzo isn't even what you would use in a higher prime
> limit,
> > > for starters. To get a linear temperament,
> >
> > Stop right there. You're changing the subject. We were only
talking
> > about bimonzos.
>
> Your paper does not introduce bimonzos for 11-limit planar
> temperaments. The only time they get used, they are used for linear
> temperaments.

Once again, the initial exposition of a "bimonzo" (not what I call
it) will be in the context of 12-equal, not a linear temperament.

> > And we weren't only talking about linear
> > temperaments.
>
> I saw nothing about planar temperaments.

I'm going to give 7-limit "planar" temperament very brief coverage,
but they'll be mentioned, certainly.

> > Oh. I was only testing the 5-limit comma, {2,3,5), and I liked
the
> > fact that you didn't have to selectively mess with the exponents,
> the
> > way you do with the bival.
>
> Commas are exellent for 5-limit temperaments,

I'm talking about 7-limit here.

> or for codimension one
> in general. However, most of the time that isn't what we are
looking
> at. In any case, the bival for a 5-limit linear temperament does,
in
> fact, give the mapping;

I know that . . . :)