Tuning-Math Digests messages 8425 - 8449

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Message: 8425

Date: Thu, 20 Nov 2003 22:36:00

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
> 
> > "m" here is the grade of the object, i.e. the number of nested
> > brakets.  A more intuitive (for me) alternative to m(m+1)/2 is
> > Ceiling(m/2). If the sum of the indices plus this quantity is even
> > then you negate it when complementing.
> 
> That's an interesting short cut.  The way I've been doing it is:
> 
> Join the old basis on the left and the new basis on the right to 
get a 
> list of all primitive bases (or whatever they are).  If this is an 
odd 
> permuatation, negate the coefficient.
> 
> This assumes all bases are being stored in numerical order of their 
> components (what's all this talk about alphabetical order?)
> 
> You can test for an odd permutation by swapping adjacent pairs of 
> primitive bases until their in the right order, and it's an odd 
> permutation if you did an odd number of swaps.  This follows from 
the 
> antisymmetry of the wedge product, which is the main thing you need 
to 
> know about it.
> You can also compare every pair of numbers in the basis, 
> and it's an odd permutation iff an odd number of them are the wrong 
way 
> round.
> 
> 
>                    Graham

Yes, the odd vs. even permutation thing is what (I think) Gene 
originally stated in this thread, and seems the clearest and most 
general way to think about it.

What does 'primitive basis' mean? Mathworld doesn't seem to have this 
usage . . .


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Message: 8426

Date: Thu, 20 Nov 2003 22:36:49

Subject: Re: Finding the complement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
> 
> > It should be mentioned that taking the complement of the complement
> > doesn't always give you back what you started with, sometimes it's the
> > negative of what you started with. So in those cases it's analogous to
> > multiplying by i (the square root of -1). This depends on the
> > dimension and the grade. But taking the complement four-times always
> > gives you back exactly what you started with.
> 
> Are you sure?  Do you have an example?

Yes. Disturbing isn't it? It occurs only for all odd grades in all
even dimensions (where the dimension is the index of the limiting
prime). So the simplest case is for a 3-limit vector (2D grade 1). The
complement of <a b] is [-b a>, and so the complement of [-b a> must be
<-a -b] which is -<a b]

~~<a b]
= ~[-b a>
= <-a -b]
= -<a b]

The fact that it occurs for 2D makes it clear there is no trick of
reordering indices that is going to get rid of it. The next ocurrence
is for 7-limit vectors and trivectors (4D grades 1 and 3).

If you look at the 13-limit examples Gene gave in this thread, these
are 6D so you should find that the vector, trivector and 5-vector
complements have the same property.


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Message: 8427

Date: Thu, 20 Nov 2003 22:39:09

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > yes, as you know i (and especially graham) like that idea very 
much --
> >  BUT 88cET has no octaves!
> 
> You can make 88cET equivalent with respect to some other interval --
 
> like 88 cents for example.
> 
> 
>                         Graham

Well, that would mean that one hears every pitch of this tuning as 
the same pitch class -- pretty absurd, really. I think octave 
similarity may never go away, so that any two notes whose ratio is an 
approximations to a power of 2 in such tunings will be heard as 
somewhat 'similar'.


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Message: 8428

Date: Thu, 20 Nov 2003 22:42:58

Subject: Re: Finding the compliment

From: Paul Erlich

> This is what Browne calls the Euclidean compliment,

I tried to compliment you before but maybe i need to find the right 
compliment . . . oh, you're talking about the compl*e*ment!


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Message: 8429

Date: Thu, 20 Nov 2003 22:45:45

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > Grassman himself apparently used a prefix vertical bar. John 
Browne
> > uses a horizontal bar above the symbol (or above a whole 
expression)
> > exactly as you describe for logical complements. But this is 
usually
> > translated to a prefix tilde ~ in ASCII, and it has the advantage 
of
> > looking similar  to a minus sign - which you say is more 
analogous,
> > but is different from a prefix minus sign which would have the 
more
> > obvious interpretation of negating _all_ the coefficients (and not
> > reversing their order or the brakets).
> 
> I'm willing to adopt a prefix tilde and not a postfix asterisk. 
>Paul?

sure.


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Message: 8430

Date: Thu, 20 Nov 2003 22:52:44

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> > > OK -- but it's interesting to note that the cross product 
> immediately 
> > > gives you the quantity of interest in 3D, regardless of indexing 
> > > conventions. 
> > 
> > Paul. You must have missed where I explained that the cross-product
> > stays the same no matter what the indexing conventions, because the
> > wedge-product and the complement change in "complementary" ways when
> > you change the indexing and A(x)B = ~(A^B).
> 
> I don't know why you think I missed that, because (even if I did) 
> it's perfectly clear to me, and I was never confused about that. It 
> was the remark that followed that one which was my main point.

Sorry Paul. I must have been reading my own confusion into what you
wrote. I assumed you were hoping for an indexing convention that would
make the 3D complement involve no changes of sign or reversals of
ordering.

> > > The GABLE tutorial claims that cross products are
> > > useless and should be dispensed with since geometric algebra has
> > > better ways of solving all the problems that the cross product is
> > > used for. I don't know . . .


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Message: 8431

Date: Thu, 20 Nov 2003 23:02:24

Subject: Re: Finding the wedge product?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> the coefficient is 5.  The result's value for e1e2 is 1+5=5.

1*5=5, you musta meant.


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Message: 8432

Date: Thu, 20 Nov 2003 23:02:49

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: 
> Yes, the odd vs. even permutation thing is what (I think) Gene 
> originally stated in this thread, and seems the clearest and most 
> general way to think about it.

Yes it's the most general and fundamental, but not the most practical
for efficient computation, whether by human or machine. Certainly for
humans, counting odd numbers in a numerically-ordered compound index
will be far quicker and less error-prone.

> What does 'primitive basis' mean? Mathworld doesn't seem to have this 
> usage . . .

I think I used "simple basis" for the same thing, meaning a basis
whose elements are not compounded of other basis elements, and
therefore have a single-digit index in the indexing scheme I'm using.
i.e. the basis of the vector, not the bi-vector or any higher grade
multivector.


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Message: 8433

Date: Thu, 20 Nov 2003 23:05:17

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Dave Keenan wrote:
> 
> > It should be mentioned that taking the complement of the 
complement
> > doesn't always give you back what you started with, sometimes 
it's the
> > negative of what you started with. So in those cases it's 
analogous to
> > multiplying by i (the square root of -1). This depends on the
> > dimension and the grade. But taking the complement four-times 
always
> > gives you back exactly what you started with.
> 
> Are you sure?  Do you have an example?
> 
> 
>                    Graham

that's easy -- in 3-dimensional space, the dual of e1^e2^e3 is 1, 
while the dual of 1 is -e1^e2^e3.


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Message: 8434

Date: Thu, 20 Nov 2003 23:07:30

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

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

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


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Message: 8435

Date: Thu, 20 Nov 2003 23:10:34

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> > > dimension and the grade. But taking the complement four-times 
always
> > > gives you back exactly what you started with.
> > 
> > Are you sure?  Do you have an example?
> 
> Yes. Disturbing isn't it? It occurs only for all odd grades in all
> even dimensions

it occurs in odd dimensions too.

> (where the dimension is the index of the limiting
> prime). So the simplest case is for a 3-limit vector (2D grade 1). 
The
> complement of <a b] is [-b a>, and so the complement of [-b a> must 
be
> <-a -b] which is -<a b]
> 
> ~~<a b]
> = ~[-b a>
> = <-a -b]
> = -<a b]
> 
> The fact that it occurs for 2D makes it clear there is no trick of
> reordering indices that is going to get rid of it. The next 
ocurrence
> is for 7-limit vectors and trivectors (4D grades 1 and 3).

you missed 5-limit scalars and pseudoscalars (3D grades 0 and 3).


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Message: 8436

Date: Thu, 20 Nov 2003 23:12:09

Subject: Re: Vals?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote: 
> > Yes, the odd vs. even permutation thing is what (I think) Gene 
> > originally stated in this thread, and seems the clearest and most 
> > general way to think about it.
> 
> Yes it's the most general and fundamental, but not the most 
practical
> for efficient computation, whether by human or machine. Certainly 
for
> humans, counting odd numbers in a numerically-ordered compound index
> will be far quicker and less error-prone.
> 
> > What does 'primitive basis' mean? Mathworld doesn't seem to have 
this 
> > usage . . .
> 
> I think I used "simple basis" for the same thing, meaning a basis
> whose elements are not compounded of other basis elements, and
> therefore have a single-digit index in the indexing scheme I'm 
using.
> i.e. the basis of the vector, not the bi-vector or any higher grade
> multivector.

so the vector or 1-vector basis, yes?


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Message: 8437

Date: Thu, 20 Nov 2003 23:15:04

Subject: Re: Finding the compliment

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> > This is what Browne calls the Euclidean compliment,
> 
> I tried to compliment you before but maybe i need to find the right 
> compliment . . . oh, you're talking about the compl*e*ment!

Come to think of it, what would a Euclidean compliment be? Perhaps
something like, "My, your triangles are looking very congruent this
morning Mrs Aristotle". :-)


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Message: 8438

Date: Thu, 20 Nov 2003 23:33:50

Subject: Re: Finding the complement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> > Dave Keenan wrote:
> > 
> > > It should be mentioned that taking the complement of the 
> complement
> > > doesn't always give you back what you started with, sometimes 
> it's the
> > > negative of what you started with. So in those cases it's 
> analogous to
> > > multiplying by i (the square root of -1). This depends on the
> > > dimension and the grade. But taking the complement four-times 
> always
> > > gives you back exactly what you started with.
> > 
> > Are you sure?  Do you have an example?
> 
> that's easy -- in 3-dimensional space, the dual of e1^e2^e3 is 1, 
> while the dual of 1 is -e1^e2^e3.

No, that second one is not correct.
 
In other words you're saying that in 3D the complement of 1 is
<<<-1]]]. By the rules posted recently, the coefficient is negated if
and only if the sum of the indices plus Ceiling(g/2) is odd, where g
is the grade. The grade of a scalar is zero and the sum of the indices
is zero (since there aren't any). So you do not negate.


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Message: 8439

Date: Thu, 20 Nov 2003 23:46:38

Subject: Re: Finding the complement

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote: 
> you missed 5-limit scalars and pseudoscalars (3D grades 0 and 3).

I don't think so.

See page 10 of
http://www.ses.swin.edu.au/homes/browne/grassmannalgebra/book/bookpdf/TheComplement.pdf - Ok *


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Message: 8440

Date: Fri, 21 Nov 2003 01:50:07

Subject: Re: Definition of val etc.

From: Dave Keenan

Gene,

It looks like you're mostly still objecting to my lack of mathematical
rigour, even when this is clearly being done in favour of educational
efficiency. I really hoped we had got beyond that.

There's room for both of us (both kinds of definition), really there is.

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> {{A "prime exponent mapping", sometimes shortened to "prime mapping",
> "exponent mapping", "mapping" or simply "map", is a list of numbers
> (integers) enclosed in < ... ] that tell you how a particular
> temperament maps each prime number (up to some limit) to numbers of a
> particular "generator" in that temperament.}}
> 
> This assumes that all such mappings are (equal, and you need to say 
> that) temperaments, which is not true.

How does it assume that? In the case of linear or higher-D
temperaments we have more than one generator. The mapping from primes
to a single one of those generators is still a val isn't it?

> Also, simply calling it 
> a "map" won't work as a specific shorthand, since that already has a 
> well-established meaning, as another word for "function" which is 
> more often used in some contexts ("homomorphic map" being one 
> example.) 

Huh?

Isn't this your definition of "val"?:
Definitions of tuning terms: val, (c) 2001 by Joe Monzo *

It starts: "A map ... ". 

How many other kinds of map do we use in this application of Grassman
algebra, or in tuning theory in general?

> I find "prime exponent mapping" too clumsy, too confusing, 
> and too verbose, and have no plans to use the term.

Sure it's clumsy and verbose, but it's _meaningful_. I won't use it
most of the time either (I'll use "map" or "mapping"), but what could
be more confusing for a newbie than a term that carries absolutely no
meaning for them whatsoever, except maybe as a person's name. (You
reading, Monz?) I should think it would at least be clear from "prime
exponent mapping" that whatever it is, it maps prime exponents to
something (and from "map" that it maps something to something).

> {{The prime numbers here represent frequency ratios.}}
> 
> This is at best confusing; the prime numbers are prime numbers and 
> don't represent anything else. Tuning is another matter.

I completely fail to understand how you could imagine that tuning is
"another matter" in a tuning dictionary. What else could the primes
represent, in a tuning dictionary.

> {{When an interval is represented in the complementary form...}}
> 
> "Complimentary form" is not a good phrase to use here.

I agree, which is why I wrote "compl_e_mentary form".

But assuming you don't like that either, please tell me why? You might
suggest alternatives.

> {{...as a prime-exponent-vector, we can find the number of generators
> corresponding to it in some temperament by multiplying each number in
> the temperament's map by the corresponding number in the vector, and
> adding up the results.}}
> 
> This is assuming the mapping in question defines an equal temperament 
> (and again leaves out the word equal), which is hardly always the 
> case.

As I said, It does not assume equal temperaments at all. It applies
equally well to finding the number of fourth generators for meantone
(or the number of octave "generators").

Perhaps a more valid criticism of this definition is that it excludes
prime mapping _matrices_, since these give you the counts of _all_ the
generators at once. But these can be seen as a stack of prime-mapping
pseudo-vectors (vals) one above the other, right?

So we should extend the definition of prime-exponent-mapping and all
its abbreviations (not including "val"), so that it includes these
matrices. It will usually be clear from the context, and from the
notation, whether one is talking about a (pseudo-)matrix or a
(pseudo-)vector (val). If necessary, one can distinguish them by using
the words "matrix" and "vector".

And then my proposed preamble to the definition of val will need to
say "When applied to tuning it usually represents a prime exponent
mapping for a single generator of a temperament". With the appropriate
links.

> {{In mathematical terms this is called the dot-product, scalar-
> product or inner-product of the map with the vector.}}
> 
> This is unfortunately the case--unfortunate, in that these in most 
> contexts mean a product defined on a single vector space, not on a 
> vector space with its dual. As if that were not enough, "interior 
> product" in the context of exterior algebra has a specialized meaning 
> that we probably don't want to mess with. What about simply calling 
> it the bracket and leaving it at that?

Gene, have you ever heard of the Principle of Parsimony, otherwise
known as Ockham's Razor?

"Entia non sunt multiplicanda praeter necessitatum"

"Do not multiply entities beyond necessity"

I agree it is necessary to distinguish this operation from a "true"
dot-product in pure math and maybe in other applications, but since it
is the only way we're using it in tuning, there is no need to confuse
people with distinctions irrelevant to their application. It's a
tuning dictionary. The actual manipulations of the numbers (the button
presses on the calculator or the formulae in the spreadsheet) are the
same as for a dot product, which some readers may at least have a
vague memory of from high school, or be able to look up in an old
textbook.

If the reader's education proceeds in this area, they will eventually
come to understand such distinctions, but nothing is gained by trying
to include them all from the start.

This is the difference between something that aims to educate or
introduce people to something new, as opposed to a repository of
precise definitions for reference by existing practitioners.

I note that mathworld.com is pretty much one of the latter, which is
why most of its definitions are next-to incomprehensible to a
non-mathematician. I would hope that Monz's tuning dictionary would
not become like that.

In education we start by introducing simplified versions of things.
Often they are _so_ simplified that an experienced practitioner could
be forgiven for being horrified at the _lies_ being told. But that was
one of the geniuses of Richard Feynman as a teacher of one of the most
difficult and heavily mathematical subjects, quantum mechanics. He
knew exactly what lies to tell (simplifications to make), and when,
and how to appeal to the reader's/listener's existing knowledge and
intuitions.

I believe I've managed to avoid telling any actual lies in my proposed
definitions in this thread, although I have of course left many things
unsaid.


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Message: 8442

Date: Fri, 21 Nov 2003 23:41:43

Subject: Re: Finding the complement

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Then the dual must not be the same thing as the Euclidean 
> complement.
> 
> What dual are we talking about?

I was getting my information from the GABLE program and from this 
tutorial:

http://carol.science.uva.nl/~leo/GABLE/tutorial.pdf - Ok *

see page 18.


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Message: 8443

Date: Fri, 21 Nov 2003 02:15:15

Subject: Re: Definition of val etc.

From: Dave Keenan

Gene,

Feel free to give us a pure-math definition of "map" to include along
with the tuning-related stuff in the tuning dictionary entry. Or how
about we just include this link?

Map -- from MathWorld *


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Message: 8445

Date: Fri, 21 Nov 2003 23:45:45

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

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

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


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Message: 8447

Date: Fri, 21 Nov 2003 16:52:34

Subject: Re: "does not work in the 11-limit"

From: Manuel Op de Coul

George wrote:
>I wanted to see if I could create
>midi files (consisting of only a single track) from scratch in Scala
>(which would save me the trouble of calculating and manually
>inserting pitch-bends), which I could then import into Cakewalk (one
>track at a time).

Ah, I assumed you were using Cakewalk to enter the notes more quickly,
but you want to use Scala to enter the notes, and use Cakewalk to adjust
the tuning at places afterwards. Well, this is a use case I hadn't
envisioned, since with Scala you can change the tuning quickly, but
typing note commands is very slowly.

>Your Scala documentation indicates that pitch-bend
>events are minimized, so that you are constantly *changing channels*
>from one note to the next (rather than inserting *pitch-bend events*
>for a single channel).

This is not entirely true anymore, I forgot to update the documentation
for that. There are also possibly program change and parameter change
events involved in channel switching. So minimising pitch bend events
doesn't make sense if it causes many more other messages.

There may be a way to do what you want but I've never tried it.
You can exclude midi channels from being used. So if you exclude all
channels except the first for the first track, then generate the
midi file for that track and for the next track exclude all
channels except the second one, generate that, etc.
I don't see why that wouldn't work.

Generating MTS from .seq files isn't a good solution because it keeps
the channels, but switches the note numbers on a round-robin
basis. That will be even more confusing to look at in Cakewalk :-)

But perhaps still the most efficient solution would be to discard
Cakewalk from the process, and do all changes in one seq file, not
looking at the midi file.

Manuel


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Message: 8448

Date: Fri, 21 Nov 2003 23:44:48

Subject: Re: Finding Generators to Primes etc

From: Paul Erlich

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

yes.

> ~ is ?  Complement?

yes.

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

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

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

Graham just explained that.


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