>>> 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).
Message: 8482 - Contents - Hide Contents
Date: Fri, 21 Nov 2003 23:39:04
Subject: Re: Finding Generators to Primes etc
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> It would be cool if you or someone could give an example of the
>> number crunching used to, say, get 81/80 from 12&19 Temperaments.
>> Can this be done using matrices? I know the wedge product of the
>> comma is equal to the wedge product of the val.. but still don't
see
>> how you get from 12&19 TO 81/80...
>
> The other Paul demonstrated this recently -- you take the cross
> product of two vals. So
>
> < 12 19 28 |
>
> is h12 and
>
> < 19 30 44 |
>
> is h19. Except there's something about using the transpose of
> one of them to get it into a form where the cross product will
> give you a monzo. Which in this case is
>
> | -4 4 -1 > = 81/80
>
> Do I have that right, guys?
The numbers are right, but you don't use the transpose of one of them.
Message: 8483 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 12:03:47
Subject: Re: Finding the wedge product?
From: Dave Keenan
That's great Graham. I think I get it now. Let me try feeding it back
in a different way so you can tell me if I've got it right, and so
others may have another chance at following it.
Lets first take the simplest case worth considering. The wedge product
of two 3-limit (2D) vectors.
[a1 a2> ^ [b1 b2>
The procedure is to first list every product of a coefficient from A
with a coefficient from B, i.e. their ordinary scalar products. So
with 2 coefficients in each there will be 2x2 = 4 products to
consider, a1*b1, a1*b2, a2*b1, a2*b2.
As you calculate each product, combine the indices of the two
coefficients to make a compound index for it. It is important to keep
the indices in their original order at this stage. So we have
(a1*b1)11, (a1*b2)12, (a2*b1)21, (a2*b2)22. The numbers after the
parentheses are the compound indexes. This is not proper mathematical
notation, but just a way to keep track of our intermediate results.
There are certain rules about what to to with each product now,
depending on its compound index. There are 3 possibilities:
1. If the indexes have a digit in common then ignore it. Just throw
the product away. So we throw away (a1*b1)11 and (a2*b2)22.
2. Otherwise if the digits in the compound index are already in
alphabetical order, do nothing. So (a1*b2)12 is just fine as it is.
3. Otherwise if they are not in alphabetical order then put them in
alphabetical order, but you must do this in simple stages. At each
stage you are only allowed to make two digits swap places, and every
time you do this you must negate the product. So if, by the time the
digits are in alphabetical order you have swapped digits an odd number
of times, then it will be negated. If an even number of swaps were
required then it will remain as it was. So (a2*b1)21 becomes (-a2*b1)12.
Now find any products that have the same index and add them together.
So we have only (a1*b2 - a2*b1)12.
Now list all these sums in alphabetical order of their indices, inside
as many brackets as the sum of the number of brackets in the two
arguments, and pointing in the same direction. I assume the wedge
product is only defined for values having their brackets pointing the
same way.
So our answer is
[[(a1*b2 - a2*b1)>>
Now lets try something more messy. A 7-limit (4D) vector wedged with a
7-limit bivector. This might represent combining a third comma with
two that have already been combined, as an intermediate result on the
way to finding the ET mapping where these all vanish.
[a1 a2 a3 a4> ^ [[b12 b13 b14 b23 b24 b34>>
We first make the list of products of all pairs, with their compound
indices.
product index
a1*b12 112
a1*b13 113
a1*b14 114
a1*b23 123
a1*b24 124
a1*b34 134
a2*b12 212
a2*b13 213
a2*b14 214
a2*b23 223
a2*b24 224
a2*b34 234
a3*b12 312
a3*b13 313
a3*b14 314
a3*b23 323
a3*b24 324
a3*b34 334
a4*b12 412
a4*b13 413
a4*b14 414
a4*b23 423
a4*b24 424
a4*b34 434
Now we get rid of all those with two digits the same. Of course once
you've got the idea, you wouldn't even bother writing them down in the
first place. This leaves.
product index
a1*b23 123
a1*b24 124
a1*b34 134
a2*b13 213
a2*b14 214
a2*b34 234
a3*b12 312
a3*b14 314
a3*b24 324
a4*b12 412
a4*b13 413
a4*b23 423
Now we do the old switcheroo on the ones that aren't already in
alphabetical order, and negate them if we have to do an odd number of
switches. And we end up with:
product index
a1*b23 123
a1*b24 124
a1*b34 134
-a2*b13 123
-a2*b14 124
a2*b34 234
a3*b12 123
-a3*b14 134
-a3*b24 234
a4*b12 124
a4*b13 134
a4*b23 234
Now we sum the products with the same index.
product index
a1*b23 + a3*b12 - a2*b13 123
a1*b24 - a2*b14 + a4*b12 124
a1*b34 - a3*b14 + a4*b13 134
234
Now we list them in alphabetical order of index inside the right
number of brackets.
[[[a1*b23+a3*b12-a2*b13 a1*b24-a2*b14+a4*b12 a1*b34-a3*b14+a4*b13
a2*b34-a3*b24+a4*b23>>>
Voila!
Message: 8484 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 13:58:47
Subject: Re: Definition of val etc.
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> I wish you guys wouldn't argue over the inclusion of the term
> "val". Dave, it isn't this that causes a problem.
I'm glad that it didn't ever cause a problem for you, but it sure as
hell cause one for me. It had me bluffed for a very long time, so I'm
guessing it may be a problem for others in future too.
>> [0 -1 1>
>>
>> We can calculate the individual dot-products, for each row in turn, or
>> we can use software that has matrix operations (e.g. Microsoft Excel)
>> and simply find the matrix-product of the mapping matrix with the
>> transpose of the exponent vector.
>
> Perfect example of what not to do. Introduce the word "transpose"
> without saying what the hell it is. It doesn't matter what word you
> use if you don't explain it.
Well the transpose is only relevant if you're going to do it using
matrix operations in software like Mathematica, Maple, Matlab, Octave
(free) or Excel. And if you're doing this you can read their help to
find out about transpose.
I've already told you you can do it by repeated dot products, which I
explained how to do (one of) above. But I should maybe give more
detail on this.
It's would be easy enough to explain transpose in this dictionary
entry if you really think I should. However, if I have to explain
transpose here, then presumably I have to explain "matrix product"
too? This would be more tedious.
However, I suppose we could give the Excel formulae in a Monz
dictionary entry, considering it to be a sort of lowest common
denominator among math tools.
The transpose is a purely "cosmetic" operation. There's no arithmetic
involved. It simply means to rotate the vector's list of numbers by 90
degrees on the page, so what was a left to right "row" vector is now a
top to bottom "column" vector or vice versa. This only matters when a
vector is interpreted as a matrix. Matrix operations give different
results depending whether a vector is treated as a 1 by n matrix or an
n by 1 matrix.
>
>> <1 2 4] [ 0 <2
>> <0 -1 -4] -1 = -3>
>> 1>
>>
>> The result is a column vector <2 -3>
>
> And how did you get that result?
Good question. I agree this could use more detail.
Let's try writing the matrix equation like this.
<1 2 4] * [ 0 -1 1>+ = <2 -3>+
<0 -1 -4]
The infix "*" is the matrix product operator and the postfix "+" is
the transpose. Just think of the "+" as a little picture of a vector
being rotated 90 degrees on the page, or as a superscript lowercase
"t" for "transpose".
In Excel we could select two cells one-above-the-other, where we want
the result, and type the formula =MMULT(m,TRANSPOSE(v)) and hit
Ctrl-Shift-Enter. You would of course replace m and v by the
spreadsheet ranges corresponding to the mapping matrix and exponent
vector respectively.
Now lets look at doing it by hand, without using matrix multiplication.
Remember the dot product of a mapping with a vector is defined as
<a1 a2 ... an] . [b1 b2 ... bn>
= a1*b1 + a2*b2 + ... an*bn
Now with multiple generators we have
<1 2 4] * [ 0 -1 1>+
<0 -1 -4]
Which can simply be done as two dot products
<1 2 4] . [ 0 -1 1> = 0 - 2 + 4 = 2
<0 -1 -4] . [ 0 -1 1> = 0 + 1 - 4 = -3
For convenience we can group these as a vector in angle brackets
<2 -3>. And in case you've forgotten by now, this represents 2 octave
"generators" up and 3 tempered-perfect-fourth generators down, as the
meantone approximation of the ratio 5/3.
>
>
> This stuff clearly isn't that hard unless you make it hard. Mainly
> by *leaving out* all-important definitions and examples.
I totally agree.
Message: 8485 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 14:37:06
Subject: Re: Finding Generators to Primes etc
From: Dave Keenan
--- 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?
Pretty much. Right answer anyway. Although I think most of us agree we
should use the square brackets instead of the vertical bar, except if
we write a "dot product" as < ... | ... >. But I have to say I can't
see a lot of point in writing them that way.
Forget about transpose here. You take the wedge product of the two ET
mappings to get a "bi-mapping" (bi-val) << -1 -4 -4 ]] and then you
take the complement of that to get the exponent vector for the comma
that vanishes. In 3D only (i.e. 5-limit), that whole sequence happens
to be the same as simply taking the cross-product.
> ^ is the wedge product.
Yes.
For 5-limit this is
<a1 a2 a3] ^ <b1 b2 b3]
= << a1*b2-a2*b1 a1*b3-a3*b1 a2*b3-a3*b2 ]]
How to calculate it in general is described in the thread starting here
Yahoo groups: /tuning-math/message/7854 * [with cont.]
> ~ is ?
The complement.
For 5-limit bi-mappings it is simply
~<<c12 c13 c23]]
= [c23 -c13 c12> (reverse the order and negate the middle one)
How to calculate it in general, is described in the thread starting here
Yahoo groups: /tuning-math/message/7845 * [with cont.]
> So the wedging with
> a complement is the same as crossing? Please answer each question,
> I'm just guessing.
Not quite. The complement of the wedging is the same as crossing (but
crossing is only defined for 3D (5-limit)).
Sorry if I'm repeating stuff other people have already told you.
> Oh, and can anyone explain in one sentence why I should care about
> bi- and tri- things?
Because bi- and tri- mappings (vals) represent linear and planar
temperaments respectively, in a way that is independent of any
particular choice of generators or vanishing commas.
Message: 8486 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 22:32:48
Subject: Re: Finding the wedge product?
From: Carl Lumma
>Oops. Something went missing near the end there. It should have been:
Can you post the entire corrected thing?
-C.
Message: 8487 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 00:17:51
Subject: Re: Definition of val etc.
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
> wrote:
>
>> Please give a non-trivial example of the use of a val in regard to
>> tuning, where it doesn't have anything to do with any temperament.
>
> This I already did--the exponent for some particular prime (the
> p-adic valuation for prime p) provides an example.
BWDIMAAATT?
> Another would be
> the vals which turn up in connection with notation systems such as
> you have been working on.
Yes. They could be applied there, and you're right that they would not
describe a temperament in the sense of a set of pitches, but rather a
set of symbols for pitch, but this is so closely related to
temperament that I have no problem with using temperament terminology
and referring to the symbol components (shafts, flags and accents) as
"generators" of the symbols. In fact the analogy is so obviously
beneficial to understanding that I've already done so.
> "Generator" should only come in as an example, not as a part of the
> definition. Otherwise, the defintion isn't correct.
It wouldn't be correct as a definition of the purely abstract
mathematical object. But since this is the interpretation it will have
in 99% of cases in tuning, that's the best way to explain it.
It is _far_ easier for most people to first understand such an object
according to how it is _used_ in their area of application, and worry
about the abstraction later, for example if they need to apply it in
some other area. But even then, they may work directly from one
application to the other by analogy.
Pure math object
^ \
/ \
abstraction application
/ \
/ v
Application 1 --analogy--> Application 2
Pure mathematics, which attempts to deal directly with the abstract
objects, is an extremely noble and valuable pursuit, but you shouldn't
assume that everyone is interested in being immersed in all its
details and terminology. Most just want you to give them some tools
they can use in their application area and tell them how to use them,
in the most obvious terms (terms that come from their application area).
How do you think a car mechanic would feel if, on being presented with
a spanner by its designer, he was told:
"Now you probably think this is a spanner. But spanners are tools
whose only purpose is the undoing or doing-up of nuts and bolts. This
tool is in fact far more general than that, and is capable of rotating
any objects having two opposing parallel faces that fit neatly within
its jaws, whether or not they have a helical threads. There wasn't any
short name for opposing-parallel-face-rotator in the literature so
we've coined the term "velma". This may sound like a randomly chosen
girl's name, but it actually comes from "velocity matching"."
The mechanic would probably be tempted to show the spanner designer
certain other "applications" of his spanner that he might not have
thought of. :-)
> I meant tuning maps--that is, for example, maps from temperaments to
> real numbers, determined by giving a specific value to the
> generators, which define a tuning. Even more concretely, maps to
> Hertz.
OK. Good point. So this means the definition we're working on here
(for the val as most commonly applied to tuning) should not be for
"map" (as Monz suggested), but for something more specific like "prime
exponent mapping". But we should still mention that it will often be
shortened to "map" when the meaning is clear from the context.
>>> Multivals define maps on
>>> corresponding multimonzos, which is a specifically Grassman
> algebra
>>> fact for you.
>>
>> OK. But I don't see any harm in calling these "multimaps".
>
> I do. It sounds as if it isn't a map, perhaps because it is a multi-
> valued function (which isn't, strictly speaking, a function at all.)
I assure you this sort of consideration is unlikely to bother anyone
on tuning-math, particularly when we're also talking about
"multimonzos". And since, as you've pointed out, "multi-valued
function" is nonsense, it shouldn't even detain a pure mathematician
for very long.
>> Yes. But we've never had any urge to refer to any of these by the
> term
>> "map". The terms "indexing" or "function" serve us just fine for
>> these. So there are no name conficts with "map" there that I can
> see.
>
> Why do you insist on rewriting standard mathematical terminology?
> That is asking for confusion.
I'm not rewriting it. I'm just allowing a general term to be used to
refer to one of its specific applications, when used in an application
area in which this application covers 99% of its applications.
You've agreed that they actually _are_ maps. Your complaint is that
this is not specific enough. But when it's clear from the context,
exactly what _kind_ of maps they are, where's the harm in abbreviating
in this way?
>>> Why do we need to keep arguing this stuff?
>>
>> Because you have mathematical knowledge that I don't have, and I
> have
>> some insights into how to explain things to non-mathematicians, that
>> you apparently don't have.
>
> It doesn't answer my question. Why do you seem hell-bent on tossing
> out standard mathematical terminology?
"val" and "icon" are not standard mathematical terminology. What other
terminology do you see me as "tossing out"?
>>> It's damned confusing. Is the domain the prime numbers, or some
> prime
>>> numbers?
>>
>> No question there for most tuners. We don't usually try to compute
>> things with infinite numbers of coefficients. :-)
>
> So what is it we magically determine the domain to be--some prime
> numbers? That isn't the correct answer!
So why don't you tell us what is? You know it gets very tedious when
you just say "that's wrong" and don't deign to tell us why, or suggest
something better, until someone specifically ask you to do so. I am
tempted to simply ignore such statements in future.
I say, near the beginning of my proposed definition, "prime numbers
(up to some limit)". What more do you want? Please explain.
>>> Is it the rational numbers, and does the map give prime
>>> expondents (which would mean they are p-adic valuations?) Is the
>>> mapping *from* prime exponents, and if so, how and to what?
>>
>> Yes. That's true. They may wonder if it's a mapping _from_ prime
>> exponents, or _to_ prime exponents, and what's on the other side.
> But
>> this still seems to be getting us a lot closer to the intended
> meaning
>> than a randomly chosen girl's name would. :-)
>
> "Val" comes from "valuation", and that *is* getting us nearer to
> where we want to be.
Most readers of Monz's dictionary will not have a clue what a
"valuation" is either. Feel free to include this information and a
link to the Mathworld definition of valuation, in your definition of
"val". But I've read the definition of valuation on Mathworld and am
none the wiser. I really don't think an understanding of valuations
will add anything to the ability of someone to use these things for
tuning calculations.
> What you hear are sounds, which if you are lucky are more or less
> periodic and have a frequency expressible in Hertz. There are various
> mappings involved here--p-limit rational numbers to abstract
> temperaments, temperaments (via a tuning map) to real numbers, real
> numbers representing intervals to Hertz--and you are trying to gum
> them all together into one ugly, confusing mess with this idea that
> prime numbers are a ratio of frequencies.
Huh?
When did I say prime numbers _are_ a ratio of frequencies? I'm simply
saying that's what they stand for in this application. i.e. they have
units of hertz/reference_hertz.
>> It just occurred to me. If vals are pseudo-vectors then when you
> stack
>> them up to make a matrix, surely it must actually be a pseudo-
> matrix.
>
> Who told you vals were pseudo-vectors?
I probably misinterpreted this
Pseudovector -- from MathWorld * [with cont.]
It says the cross product of two vectors is a pseudovector. Is it only
3D vals that are pseudovectors?
Message: 8488 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 08:37:20
Subject: Re: Finding Generators to Primes etc
From: Carl Lumma
>Not quite. The complement of the wedging is the same as crossing (but
>crossing is only defined for 3D (5-limit)).
Is the cross product really only defined, for anything, for 3-item
vectors?
>Sorry if I'm repeating stuff other people have already told you.
Not at all; thanks for the references.
>> Oh, and can anyone explain in one sentence why I should care about
>> bi- and tri- things?
>
>Because bi- and tri- mappings (vals) represent linear and planar
>temperaments respectively, in a way that is independent of any
>particular choice of generators or vanishing commas.
And you're the first to answer this.
-C.
Message: 8489 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 22:35:32
Subject: Re: Definition of val etc.
From: Carl Lumma
>> >ranspose
>A very good point which had completely slipped my mind in all this
>heavy mathematics.
It isn't usually used as a noun in music. And where there are
collisions the math terminology should probably be favoured because
it's more precise.
-Carl
Message: 8490 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 00:19:40
Subject: Re: Finding the complement
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
> wrote:
>> --- 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.
>
> You're probably doing something wrong, then. This is correct
> according to both the GABLE tutorial and the program itself.
Can you give a URL for this GABLE tutorial? Does it use alphabetical
ordering of indices?
Message: 8491 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 08:56:31
Subject: Re: Definition of val etc.
From: Carl Lumma
>>> >0 -1 1>
>>>
>>> We can calculate the individual dot-products, for each row in turn, or
>>> we can use software that has matrix operations (e.g. Microsoft Excel)
>>> and simply find the matrix-product of the mapping matrix with the
>>> transpose of the exponent vector.
>>
>> Perfect example of what not to do. Introduce the word "transpose"
>> without saying what the hell it is. It doesn't matter what word you
>> use if you don't explain it.
>
>Well the transpose is only relevant if you're going to do it using
>matrix operations in software like Mathematica, Maple, Matlab, Octave
>(free) or Excel.
Then you need to say so.
>And if you're doing this you can read their help to
>find out about transpose.
I typed "transpose" into Excel help and got back this single result:
>TRANSPOSE(array)
>
>Array is an array or range of cells on a worksheet that you want to
>transpose. The transpose of an array is created by using the first row
>of the array as the first column of the new array, the second row of
>the array as the second column of the new array, and so on.
Is that right for matrices too?
>I've already told you you can do it by repeated dot products, which I
>explained how to do (one of) above. But I should maybe give more
>detail on this.
Maybe.
>It's would be easy enough to explain transpose in this dictionary
>entry if you really think I should. However, if I have to explain
>transpose here, then presumably I have to explain "matrix product"
>too? This would be more tedious.
I, for one, have no idea what a "matrix product" is.
>However, I suppose we could give the Excel formulae in a Monz
>dictionary entry, considering it to be a sort of lowest common
>denominator among math tools.
I'd prefer to actually know how to do these things by hand.
>Let's try writing the matrix equation like this.
>
><1 2 4] * [ 0 -1 1>+ = <2 -3>+
><0 -1 -4]
>
>The infix "*" is the matrix product operator and the postfix "+" is
>the transpose. Just think of the "+" as a little picture of a vector
>being rotated 90 degrees on the page, or as a superscript lowercase
>"t" for "transpose".
>
>In Excel we could select two cells one-above-the-other, where we want
>the result, and type the formula =MMULT(m,TRANSPOSE(v)) and hit
>Ctrl-Shift-Enter. You would of course replace m and v by the
>spreadsheet ranges corresponding to the mapping matrix and exponent
>vector respectively.
>
>Now lets look at doing it by hand, without using matrix multiplication.
>
>Remember the dot product of a mapping with a vector is defined as
>
><a1 a2 ... an] . [b1 b2 ... bn>
>= a1*b1 + a2*b2 + ... an*bn
>
>Now with multiple generators we have
>
><1 2 4] * [ 0 -1 1>+
><0 -1 -4]
>
>Which can simply be done as two dot products
>
><1 2 4] . [ 0 -1 1> = 0 - 2 + 4 = 2
>
><0 -1 -4] . [ 0 -1 1> = 0 + 1 - 4 = -3
>
>For convenience we can group these as a vector in angle brackets
><2 -3>. And in case you've forgotten by now, this represents 2 octave
>"generators" up and 3 tempered-perfect-fourth generators down, as the
>meantone approximation of the ratio 5/3.
Ah, so the "matrix product" is a pairwise dot product of sorts?
-Carl
Message: 8492 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 00:54:16
Subject: Re: Finding Generators to Primes etc
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> I am thinking of buying both Mathematica and Maple, even though
Maple
>> is fairly expensive, I imagine.
>
> Both are very expensive, unless you're a student at a university.
>
> Mathematica makes Maple look like a joke, if you ask me.
Geometers and analysts tend to prefer Mathematica, and algebraists
and number theorists Maple. Set theorists don't give a good goddamn.
Maple certainly isn't a joke.
Message: 8493 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 09:00:37
Subject: Re: Finding the wedge product?
From: Carl Lumma
>That's great Graham. I think I get it now. Let me try feeding it back
>in a different way so you can tell me if I've got it right, and so
>others may have another chance at following it.
Thanks Dave. Can someone confirm this? I'm about to take it as
Gospel.
-C.
Message: 8494 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 01:07:01
Subject: Re: Finding Generators to Primes etc
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> no, look at the parentheses. the complement of the wedge product is
> the cross product (when you're dealing with a 3 dimensional
problem).
Not exactly. A cross product takes vectors to vectors (or
pseudovectors, if you are a physicist) and in fact a three
dimensional real vector space with cross product is the real Lie
algebra o(3). The complement of a wedge product of bra vectors is a
ket vector, and conversely.
Message: 8495 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 18:36:15
Subject: Re: Finding Generators to Primes etc
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>
>> no, look at the parentheses. the complement of the wedge product
is
>> the cross product (when you're dealing with a 3 dimensional
> problem).
>
> Not exactly. A cross product takes vectors to vectors (or
> pseudovectors, if you are a physicist) and in fact a three
> dimensional real vector space with cross product is the real Lie
> algebra o(3). The complement of a wedge product of bra vectors is a
> ket vector, and conversely.
*hands thrown up in air*
so why "Not exactly"???
Message: 8496 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 01:30:14
Subject: Re: Definition of val etc.
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
> It wouldn't be correct as a definition of the purely abstract
> mathematical object. But since this is the interpretation it will
have
> in 99% of cases in tuning, that's the best way to explain it.
Why not give a correct definition, and then say most of the time we
are talking about things relating to temperaments? By the way, it
occurs to me that Fokker blocks are another case where vals turn up
in ways not obviously related to temperaments.
>
> It is _far_ easier for most people to first understand such an
object
> according to how it is _used_ in their area of application, and
worry
> about the abstraction later, for example if they need to apply it in
> some other area.
It's certainly not easier for mathematicians, who are going to focus
on the definition a little like the way a lawyer focuses on the exact
wording of the law. With any luck, most readers won't be
mathematicians, or even lawyers, of course.
> There wasn't any
> short name for opposing-parallel-face-rotator in the literature so
> we've coined the term "velma". This may sound like a randomly chosen
> girl's name, but it actually comes from "velocity matching"."
The story I heard was that Andre Weil called adeles "adeles" because
that was the name of his girlfriend. I do not now nor ever have had a
girlfriend named "Val"; it really did derive from "valuation".
> The mechanic would probably be tempted to show the spanner designer
> certain other "applications" of his spanner that he might not have
> thought of. :-)
>
>> I meant tuning maps--that is, for example, maps from temperaments
to
>> real numbers, determined by giving a specific value to the
>> generators, which define a tuning. Even more concretely, maps to
>> Hertz.
>
> OK. Good point. So this means the definition we're working on here
> (for the val as most commonly applied to tuning) should not be for
> "map" (as Monz suggested), but for something more specific
like "prime
> exponent mapping". But we should still mention that it will often be
> shortened to "map" when the meaning is clear from the context.
Only if it could also be shorted to "function." That is, the specific
val in question is "a map", but the set of vals is a subset of the
class of maps.
> I assure you this sort of consideration is unlikely to bother anyone
> on tuning-math, particularly when we're also talking about
> "multimonzos". And since, as you've pointed out, "multi-valued
> function" is nonsense, it shouldn't even detain a pure mathematician
> for very long.
"Multi-valued function" isn't nonsense, but terminology of long
standing in complex analysis. What the hell a multimap is I don't
know, but a multival we could give a defintion to.
>>> Yes. But we've never had any urge to refer to any of these by
the
>> term
>>> "map". The terms "indexing" or "function" serve us just fine for
>>> these. So there are no name conficts with "map" there that I
can
>> see.
>>
>> Why do you insist on rewriting standard mathematical terminology?
>> That is asking for confusion.
>
> I'm not rewriting it.
Of course you are. You are taking standard usage and tossing it into
the trash, and replacing it with the Dictionary According to Dave,
and I object.
> You've agreed that they actually _are_ maps. Your complaint is that
> this is not specific enough. But when it's clear from the context,
> exactly what _kind_ of maps they are, where's the harm in
abbreviating
> in this way?
Fine. Let's call the sine function simply "function", since it is a
function.
> "val" and "icon" are not standard mathematical terminology. What
other
> terminology do you see me as "tossing out"?
"Map".
>> So what is it we magically determine the domain to be--some prime
>> numbers? That isn't the correct answer!
>
> So why don't you tell us what is? You know it gets very tedious when
> you just say "that's wrong" and don't deign to tell us why, or
suggest
> something better, until someone specifically ask you to do so. I am
> tempted to simply ignore such statements in future.
The domain is what I've repeatedly said it was--normally the p-limit
positive rational numbers, or sometimes if one likes (but let's avoid
that) all the positive rational numbers, or in some cases some other
group of rational numbers (eg. generated by {2,3,7}.)
>> When did I say prime numbers _are_ a ratio of frequencies? I'm
simply
> saying that's what they stand for in this application. i.e. they
have
> units of hertz/reference_hertz.
They do not. They can be *mapped* to frequencies, if you give a
mapping. When you do this, you may find your mapping sends 3 not to a
frequency 3 times the base frequency, but to some other freqency. You
may also find it convenient (and we do find it convenient when
discussing linear, etc temperaments) not to map to frequencies or to
any other real numbers (cents, etc) at all.
> It says the cross product of two vectors is a pseudovector. Is it
only
> 3D vals that are pseudovectors?
Argh. Let's leave pseudovectors out of it.
Message: 8497 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 18:43:52
Subject: Re: Definition of val etc.
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
>> It says the cross product of two vectors is a pseudovector. Is it
> only
>> 3D vals that are pseudovectors?
>
> Argh. Let's leave pseudovectors out of it.
WHAT!!! Why pull the rug out from under me? I wish you had commented
when I posted this:
Yahoo groups: /tuning-math/message/7798 * [with cont.]
I took this as an important step in my learning about bra and ket
vectors. So I should forget about it? And if so, why?
Message: 8498 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 01:31:50
Subject: Re: Finding Generators to Primes etc
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> Geometers and analysts tend to prefer Mathematica, and algebraists
> and number theorists Maple. Set theorists don't give a good
goddamn.
> Maple certainly isn't a joke.
Full disclosure--the Maple code for finding Galois groups is partly
based on stuff I did.
Message: 8499 - Contents - Hide Contents
Date: Sat, 22 Nov 2003 18:47:12
Subject: Re: Definition of val etc.
From: monz
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>
>>> We can calculate the individual dot-products, for
>>> each row in turn, or we can use software that has
>>> matrix operations (e.g. Microsoft Excel) and simply
>>> find the matrix-product of the mapping matrix with
>>> the transpose of the exponent vector.
>>
>> Perfect example of what not to do. Introduce the word
>> "transpose" without saying what the hell it is. It
>> doesn't matter what word you use if you don't explain it.
>
> Well the transpose is only relevant if you're going to
> do it using matrix operations in software like Mathematica,
> Maple, Matlab, Octave (free) or Excel. And if you're doing
> this you can read their help to find out about transpose.
are we still hashing out what i should put into the Dictionary?
i hope so ... unfortunately i'm understanding very little
of what's been posted here in the last week.
anyway, let's please be careful about using the word
"transpose" in these definitions. it already has a firmly
established meaning to musicians, and you guys are using
a different (mathematical) definition of it now.
-monz
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