Tuning-Math Digests messages 8275 - 8299

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Message: 8275

Date: Sat, 15 Nov 2003 00:04:27

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >>>>>>I started with six rationals and ended up with 6 
> >>>>>>integers.  What's the problem?
> >>>>>
> >>>>>Are your integers consecutive?
> >>>> 
> >>>> No, and that's part of the def. of standard val, but what
> >>>> motivates it?
> >
> >what does that have to do with the definition of standard val?
> 
> Sorry, it doesn't.  I forgot the definition doesn't mention
> consecutive.  It's just this particular case.
> 
> Wait... is this true: 'For a scale with card k, if there is
> no standard val with n=k that consistently maps the scale, the
> scale is not a Constant Structure.'

Carl,

Drop the word "standard". There's absolutely no relationship between
"standard" vals and Constant Structure. Sorry if anything I said
mislead you in that direction.


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Message: 8276

Date: Sat, 15 Nov 2003 05:42:38

Subject: Re: Vals?

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:
> 
> > How does that generalise to other than the 5-limit? i.e. vectors 
> with
> > other than 3 components?
> 
> In the 7-limit, the wedge product of two monzos is a 6D wedge product 
> vector, (which is the two intervals are commas gives us on reduction 
> a wedgie for a temperament)

How do you reduce it?

Is there a direct interpretation of the coefficients of the 6D
wedge-product in tuning terms, either before or after the reduction?
As there is in the 3D case?

> wedging it with a monzo again gives us a 
> val. The wedge product of two vals (I'm assuming things are set up 
> the way I define them) gives us, once again, a 6D wedge product 
> vector, (which if the two vals are et vals gives us on reduction a 
> wedgie for a temperament) wedging it with a val again gives us a 
> monzo. This has to be done carefully in terms of basis elements to 
> make the equivalencies work.

I think this is coming a bit too fast for me yet.


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Message: 8277

Date: Sat, 15 Nov 2003 19:33:14

Subject: Re: Vals?

From: Graham Breed

Paul Erlich wrote:
> similarly, if i take the (transpose of the?) val for 12-equal:
> 
> |12 19 28>
> 
> and take the cross product with the val for 22-equal:
> 
> <22 35 51|
> 
> i get the monzo for the diaschisma, the interval that vanishes in 
> both tunings:
> 
>    [-11     4     2]
> 
> again, not sure what's going on notationally, but the numbers 
> work . . .

As Gene's said, this should be written

<12 19 28] ^ <22 35 51] = <-11 4 2]

meaning the wedge product of the two vals is equivalent to that monzo. 
Here's how you write it using my Python module:

 >>> from temper import Wedgie as Val
 >>> (Val((12,19,28))^Val((22,35,51))).complement().flatten()
(-11, 4, 2)

That the interval vanishes in both tunings can be expressed by the 
brakets (or whatever the products are called without complex numbers) 
equalling zero:

<12 19 28 | -11 4 2> = 0

<22 35 51 | -11 4 2> = 0

to check:

 >>> Monzo=Val
 >>> int(Val((22,35,51))^~Monzo((-11,4,2)))
0
 >>> int(Val((12,19,28))^~Monzo((-11,4,2)))
0

(If the module knew the difference between covariant an contravariant 
vectors (as one version did) you wouldn't need that ~ .)

The complement (~ or .complement()) is not the same as a matrix 
transpose.  Which way round you do the wedge product only affects the 
sign of the result.


>>the symbol normally indicates the cross-product, which is extremely 
>>useful in tuning: for example, if i take the monzo for the 
> 
> diaschisma
> 
>>[-4 4 -1>
>>
>>and cross it with the (transpose of the?) monzo for the syntonic 
> 
> comma
> 
>><-11 4 2]
>>
>>i get the val for the et where they both vanish:
>>
>>[12    19    28]
>>
>>not sure how gene would do this notationally, probably i did 
>>something terrible, but without it i could not have made those 
>>charts . . .

 >>> (Monzo((-4,4,-1))^Monzo((-11,4,2))).complement().flatten()
(12, 19, 28)



                   Graham


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Message: 8278

Date: Sat, 15 Nov 2003 00:05:23

Subject: Re: Vals?

From: Dave Keenan

Perhaps I should have said: There's absolutely no relationship between
the "standardness" of the vals and Constant Structure.


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Message: 8279

Date: Sat, 15 Nov 2003 06:33:26

Subject: Re: Vals?

From: monz

hi Gene,


--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> 
> In the 7-limit, the wedge product of two monzos is a
> 6D wedge product vector, (which is the two intervals are
> commas gives us on reduction a wedgie for a temperament)
> wedging it with a monzo again gives us a val. The wedge
> product of two vals (I'm assuming things are set up 
> the way I define them) gives us, once again, a 6D wedge
> product vector, (which if the two vals are et vals gives
> us on reduction a wedgie for a temperament) wedging it
> with a val again gives us a monzo. This has to be done
> carefully in terms of basis elements to make the
> equivalencies work.



i sure wish i knew what the hell this was all about.
especially since my name is being used as a term all thru it.

you guys (Gene, paul, Dave) lost me on this long ago.
but it sure seems interesting.



-monz


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Message: 8281

Date: Sat, 15 Nov 2003 00:32:02

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:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> > > wrote:
> > > 
> > > > If we are told that the mapping is for a tET then _which_ tET 
> it is
> > > > for can be read straight out of the mapping, as the coefficient 
> for
> > > > the prime 2 (the first coefficient). And the generator is 
> simply one
> > > > step of that tET.
> > > 
> > > just wondering why you keep saying "tET" -- 'If we are told that 
> the 
> > > mapping is for a tone equal temperament then . . .' ??
> > 
> > I agree it's awkward. Carl objected so vehemently to EDO and I 
> wanted
> > to reserve ET for the most general term (including EDOs ED3s cETs).
> > Perhaps this would be a misuse of ET. Do we have some other term for
> > the most general category of 1D temperaments, i.e. any single
> > generator temperament whether or not it is an integer fraction of 
> any
> > ratio? I guess "1D-temperament" will do.
> > 
> > > actually, > and < fit together and create a X (as in times) !
> > 
> > Oops. Well we could interpret that as the matrix-product as opposed 
> to
> > the scalar-product (dot-product), but I don't know of any meaning 
> for
> > that in tuning.
> 
> the symbol normally indicates the cross-product, which is extremely 
> useful in tuning: for example, if i take the monzo for the diaschisma
> 
> [-4 4 -1>
> 
> and cross it with the (transpose of the?) monzo for the syntonic comma
> 
> <-11 4 2]

Should have been [-11 4 2>

> i get the val for the et where they both vanish:
> 
> [12    19    28]

Now you could write <12 19 28]

That's magic! I never knew that! But of course if someone ever said it
before I wouldn't have understood it since I didn't have a clue what a
val was.

So [-4 4 -1> (x) [-11 4 2> = <12 19 28]

Where (x) is a rather poor ASCII version of the cross-product
operator. Is there a standard ASCII version of that. And while we're
at it how about an ASCII version of the matrix transpose operator.
"^T" ? It's obviously bad having letters in operator symbols since
they invite confusion with variables.

For non-math types:

The cross-product of vectors
<a1 a2 a3} and <b1 b2 b3] 
is [a2b3-a3u2 a3b1-a1b3 a1b2-a2b1>


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Message: 8282

Date: Sat, 15 Nov 2003 07:16:21

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> i sure wish i knew what the hell this was all about.
> especially since my name is being used as a term all thru it.
> 
> you guys (Gene, paul, Dave) lost me on this long ago.
> but it sure seems interesting.

I think I'll eventually be able to explain it in a way you can
understand it. But it wouldn't do to try until I'm sure I've actually
got it all sorted out myself. Paul could probably do the job too.


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Message: 8284

Date: Sat, 15 Nov 2003 00:34:56

Subject: Re: Vals?

From: Dave Keenan

I should probably have made it clearer by writing:

The cross-product of vectors

<a1 a2 a3} 

and 

<b1 b2 b3] 

is 

[a2*b3-a3*b2 a3*b1-a1*b3 a1*b2-a2*b1>

How does that generalise to other than the 5-limit? i.e. vectors with
other than 3 components?


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Message: 8286

Date: Sat, 15 Nov 2003 01:05:29

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> similarly, if i take the (transpose of the?) val for 12-equal:
> 
> |12 19 28>
> 
> and take the cross product with the val for 22-equal:
> 
> <22 35 51|
> 
> i get the monzo for the diaschisma, the interval that vanishes in 
> both tunings:
> 
>    [-11     4     2]

I never knew this either! 

Although your use of notation sucks, as you suggested it might.

I'd write 

<12 19 28] (x) <22 35 51] = [-11 4 2>

I don't think the bra and ket notation was particularly designed to
help us with what can be crossed with what and what the result is.
Although we can see that ]< and >[ are both cross products while ][
(which can be relaced by | is the dot product and >< is the matrix
product.


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Message: 8288

Date: Sat, 15 Nov 2003 01:07:53

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:
> 
> > > let me just repeat dave and say that this has *nothing* to do 
> with 
> > > the definition of vals -- it's a separate question that you can 
> > > safely ignore if you want to understand vals.
> > 
> > I was quite aware of that.
> 
> you should be, because as i said, i was just repeating you!
> 
> > I was merely trying to answer Carl's questions.
> 
> me too!

Sorry. I read it as "just let me repeat, dave". i.e. I thought you
were talking to me. Duh.


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Message: 8290

Date: Sat, 15 Nov 2003 01:33:27

Subject: Re: Vals?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> > When we go beyond 1D
> > temperaments we have prime-mappings which are matrices (one row per
> > generator) and we multiply that by the transpose of a ratio's
> > prime-exponent-vector (monzo) to get a vector giving the count of 
> each
> > generator.
> 
> can you show an example? obviously i'm plenty confused as to how to 
> correctly notate these things . . .

Me too, since I want it to be generalisable to matrices, and it seems
Gene doesn't care about that.

Here's a 5-limit mapping matrix for meantone (call it "M") in one
possible notation.

<1  2  4]
<0 -1 -4]

or on one line <1 2 2; 0 -1 -4]

The first row related the primes to the octave generator, the second
row relates them to the fourth generator. In Gene's terminology, each
row is a val.

And let "a" be a prime-exponent-vector for some ratio, say 5/3
[0 -1 1>

By treating M as a single matrix instead of a pair of vectors (vals)
we can just use software that has matrix operations (even Excel) and write
M*a (in that would be Excel {=MMULT(M,a)}).

However, the fine details are that "a" has to be a column vector for
this to work, and the result will be a column vector. If we want them
to be rows we have to write (M*aT)T where T is the transpose operator.
In Excel {=TRANSPOSE(MMULT(M,TRANSPOSE(a)))}

The result is  <2, -3> meaning 2 octaves up and 3 fourth-generators down.


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Message: 8291

Date: Sat, 15 Nov 2003 20:10:13

Subject: Re: Vals?

From: Graham Breed

monz wrote:

> i sure wish i knew what the hell this was all about.
> especially since my name is being used as a term all thru it.
> 
> you guys (Gene, paul, Dave) lost me on this long ago.
> but it sure seems interesting.

We've established that the wedge product of two monzos corresponds to 
the temperament in which they vanish.  So, with Gene's notation, a comma 
and diaschisma give 12-equal.

|-4 4 -1> ^ |-11 4 2> = <12 19 28|

Well, on top of that, you can temper out 36:35, or |2 2 -1 -1>

|-4 4 -1> ^ |-11 4 2> ^ |2 2 -1 -1> = <12 19 28 34|

Which, to check with my Python module:

 >>> from temper import Wedgie as Monzo
 >>> syntonic = Monzo((-4,4,-1))
 >>> diaschisma = Monzo((-11,4,2))
 >>> septimal = Monzo((2,2,-1,-1))
 >>> (syntonic^diaschisma^septimal).complement().flatten()
(12, 19, 28, 34)

If you only temper out two commas, you get a linear temperament.

|-4 4 -1> ^ |2 2 -1 -1> = 7-limit meantone.

I don't know how to write linear temperaments as bras, but there are 
some things you can show.  For example, an octave equivalent mapping is 
like tempering out the octave.

 >>> octave = Monzo((1,0,0))
 >>> (syntonic^septimal^octave).complement().flatten()
(0, 1, 4, -2)

which means

|-4 4 -1> ^ |2 2 -1 -1> ^ |1 0 0> = <0 1 4 -2|

and (1 4 -2) is the octave-equivalent mapping for this particular 
version of meantone, where C-Bb approximates 4:7, rather than C-A#.  For 
the more accurate one, you can do

|-4 4 -1> ^ |1 2 -3 1> ^ |1 0 0> = <0 1 4 10|


                   Graham


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Message: 8298

Date: Sat, 15 Nov 2003 16:47:21

Subject: Re: Vals?

From: Carl Lumma

>But yeah. What do others think? Square brackets or vertical
>bars (pipes)?

I don't care, but I think we should standardize.

-Carl


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