inverse of matrix --> for what?

As Paul stated recently in a Tuning List post, the
three unison-vectors 50:49, 64:63, and 245:243 define
22-EDO tuning.

Rewriting those as a (3^x)*(5^y)*(7^z) matrix, we get:

matrix
| 0 2 -2 |
|-2 0 -1 |
|-5 1 2 |

Using Microsoft Excel's "minverse" function, as explained
in Graham's webpage:
http://x31eq.com/matritut.htm

decimal inverse
| 0.045454545 -0.272727273 -0.090909091 |
| 0.409090909 -0.454545455 0.181818182 |
|-0.090909091 -0.454545455 0.181818182 |

Excel's "mdeterm" function gives 22 as the determinant of
the original matrix. Multiplying the inverse of the matrix
by the determinant gives the inverse as fractional parts of 22:

fractional inverse
| 1 -6 -2 | * 1
| 9 -10 4 | --
|-2 -10 4 | 22

My questions: what does this inverse explain?
What purpose does it serve?

You all know that I prefer dealing with exact fractional numbers,
if they exist, rather than approximate floating-point decimals.
So why is this fractional inverse matrix useful?

Do these integers tell us something about 22-EDO?
Or about 22-EDO's representation of the prime-factors?

????

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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In-Reply-To: <002301c18678$e1e58f00$af48620c@dsl.att.net>
monz wrote:

> Excel's "mdeterm" function gives 22 as the determinant of
> the original matrix. Multiplying the inverse of the matrix
> by the determinant gives the inverse as fractional parts of 22:
>
> fractional inverse
> | 1 -6 -2 | * 1
> | 9 -10 4 | --
> |-2 -10 4 | 22

You shouldn't use | for the brackets. They're for determinants.

> My questions: what does this inverse explain?
> What purpose does it serve?

Each column is a generator mapping. The left hand one corresponds to the
top row of the original, 50:49, being the chromatic unison vector. That
gives a 710 cent generator that approximates 3;2, with 9 octave reduced
fifths approximating 5:4 and 2 octave reduced fourths approximating 7:4.

The next column is for 64:63 being the chromatic unison vector. As it has
a common factor of 2, you know the octave is divided into 2 equal parts.
You could set the generator as 434 cents. Then, 3 generators are a 3:2,
and 5 could be either 5:4 or 7:4 (with tritone reduction). Because 7:4
and 5:4 are the same tritone-reduced, 7:5 must be a tritone. So 7:5 and
10:7 are the same, and 50:49 is tempered out, as expected. I think this
one is Paultone.

The last column is for 245:243 tempered out. I get a 109.4 cent
generator, with a 7-limit error of 17.5 cents.

According to Gene, this:

( 1 -6 -2 )
( 9 -10 4 )
(-2 -10 4 )

is the adjoint of the original matrix, and each column is the wedge
product of the relevant commatic unison vectors.

> Do these integers tell us something about 22-EDO?
> Or about 22-EDO's representation of the prime-factors?
>
> ????

You should have left the factors of 2 in for that. Add the octave to the
matrix:

( 1 0 0 0)
( 1 0 2 -2)
( 6 -2 0 -1)
( 0 -5 1 2)

then the adjoint is

(22 0 0 0)
(35 1 -6 -1)
(51 9 -10 4)
(62 -2 -10 4)

so you now have an extra column that tells you the number of steps to each
prime interval. It's also the wedge product of the three unison vectors.

Graham

----- Original Message -----
From: <graham@microtonal.co.uk>
To: <tuning-math@yahoogroups.com>
Sent: Monday, December 17, 2001 4:49 AM
Subject: [tuning-math] Re: inverse of matrix --> for what?

> You shouldn't use | for the brackets. They're for determinants.

Wow -- thanks for clearing that up!

>
> > My questions: what does this inverse explain?
> > What purpose does it serve?
>
> Each column is a generator mapping. The left hand one corresponds to the
> top row of the original, <snip...>
>
> According to Gene, this:
>
> ( 1 -6 -2 )
> ( 9 -10 4 )
> (-2 -10 4 )
>
> is the adjoint of the original matrix, and each column is the wedge
> product of the relevant commatic unison vectors.

Thanks very much for explaining this, Graham. Now I'm at least
beginning to hope that someday I'll understand Gene's work.

Shouldn't I have Tuning Dictionary definitions for "wedge product"
and "adjoint"? Please help. ... Gene? Paul?

> then the adjoint is
>
> (22 0 0 0)
> (35 1 -6 -1)
> (51 9 -10 4)
> (62 -2 -10 4)

Looks like a typo... shouldn't the second row be (35 1 -6 -2) ?

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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In-Reply-To: <00c901c18718$c3a638a0$af48620c@dsl.att.net>
monz wrote:

> Shouldn't I have Tuning Dictionary definitions for "wedge product"
> and "adjoint"? Please help. ... Gene? Paul?

I don't know. It depends on how bloated you want it to get. They're both
linear algebra terms that have a specialist application to tuning theory.
And wedge products conceptually make the adjoint obsolete anyway. The
adjoint's only useful because it can sometimes be calculated more
efficiently if you already have a library that does inverses (or solves
systems of linear equations, which comes to the same thing). Even then,
it'll probably mean taking the inverse, multiplying by the determinant,
and rounding off to integers. So knowing it's called an "adjoint" isn't
much help.

> > then the adjoint is
> >
> > (22 0 0 0)
> > (35 1 -6 -1)
> > (51 9 -10 4)
> > (62 -2 -10 4)
>
>
> Looks like a typo... shouldn't the second row be (35 1 -6 -2) ?

Yes, looks like it, although I've lost the original calculation.

Graham

--- In tuning-math@y..., graham@m... wrote:

> The next column is for 64:63 being the chromatic unison vector.
> As it has
> a common factor of 2, you know the octave is divided into 2 equal
parts.
> You could set the generator as 434 cents. Then, 3 generators are a
3:2,
> and 5 could be either 5:4 or 7:4 (with tritone reduction). Because
7:4
> and 5:4 are the same tritone-reduced, 7:5 must be a tritone. So
7:5 and
> 10:7 are the same, and 50:49 is tempered out, as expected. I think
this
> one is Paultone.

Generator of 434 cents? I don't think so!
>
> The last column is for 245:243 tempered out.

You mean _not_ tempered out.

> I get a 109.4 cent
> generator, with a 7-limit error of 17.5 cents.

_That's_ paultone!

--- In tuning-math@y..., graham@m... wrote:

> Each column is a generator mapping. The left hand one corresponds to the
> top row of the original, 50:49, being the chromatic unison vector. That
> gives a 710 cent generator that approximates 3;2, with 9 octave reduced
> fifths approximating 5:4 and 2 octave reduced fourths approximating 7:4.

It can be done as 9/22, better as 11/27, and best of all as 20/49, where it is the 27+22 system.

> The next column is for 64:63 being the chromatic unison vector. As it has
> a common factor of 2, you know the octave is divided into 2 equal parts.
> You could set the generator as 434 cents. Then, 3 generators are a 3:2,
> and 5 could be either 5:4 or 7:4 (with tritone reduction). Because 7:4
> and 5:4 are the same tritone-reduced, 7:5 must be a tritone. So 7:5 and
> 10:7 are the same, and 50:49 is tempered out, as expected. I think this
> one is Paultone.

This is the chain-of-supermajor-thirds system, an interesting system with a unique association to the 22-et.

> The last column is for 245:243 tempered out. I get a 109.4 cent
> generator, with a 7-limit error of 17.5 cents.

This is twintone, aka Paultone.

> According to Gene, this:
>
> ( 1 -6 -2 )
> ( 9 -10 4 )
> (-2 -10 4 )
>
> is the adjoint of the original matrix, and each column is the wedge
> product of the relevant commatic unison vectors.

It's the adjoint matrix; the columns are wedge products in a 3D space of octave equivalence classes, where the wedge product becomes a cross-product. I don't recommend this point of view, which throws away some valuable information.

From the full 7-limit point of view, we can do something equivalent by taking the odd part of the commas; we then have

25/49^1/63 = [-2,4,4,0,0,0]
25/49^245/243 = [6,10,10,0,0,0]
1/63^245/243 = [1,9,-2,0,0,0]

Looking at this, we might think we have torsion in the first two examples; however

50/49^64/63 = [-2,4,4,-2,-12,11]
50/49^245/243 = [6,10,10,-5,1,2]
64/63^245/243 = [1,9,-2,-30,6,12]

The above shows we do not have torsion, and tells us other things, such as how to calculate the second column of the period matrix.

> > Do these integers tell us something about 22-EDO?
> > Or about 22-EDO's representation of the prime-factors?
> >
> > ????
>
> You should have left the factors of 2 in for that. Add the octave to the
> matrix:

Another way is to add a top row of basis vectors to the matrix; this is is the same as taking the triple wedge product:
50/49^64/63^245/243 = 22 i + 35 j + 51 k + 62 l; we see that the triple wedge product gives us the 22-et from the three commas.

In-Reply-To: <9vlrb9+8s4b@eGroups.com>
genewardsmith wrote:

> It's the adjoint matrix; the columns are wedge products in a 3D space
> of octave equivalence classes, where the wedge product becomes a
> cross-product. I don't recommend this point of view, which throws away
> some valuable information.

Is the information we lose really that valuable? Ignoring torsion, the
commatic, octave-equivalent unison vectors still give us the mapping by
generators and the number of periods to an octave. Can you go from that
to an optimum generator, and reconstruct the period mapping?

Including the chromatic unison vector gives us the number of notes in a
given MOS, and don't we have a way of getting the periodicity block as
monotonically increasing pitches? That should make it even easier.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9vlrb9+8s4b@e...>
> genewardsmith wrote:

> Is the information we lose really that valuable? Ignoring torsion, the
> commatic, octave-equivalent unison vectors still give us the mapping by
> generators and the number of periods to an octave.

We can't ignore torsion, for starters, and if we simply use the information that the cross-product supplies, we don't have enough to define the temperament, which to my mind is one of the main points of it all.

Can you go from that
> to an optimum generator, and reconstruct the period mapping?

No, although usually there will be an obvious "best" choice, I presume. What is your objection to an invariant which actually does the job, instead of only some of it? It seems a little perverse to me.

If the cross-product gives us [-1,-4,-10] it is a fair bet that we have meantone, but if the wedgie is [-1,-4,-10,-12,13,-4] then we *know* we have meantone; and not [-1,-4,-10,-8,17,-6] or
[-1,-4,-10,-10,15,-5] or something.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, December 17, 2001 2:24 PM
> Subject: [tuning-math] Re: inverse of matrix --> for what?
>
>
> --- In tuning-math@y..., graham@m... wrote:
>
> > Each column is a generator mapping. The left hand one corresponds to
the
> > top row of the original, 50:49, being the chromatic unison vector. That
> > gives a 710 cent generator that approximates 3;2, with 9 octave reduced
> > fifths approximating 5:4 and 2 octave reduced fourths approximating 7:4.
>
> It can be done as 9/22, better as 11/27, and best of all as 20/49, where
it
> is the 27+22 system.

<etc. -- snip>

Could you (or someone else?) please give an analysis similar to the one in
this post, but for 55-EDO? Thanks.

I'm especially interested in all the 5-limit unison-vectors which can
define 55-EDO.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> I'm especially interested in all the 5-limit unison-vectors which can
> define 55-EDO.

I just wrote a program yesterday which finds the 5-limit comma associated to (n, g) for the et n and generator g; here is what I get for 55:

2^90/3^26/5^21, 2^82/3^18/5^23, 2^7*3^25/5^20, 2^31*3/5^14, 2^27*3^5/5^15,
2^39/3^7/5^12, 3^27*5^7/2^59, 2^35/3^3/5^13, 2^74/3^10/5^25, 2^23*3^9/5^16,
2^19*3^13/5^17, 2^66/3^2/5^27, 2^47/3^15/5^10, 2^15*3^17/5^18, 3^19*5^9/2^51,
2^43/3^11/5^11, 3^4/2^4/5, 2^11*3^21/5^19, 3^23*5^8/2^55

Each of these 19 commas defines a linear temperament associated to 55; one, of course (3^4/2^4/5 in the notation my computer used) is
81/80.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Could you (or someone else?) please give an analysis similar to the
one in
> this post, but for 55-EDO? Thanks.
>
> I'm especially interested in all the 5-limit unison-vectors which
can
> define 55-EDO.

Hi Monz.

The three unison vectors we came up with for 22-tET were a "Minkowski-
reduced trio" . . . that is, they're essentially the three simplest
unison vectors which define 22-tET in the 7-limit. But there are many
other ways to define 22-tET in the 7-limit with three unison vectors -
- this was just the simplest way.

For 5-limit, we will only need two unison vectors to define an ET, in
this case 55-tET. One of these unison vectors should of course 81:80,
the unison vector that defines meantone. I don't know what the other
pair of the "Minkowski-reduced duo" for 55-tET is, but the choice is
essentially immaterial -- any such choice will be equivalent when
81:80 is tempered out, and when 81:80 is not tempered out, you
essentially have garbage, since 55-tET only makes sense as a
particular meantone and not as some other kind of 5-limit linear
temperament.

If you're still interested, it should be easy for you to find
candidates for the second unison vector, Monz. Just look at one of
the enharmonic equivalencies in 55-tET and express both of the notes
comprising the equivalency as a JI ratio in several different ways.
Then take the quotient of various pairs of ratios representing the
two notes, and voila -- various unison vector candidates.

I wrote,

> 55-tET only makes sense as a
> particular meantone and not as some other kind of 5-limit linear
> temperament.

By "makes sense", I of course meant "has reasonable complexity",
not "is mathematically correct". Sure, the other unison vectors Gene
described corresponds to a way of generating 55-tET from some
interval other than the fifth (fourth), but these ways entail great
complexity and are irrelevant to the historical use of 55-tET, which
was as a measuring system for a particular flavor of meantone.

(Just being the voice of grounding in reality -- there's nothing
wrong with pursuing these curiosities for their own sake, of course.)

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> For 5-limit, we will only need two unison vectors to define an ET, in
> this case 55-tET. One of these unison vectors should of course 81:80,
> the unison vector that defines meantone.

I got two of the commas on my list--one, of course, 81/80, and the other 6442450944/6103515625 = 2^31*3*5^(-14). My badness score for the associated temperament is 6590, but some of the other commas do better--in particular, 2^47 3^(-15) 5^(-10) scores 1378; which hardly compares with the score of 108 for meantone and would not make my best list, where I have a cutoff of 500, but it isn't garbage. The period matrix is

[ 0 5]
[ -2 11]
[ 3 7]

and the generators are a = 19.98/65 and b = 1/5; it really is more of a 65-et system than a 55-et system, and scores as well as it does since it is in much better tune than the 55-et itself, with errors:

3: .317
5: .228
5/3: -.040

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > For 5-limit, we will only need two unison vectors to define an
ET, in
> > this case 55-tET. One of these unison vectors should of course
81:80,
> > the unison vector that defines meantone.
>
> I got two of the commas on my list--one, of course, 81/80, and the
other 6442450944/6103515625 = 2^31*3*5^(-14). My badness score for
the associated temperament is 6590, but some of the other commas do
better--in particular, 2^47 3^(-15) 5^(-10) scores 1378; which hardly
compares with the score of 108 for meantone and would not make my
best list, where I have a cutoff of 500, but it isn't garbage. The
period matrix is
>
> [ 0 5]
> [ -2 11]
> [ 3 7]
>
> and the generators are a = 19.98/65 and b = 1/5; it really is more
of a 65-et system than a 55-et system, and scores as well as it does
since it is in much better tune than the 55-et itself, with errors:
>
> 3: .317
> 5: .228
> 5/3: -.040

So, in an evaluation of 55-tET generators, it's pretty much garbage.

So, Monz, according to Gene the simplest pair of unison vectors for
defining 55-tET is

81:80
and
6442450944:6103515625

The latter is the result of going 14 major thirds down and one
perfect fifth up. In JI, it's about 93.563 cents; and in 55-tET, it's
of course 0.000 ;)

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> So, Monz, according to Gene the simplest pair of unison vectors for
> defining 55-tET is
>
> 81:80
> and
> 6442450944:6103515625
>
> The latter is the result of going 14 major thirds down and one
> perfect fifth up. In JI, it's about 93.563 cents; and in 55-tET, it's
> of course 0.000 ;)

It's 14 major thirds down, a perfect fifth and four octaves up; in other words it tells us (5/4)^14 ~ 24 is an approximation of the
55-et, along with (3/2)^4 ~ 5.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 2:33 PM
> Subject: [tuning-math] Re: inverse of matrix --> for what?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > I'm especially interested in all the 5-limit unison-vectors
> > which can define 55-EDO.
>
> I just wrote a program yesterday which finds the 5-limit comma
> associated to (n, g) for the et n and generator g; here is
> what I get for 55:
>
> 2^90/3^26/5^21, 2^82/3^18/5^23, 2^7*3^25/5^20, 2^31*3/5^14,
> 2^27*3^5/5^15, 2^39/3^7/5^12, 3^27*5^7/2^59, 2^35/3^3/5^13,
> 2^74/3^10/5^25, 2^23*3^9/5^16, 2^19*3^13/5^17, 2^66/3^2/5^27,
> 2^47/3^15/5^10, 2^15*3^17/5^18, 3^19*5^9/2^51, 2^43/3^11/5^11,
> 3^4/2^4/5, 2^11*3^21/5^19, 3^23*5^8/2^55
>
> Each of these 19 commas defines a linear temperament associated
> to 55; one, of course (3^4/2^4/5 in the notation my computer used)
> is 81/80.

So, rewritten in a form that I'm more familiar with, that's:

where unison-vector = 2^x * 3^y * 5^z,

x y z

( 90 -26 -21 )
( 82 -18 -23 )
( 7 25 -20 )
( 31 1 -14 )
( 27 5 -15 )
( 39 -7 -12 )
(-59 27 7 )
( 35 -3 -13 )
( 74 -10 -25 )
( 23 9 -16 )
( 19 13 -17 )
( 66 -2 -27 )
( 47 -15 -10 )
( 15 17 -18 )
(-51 19 9 )
( 43 -11 -11 )
( -4 4 -1 )
( 11 21 -19 )
(-55 23 8 )

Thanks to Paul for the invaluable subsequent comments.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 3:53 PM
> Subject: [tuning-math] 55-tET (was: Re: inverse of matrix --> for what?)
>
>
> So, Monz, according to Gene the simplest pair of unison vectors for
> defining 55-tET is
>
> 81:80
> and
> 6442450944:6103515625
>
> The latter is the result of going 14 major thirds down and one
> perfect fifth up. In JI, it's about 93.563 cents; and in 55-tET, it's
> of course 0.000 ;)

What I had in mind was that there should be a pair of
unison-vectors which defines the set of acoustically implied
ratios which I put on my lattice at
<http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm>?
... assuming, of course, that in the places where two
ratios are implied equally well/badly, only one can be chosen.

I find that if I continue my diagram, the unison-vector that
"works" together with the syntonic comma (-4 4 -1 ) to close
the system at 55 tones, is the (-51 19 9 ). The 8ve-invariant
tuning of the 55th quasi-meantone pitch would be 3^19 * 5^(55/6),
which is ~10.38405963 cents higher than the starting pitch, and
the ratio it implies most closely is 3^19 * 5^9.

-monz

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> So, rewritten in a form that I'm more familiar with, that's:
>
> where unison-vector = 2^x * 3^y * 5^z,
>
> x y z
>
> ( 90 -26 -21 )
> ( 82 -18 -23 )
> ( 7 25 -20 )
> ( 31 1 -14 )
> ( 27 5 -15 ) <etc. -- snip>

And of course, Yahoo's new space-removing "feature" ruined
the careful formatting I put into that matrix, on the web-based
version of the list.

-monz

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