Tuning-Math Digests messages 2875 - 2899

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Message: 2875

Date: Sun, 30 Dec 2001 20:07:18

Subject: Re: coordinates from unison-vectors

From: monz

A plea to all who understand matrix math:


A week ago, I posted the pseudo-code for the formulas
in my Excel spreadsheet which calculates the coordinates
of a 2-dimensional periodicity-block from a given pair
of unison-vectors.  It's quoted below, with some additional
comments inserted.

The algorithm sometimes calculates the proper coordinates,
but not always.  I've tried to work from Paul's _Gentle
Introduction to Fokker Periodicity Blocks, part 3_
<A gentle introduction to Fokker periodicity blocks, part 3, *>,
but I'm just not getting it.

I am *begging* someone who knows how to do this to have
a look at my algorithm and correct it.


The two big problems:

1)
I find that sometimes in order to get the correct periodicity-block,
I have to enter one or both pairs of the unison-vector exponents
with the signs (=/-) reversed.

2)
Sometimes the blocks are centered on 1/1, as I intend
for them to be ... but sometimes they're not, with 1/1 being
plotted at a corner.


Also, it seems that sometimes the order in which I list the
unison-vectors makes a difference.  Is it supposed to?
(I don't think so...)


The code within the "LOOP" is the part that needs fixing.


Thanks in advance for any help received!


-monz



> ----- Original Message -----
>
> > From: monz <joemonz@xxxxx.xxx>
> > To: <tuning-math@xxxxxxxxxxx.xxx>
> > Sent: Sunday, December 23, 2001 12:28 AM
> > Subject: Re: [tuning-math] coordinates from unison-vectors (was: 55-tET)
> >
> >
> >
> > Here's the pseudo-code for the formulas in my spreadsheet.
> > Please feel free to correct any errors or to make the code
> > more elegant.
> >
> >
> >
> > unison-vectors =
> >
> >   (3^a) * (5^b)
> >   (3^c) * (5^d)
> >
> >
> >
> > unison-vector matrix =
> >
> >   (a b)
> >   (c d)
> >
> >
> >
> > determinant n of the matrix :
> >
> >   n  =  (a*d) - (c*b)
> >
> >
> >
> > inverse of the matrix =
> >
> >   ( d -b)
> >   (-c  a)
> >   -------
> >      n
> >
> >
> >
> > inverse coordinates p, q :
> >
> >
> >    p = 0, q = 0
> >
> >
> > LOOP
> >
> >    '-- This part finds the values p,q which fit into the unit-cube
> >
> >    if ABS(p+d) > (ABS(n)/2)
> >
> >      then p = MOD(p+d, ABS(n)) - ABS(n)
> >
> >      else p = p + d
> >
> >    end if
> >
> >
> >
> >    if ABS(q-b) > (ABS(n)/2)
> >
> >      then q = MOD(q-b, ABS(n)) - ABS(n)
> >
> >      else q = q - b
> >
> >    end if
> >
> >
> >
> > lattice coordinates x, y :
> >
> >   '-- This part transforms back to the original space
> >
> >   x = ( (q*c) + (p*a) ) / n
> >
> >   y = ( (q*d) + (p*b) ) / n
> >
> >
> >
> > END LOOP
> >
> >
>
>










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Message: 2876

Date: Sun, 30 Dec 2001 03:07:02

Subject: Re: the unison-vectordeterminant relationship

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Ah, so this is related to the tensor product? 
> 
> A wedge product can be defined as a quotient of a tensor product; 
it is an antisymmetrized tensor product, in effect.
> 
> Is there an ellipsoid 
> > associated with the wedge product??
> 
> There's a parallepiped associated to it, and you could associate an 
ellipsoid to that if you wanted to--why do you ask?

This parallelepiped -- does it exist in the lattice of notes? Perhaps 
after some transformation? (I really need to go through the GA Matlab 
tutorial you gave me -- haven't had time yet).


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Message: 2877

Date: Sun, 30 Dec 2001 20:38:34

Subject: Re: coordinates from unison-vectors

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, December 30, 2001 8:07 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors
> <Yahoo groups: /tuning-math/message/2332 *>
>
> 
> A plea to all who understand matrix math:
> 
> 
> A week ago, I posted the pseudo-code for the formulas
> in my Excel spreadsheet which calculates the coordinates
> of a 2-dimensional periodicity-block from a given pair
> of unison-vectors.  <etc.>


I realized that if I elaborate more on what's going on
in my spreadsheet, I'm more likely to get help.

The first part of the "LOOP" treats the unison-vectors
as boundaries of a unit-cube, and calculates values p,q for
the coordinates within that unit-cube, on the transformed lattice.
I think this is working OK... but if anyone wants to check...

The second part of the "LOOP" transforms these coordinates
back to the original lattice, as values x,y.  This is the
part that is definitely *not* working all the time.

Paul has already mentioned that I should be dividing by
the determinant n again at the end, but it doesn't work
at all if I leave that out.  I don't understand why he says
that either, because the inverse matrix has the determinant
as the denominator.  I think Paul is confused because he
uses the decimal fractional values of the adjoint, whereas
I keep the exponents in integer form and require the determinant
denominator.  But of course, I'm way out of my league here and
could definitely be very wrong...



-monz


 



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Message: 2878

Date: Sun, 30 Dec 2001 22:16:08

Subject: Re: coordinates from unison-vectors

From: monz

Wow... big, *BIG*  Oops!... my bad!

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, December 30, 2001 8:38 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors
>

> ...
> Paul has already mentioned that I should be dividing by
> the determinant n again at the end, ...


That should be "Paul has already mentioned that I

should *NOT* be dividing by the determinant n again
       ^^^^^
at the end,..."



-monz






  



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Message: 2879

Date: Sun, 30 Dec 2001 03:30:15

Subject: Re: the unison-vectordeterminant relationship

From: genewardsmith

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

> This
parallelepiped -- does it exist in the lattice of notes? 

It's simply the parallepiped defined by the vectors we are wedging
together, so yes. The wedge of two vectors is the directed area of the
associated parallogram, of three vectors the directed volume of the
associated parallepiped, and so forth.


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Message: 2880

Date: Sun, 30 Dec 2001 03:48:29

Subject: Re: the unison-vectordeterminant relationship

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, December 29, 2001 6:52 PM
> > Subject: [tuning-math] Re: the unison-vector<-->determinant 
relationship
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > 
> > > The 7/25-comma meantone is lattice right down the middle of
> > > the symmetrical periodicity-block.
> > 
> > Monz, I can't view your .xls file right now, and I'm wondering 
what 
> > it means. I've seen quite a few of your lattices for this topic, 
but 
> > I have to [_sic_: no] clue as to how to picture what you're 
describing
> > above.
> 
> 
> OK, let's go back to 1/6-comma meantone, since that's what I've been
> mostly working with here.
> 
> On my diagram here:
> JustMusic software,  (c) 1999 by Joseph L. Monzo *
> (you can click on the diagram to open a big version),
> you can see that 1/6-comma meantone does not run exactly
> down the center of the (19 9),(4 -1) periodicity-block.

Right, and if you used a different unison vector besides (19 9) to 
define your JI block, you would also see this.

> There are other fraction-of-a-comma meantones which come
> closer to the center, and it seems to me that the one which
> *does* run exactly down the middle is 8/49-comma.
> 
> Is this derivable from the [19 9],[4 -1] matrix?

You should find that the interval corresponding to (19 9), AS IT 
APPEARS in 8/49-comma meantone, is a very tiny interval.

> Is there
> any kind of significance to it?

In my opinion, no.
> 
> It seems to me that a meantone chain that would run down the
> center of a periodicity-block would have the smallest overall
> deviation from the most closely implied JI ratios in the
> periodicity-block, assuming that the JI lattice is wrapped
> into a cylinder.  Yes?

Assuming the lattice is not wrapped into a cylinder, it might make 
some infinitesimal difference. If the lattice is wrapped around a 
cylinder, as it should be if you're talking about an actual meantone 
tuning, then you're implying an infinite number of JI ratios no 
matter what meantone tuning you're using.


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Message: 2881

Date: Sun, 30 Dec 2001 03:49:29

Subject: Re: the unison-vectordeterminant relationship

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > This parallelepiped -- does it exist in the lattice of notes? 
> 
> It's simply the parallepiped defined by the vectors we are wedging 
>together, so yes. The wedge of two vectors is the directed area of 
>the associated parallogram, of three vectors the directed volume of 
>the associated parallepiped, and so forth.

So the wegde product is simply the boundary of the Fokker periodicity 
block??


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Message: 2882

Date: Sun, 30 Dec 2001 04:07:07

Subject: Re: the unison-vectordeterminant relationship

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, December 29, 2001 7:48 PM
> > Subject: [tuning-math] Re: the unison-vector<-->determinant 
relationship
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > There are other fraction-of-a-comma meantones which come
> > > closer to the center, and it seems to me that the one which
> > > *does* run exactly down the middle is 8/49-comma.
> > > 
> > > Is this derivable from the [19 9],[4 -1] matrix?
> > 
> > You should find that the interval corresponding to (19 9), AS IT 
> > APPEARS in 8/49-comma meantone, is a very tiny interval.
> 
> 
> Ah... so then 8/49-comma meantone does *not* run *exactly*
> down the middle.  How could one calculate the meantone which
> *does* run exactly down the middle?

It's 55-tET. 
> 
> Right, I understand that.  But since any given meantone interval
> can only be closest to *one* particular 5-limit JI ratio,

While functioning as an infinite number.

> which
> should fall within the periodicity-block, the vector of the meantone
> chain will *still* imply a unique periodicity-block, will it not?

I don't get it. What's the vector of the meantone chain? Is (19 9) an 
example?

> (assuming that the periodicity-block is replicated a comma away
> as one travels around the cylinder)

Assuming to the answer to the last question it "yes", then I'd 
say, "no, one would get a "strip" rather than a periodicity block", 
but then of course I'd be ignoring your "since any given meantone 
interval can only be closest to *one* particular 5-limit JI ratio". 
If I take that part seriously, I have two comments:

(a) You are NOT, with your current method, mapping identical meantone 
intervals to identical JI ratios, and

(b) if you really meant "pitches" rather than "intervals", I'd argue 
that the mappings you are producing involve a rather arbitrary rule, 
and don't reflect the musical properties of the meantone tunings. The 
only case in which they would is if you specifically knew you were 
not going to use any of the consonances that "wrap" around the block, 
AND you were interested in using a simultaneous JI tuning with the 
meantone that would minimize the _pitch_ differences between the two -
- a very contrived scenario.


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Message: 2883

Date: Sun, 30 Dec 2001 04:40:54

Subject: Nonatonic scales

From: genewardsmith

I've put the ones with connectivity of two in front; the number is the
edge connectivity. I'm afraid I was lazy and didn't try to find a good
mode for these; it would be interesting to see what the lattice
diagrams of these scales look like, and that would tell us a good
choice for 1/1.

[1, 25/24, 10/9, 32/27, 4/3, 25/18, 40/27, 5/3, 16/9]
[25/24, 16/15, 16/15, 9/8, 25/24, 16/15, 9/8, 16/15, 9/8]   2

[1, 25/24, 10/9, 5/4, 125/96, 25/18, 25/16, 5/3, 15/8]
[25/24, 16/15, 9/8, 25/24, 16/15, 9/8, 16/15, 9/8, 16/15]   2

[1, 25/24, 10/9, 5/4, 4/3, 25/18, 40/27, 5/3, 16/9]
[25/24, 16/15, 9/8, 16/15, 25/24, 16/15, 9/8, 16/15, 9/8]   2

[1, 25/24, 10/9, 5/4, 4/3, 25/18, 25/16, 5/3, 16/9]
[25/24, 16/15, 9/8, 16/15, 25/24, 9/8, 16/15, 16/15, 9/8]   2

[1, 25/24, 625/576, 125/108, 125/96, 25/18, 25/16, 5/3, 15/8]
[25/24, 25/24, 16/15, 9/8, 16/15, 9/8, 16/15, 9/8, 16/15]   1

[1, 25/24, 10/9, 125/108, 100/81, 25/18, 40/27, 5/3, 16/9]
[25/24, 16/15, 25/24, 16/15, 9/8, 16/15, 9/8, 16/15, 9/8]   1

[1, 25/24, 10/9, 125/108, 125/96, 25/18, 40/27, 5/3, 16/9]
[25/24, 16/15, 25/24, 9/8, 16/15, 16/15, 9/8, 16/15, 9/8]   1

[1, 25/24, 10/9, 32/27, 100/81, 25/18, 40/27, 5/3, 16/9]
[25/24, 16/15, 16/15, 25/24, 9/8, 16/15, 9/8, 16/15, 9/8]   1

[1, 25/24, 10/9, 32/27, 4/3, 25/18, 40/27, 128/81, 16/9]
[25/24, 16/15, 16/15, 9/8, 25/24, 16/15, 16/15, 9/8, 9/8]   1

[1, 25/24, 10/9, 32/27, 4/3, 25/18, 25/16, 5/3, 16/9]
[25/24, 16/15, 16/15, 9/8, 25/24, 9/8, 16/15, 16/15, 9/8]   1

[1, 25/24, 10/9, 32/27, 4/3, 64/45, 40/27, 5/3, 16/9]
[25/24, 16/15, 16/15, 9/8, 16/15, 25/24, 9/8, 16/15, 9/8]   1

[1, 25/24, 10/9, 32/27, 4/3, 64/45, 8/5, 5/3, 16/9]
[25/24, 16/15, 16/15, 9/8, 16/15, 9/8, 25/24, 16/15, 9/8]   1

[1, 25/24, 10/9, 32/27, 4/3, 3/2, 8/5, 5/3, 16/9]
[25/24, 16/15, 16/15, 9/8, 9/8, 16/15, 25/24, 16/15, 9/8]   1

[1, 25/24, 10/9, 5/4, 125/96, 25/18, 25/16, 5/3, 16/9]
[25/24, 16/15, 9/8, 25/24, 16/15, 9/8, 16/15, 16/15, 9/8]   1

[1, 25/24, 10/9, 5/4, 4/3, 25/18, 40/27, 5/3, 15/8]
[25/24, 16/15, 9/8, 16/15, 25/24, 16/15, 9/8, 9/8, 16/15]   1

[1, 25/24, 10/9, 5/4, 4/3, 64/45, 8/5, 128/75, 16/9]
[25/24, 16/15, 9/8, 16/15, 16/15, 9/8, 16/15, 25/24, 9/8]   1


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Message: 2884

Date: Sun, 30 Dec 2001 05:02:10

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>>>It's a matter of _how_ clear. Typically, according to Jacky, the
>>>2nd and 3rd partials are about 50 cents from their harmonic
>>>series positions. That spells increased entropy (yes, timbres
>>>have entropy),
>> 
>> Which is why I suggested that the plug-in-the-fundamentals-only
>> shortcut shouldn't be applied.
> 
>Why not? Afraid of a little assymetry?

Only when it's spelled like that.  ;)

Not sure what you mean.  The reason I suggested the shortcut not
be applied for inharmonic timbres is because... it is a shortcut.
Which assumes you have clearly resolved fundamentals.  No?

>By the way Carl, have you tried any actual _listening
>experiments_ yet?

You mean with a synthesizer?  As I explained, I don't have the
right gear -- I've got an additive synth that's stuck in JI.

What do you have in mind?  I'm not clear how one would go about
testing anything that's been said here.

>The gamelan scales sound like they contain a rough major
>triad and a rough minor triad, forming a very rough major
>seventh chord together, plus one extra note -- don't they?

Yes, to me, pelog sounds like a I and a III with a 4th in the
middle.  But the music seems to use a fixed tonic, with not
much in the way of triadic structure.  Okay, let's take a
journey...

"Instrumental music of Northeast Thailand"

Characteristic stop rhythm.  Harmonium and marimba-sounding
things play major pentatonic on C# (A=440) or relative minor
on A#.  Scale is treated like a chord, not melodically -- tone
cluster on harmonium for drone, melody is essentially a scale
'arpeggio' figure centered on notes of the scale (usually three
notes are used as centers of this pattern, sometimes they form
a 1st-inversion minor triad).

"JAVA Tembang Sunda" (Inedit)

This is unlike the gamelan music I've heard (it's a plucked
string ensemble with vocalists and flute).  Jeez, I forgot I
had this CD!  There _is_ I -> III, and even I -> IV motion
here.

"Gamelan Semar Pagulingan from Besang-Ababi/Karangasem
Music from Bali"

I suppose there is some argument for triadic structure here
too, but if I hadn't heard the last disc beforehand, I'd
say they were just doing the 'start the figure on different
scale members' thing, as in the first disc.  I don't know
Paul, this is not life as we know it (or hear it).  I still
say there's nothing here that would turn up an optimized
5-limit temperament!  This music bores the hell out of me
when I'm not going for the glassy partial soup that I love
so much.

"The Gamelan of Cirebon"

There's less triadic inuendo here.  Their low 'phones have
a more resolved timbre than the Balinese, above, ensemble
had.  Also, I haven't heard pelog on this disc.  Seems to be
sorog (one chinese, one pelog tetrachord).

I guess it all depends if you consider these tonic changes
or just points of symmetry in a melisma (sp?).  If there's
anything that would produce an optimum 5-limit temperament
in there either way, I'll eat my shoe.

-Carl


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Message: 2885

Date: Sun, 30 Dec 2001 00:00:09

Subject: Re: non-uniqueness of a^(b/c) type numbers

From: paulerlich

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

> No, I don't think I can give an example of that either.
> But does it matter?
> 
> I'm simply having a hard time understanding how
> a^(b/c) can possibly equal d^(e/f) EXACTLY.
> 
> As far as I can see, there are no six integers
> that will satisfy that equation.

You mean, if a and d are prime? Looks about right. Now let's look at 
a more relevant example, how about 7/26-comma meantone. How did you 
express the fifth and the major third of this tuning again?


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Message: 2886

Date: Sun, 30 Dec 2001 08:38:46

Subject: Decatonic scales

From: genewardsmith

Here they be:

[1, 81/80, 27/25, 6/5, 243/200, 162/125, 36/25, 8/5, 81/50, 9/5]
[81/80, 16/15, 10/9, 81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 10/9]   2

[1, 81/80, 27/25, 6/5, 243/200, 27/20, 36/25, 8/5, 81/50, 9/5]
[81/80, 16/15, 10/9, 81/80, 10/9, 16/15, 10/9, 81/80, 10/9, 10/9]   2

[1, 81/80, 27/25, 6/5, 243/200, 27/20, 3/2, 243/160, 81/50, 9/5]
[81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 81/80, 16/15, 10/9, 10/9]   2

[1, 81/80, 27/25, 6/5, 243/200, 27/20, 3/2, 243/160, 27/16, 9/5]
[81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 81/80, 10/9, 16/15, 10/9]   2

[1, 81/80, 27/25, 6/5, 243/200, 27/20, 3/2, 8/5, 81/50, 9/5]
[81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 16/15, 81/80, 10/9, 10/9]   2

[1, 81/80, 27/25, 6/5, 243/200, 162/125, 36/25, 729/500, 81/50, 9/5]
[81/80, 16/15, 10/9, 81/80, 16/15, 10/9, 81/80, 10/9, 10/9, 10/9]   1

[1, 81/80, 27/25, 6/5, 243/200, 27/20, 36/25, 729/500, 81/50, 9/5]
[81/80, 16/15, 10/9, 81/80, 10/9, 16/15, 81/80, 10/9, 10/9, 10/9]   1

[1, 81/80, 27/25, 6/5, 32/25, 162/125, 36/25, 8/5, 81/50, 9/5]
[81/80, 16/15, 10/9, 16/15, 81/80, 10/9, 10/9, 81/80, 10/9, 10/9]   1

[1, 81/80, 27/25, 6/5, 4/3, 27/20, 36/25, 8/5, 16/9, 9/5]
[81/80, 16/15, 10/9, 10/9, 81/80, 16/15, 10/9, 10/9, 81/80, 10/9]   1

[1, 81/80, 27/25, 6/5, 4/3, 27/20, 3/2, 243/160, 27/16, 9/5]
[81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 81/80, 10/9, 16/15, 10/9]   1

[1, 81/80, 27/25, 6/5, 4/3, 27/20, 3/2, 8/5, 16/9, 9/5]
[81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 16/15, 10/9, 81/80, 10/9]   1

[1, 81/80, 27/25, 6/5, 4/3, 27/20, 3/2, 5/3, 16/9, 9/5]
[81/80, 16/15, 10/9, 10/9, 81/80, 10/9, 10/9, 16/15, 81/80, 10/9]   1

[1, 81/80, 9/8, 729/640, 81/64, 27/20, 3/2, 243/160, 27/16, 9/5]
[81/80, 10/9, 81/80, 10/9, 16/15, 10/9, 81/80, 10/9, 16/15, 10/9]   1


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Message: 2887

Date: Sun, 30 Dec 2001 00:06:31

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > > 
> > > Simple neutral thirds? as opposed to the complex ones above?
> > 
> > If 25/24 is a unison, then 6/5~5/4, and that is the basis of this 
> temperament.
> 
> Sure. But if we call this temperament "neutral thirds temperament" 
> without qualification, this conflicts with the usage in Graham's 
> catalog

Really? Why so? He also uses a generator of a neutral third in 
his "neutral thirds temperament".


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Message: 2888

Date: Sun, 30 Dec 2001 09:38:06

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Have you ever looked at it?  It's basically everything that ever
> entered anybody's fancy.  There are scales in there named after
> me I don't even remember making up.

Not really--I don't think any of my scales are in there.


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Message: 2889

Date: Sun, 30 Dec 2001 00:10:09

Subject: Re: the unison-vectordeterminant relationship

From: genewardsmith

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

> ... you can see that the 1/6-comma meantone vector is has an
> off-kilter relationship with 5-limit JI.  Two different linear
> algebras
simultaneous at work, right?  (I sure hope so...)

If you look at u^v, then it is linear in u *and* v, hence bilinear.
The same is true of a related doodad called the tensor product, and of
the dot product, which is a bilinear form. So all this stuff is called
"multilinear algebra", except in contexts where something called the
Clifford algebra comes to the fore and we have "geometric algebra"
instead.

> 
> OK, fix what's wrong with this, if anything...
> 
> One can find an infinity of closer and closer 
> fraction-of-a-comma meantone representations of that
> 5-limit JI periodicity-block, which would be represented
> on my lattice here as shiftings of the angle of the vector
> representing the meantone.  But one could never find one
> that would go straight down the middle of the symmetrical
> periodicity-block.  
> 
> Now, what does this mean?

I'm not following you--why not spell it out, giving the block in
question?


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Message: 2890

Date: Sun, 30 Dec 2001 10:58:20

Subject: Classes of 7-limit scales

From: genewardsmith

Here are the classes of 7-limit epimorphic superparticular scales. The
larger ones are far beyond the limits of computation so far as
classifying all of the connected scales. I think a subset of these,
with some kind of propriety condition, and with the number of scale
degrees not too large, is what needs to be considered.

4

[64/63, 21/20, 5/4, 3/2]
[1, 1, 1, 1]

[36/35, 28/27, 5/4, 3/2]
[1, 1, 1, 1]

[36/35, 10/9, 7/6, 3/2]
[1, 1, 1, 1]

[36/35, 7/6, 5/4, 4/3]
[1, 1, 1, 1]

[28/27, 15/14, 6/5, 3/2]
[1, 1, 1, 1]

[21/20, 10/9, 8/7, 3/2]
[1, 1, 1, 1]

[21/20, 8/7, 5/4, 4/3]
[1, 1, 1, 1]

[16/15, 15/14, 7/6, 3/2]
[1, 1, 1, 1]

[15/14, 7/6, 6/5, 4/3]
[1, 1, 1, 1]

[8/7, 7/6, 6/5, 5/4]
[1, 1, 1, 1]

5

[225/224, 126/125, 10/9, 4/3]
[1, 1, 1, 2]

[225/224, 28/27, 6/5, 4/3]
[1, 1, 2, 1]

[225/224, 21/20, 16/15, 4/3]
[1, 1, 1, 2]

[225/224, 16/15, 7/6, 3/2]
[1, 2, 1, 1]

[126/125, 15/14, 10/9, 3/2]
[1, 1, 2, 1]

[81/80, 28/27, 15/14, 4/3]
[1, 1, 1, 2]

[64/63, 21/20, 6/5, 5/4]
[1, 1, 1, 2]

[49/48, 8/7, 6/5, 5/4]
[1, 2, 1, 1]

[36/35, 28/27, 6/5, 5/4]
[1, 1, 1, 2]

[36/35, 21/20, 10/9, 3/2]
[1, 1, 2, 1]

[36/35, 16/15, 7/6, 5/4]
[1, 1, 1, 2]

[36/35, 15/14, 7/6, 4/3]
[1, 1, 2, 1]

[36/35, 8/7, 7/6, 5/4]
[1, 1, 2, 1]

[28/27, 15/14, 6/5, 5/4]
[1, 1, 2, 1]

[25/24, 8/7, 7/6, 6/5]
[1, 1, 1, 2]

[21/20, 16/15, 8/7, 5/4]
[1, 1, 1, 2]

[21/20, 8/7, 7/6, 5/4]
[1, 2, 1, 1]

[15/14, 10/9, 7/6, 6/5]
[1, 1, 1, 2]

[15/14, 8/7, 7/6, 6/5]
[1, 1, 2, 1]

6

[2401/2400, 15/14, 8/7, 5/4]
[1, 1, 3, 1]

[2401/2400, 15/14, 8/7, 4/3]
[1, 2, 2, 1]

[225/224, 64/63, 21/20, 4/3]
[1, 1, 2, 2]

[225/224, 28/27, 10/9, 6/5]
[1, 1, 1, 3]

[81/80, 10/9, 8/7, 7/6]
[1, 1, 2, 2]

[64/63, 21/20, 8/7, 5/4]
[1, 2, 1, 2]

[49/48, 36/35, 10/9, 3/2]
[1, 2, 2, 1]

[49/48, 25/24, 8/7, 6/5]
[1, 1, 2, 2]

[49/48, 21/20, 8/7, 5/4]
[1, 1, 3, 1]

[49/48, 16/15, 15/14, 3/2]
[1, 2, 2, 1]

[36/35, 28/27, 7/6, 5/4]
[2, 1, 1, 2]

[36/35, 25/24, 7/6, 4/3]
[2, 1, 2, 1]

[36/35, 15/14, 8/7, 7/6]
[1, 1, 1, 3]

[36/35, 10/9, 7/6, 5/4]
[2, 1, 2, 1]

[28/27, 25/24, 15/14, 6/5]
[1, 1, 1, 3]

[28/27, 15/14, 7/6, 6/5]
[1, 2, 1, 2]

[25/24, 21/20, 8/7, 4/3]
[1, 2, 2, 1]

[21/20, 16/15, 15/14, 5/4]
[1, 2, 1, 2]

[21/20, 15/14, 10/9, 6/5]
[1, 1, 2, 2]

[21/20, 15/14, 8/7, 7/6]
[1, 1, 2, 2]

[21/20, 10/9, 8/7, 5/4]
[2, 1, 2, 1]

[16/15, 15/14, 7/6, 6/5]
[1, 2, 2, 1]

7

[4375/4374, 126/125, 15/14, 6/5]
[1, 1, 2, 3]

[2401/2400, 15/14, 8/7, 7/6]
[1, 2, 3, 1]

[225/224, 126/125, 10/9, 6/5]
[1, 1, 3, 2]

[225/224, 28/27, 15/14, 6/5]
[1, 2, 1, 3]

[225/224, 21/20, 16/15, 5/4]
[1, 1, 3, 2]

[126/125, 28/27, 15/14, 3/2]
[1, 2, 3, 1]

[81/80, 50/49, 28/27, 4/3]
[2, 1, 2, 2]

[64/63, 25/24, 21/20, 6/5]
[1, 2, 1, 3]

[64/63, 21/20, 9/8, 5/4]
[2, 2, 1, 2]

[50/49, 49/48, 8/7, 6/5]
[1, 2, 2, 2]

[50/49, 21/20, 16/15, 5/4]
[1, 2, 2, 2]

[50/49, 21/20, 10/9, 6/5]
[1, 2, 2, 2]

[50/49, 21/20, 8/7, 7/6]
[1, 2, 2, 2]

[49/48, 36/35, 8/7, 5/4]
[2, 1, 3, 1]

[49/48, 15/14, 8/7, 6/5]
[2, 1, 3, 1]

[36/35, 28/27, 25/24, 6/5]
[1, 1, 2, 3]

[36/35, 28/27, 9/8, 5/4]
[2, 2, 1, 2]

[36/35, 25/24, 8/7, 7/6]
[2, 1, 1, 3]

[36/35, 16/15, 15/14, 7/6]
[1, 1, 2, 3]

[36/35, 15/14, 10/9, 7/6]
[2, 1, 1, 3]

[36/35, 10/9, 9/8, 7/6]
[2, 2, 1, 2]

[28/27, 15/14, 9/8, 6/5]
[2, 2, 1, 2]

[25/24, 21/20, 8/7, 7/6]
[1, 2, 3, 1]

[21/20, 10/9, 9/8, 8/7]
[2, 2, 1, 2]

[16/15, 15/14, 9/8, 7/6]
[2, 2, 1, 2]

8

[2401/2400, 49/48, 15/14, 8/7]
[1, 1, 2, 4]

[2401/2400, 16/15, 15/14, 4/3]
[1, 2, 4, 1]

[225/224, 49/48, 16/15, 3/2]
[2, 1, 4, 1]

[225/224, 28/27, 16/15, 6/5]
[2, 2, 1, 3]

[225/224, 16/15, 7/6, 6/5]
[2, 3, 2, 1]

[126/125, 64/63, 25/24, 6/5]
[1, 1, 3, 3]

[126/125, 25/24, 8/7, 4/3]
[2, 3, 2, 1]

[126/125, 15/14, 10/9, 5/4]
[2, 2, 3, 1]

[81/80, 49/48, 10/9, 8/7]
[1, 2, 1, 4]

[81/80, 36/35, 10/9, 7/6]
[1, 2, 3, 2]

[81/80, 21/20, 10/9, 8/7]
[1, 2, 3, 2]

[64/63, 21/20, 15/14, 5/4]
[2, 3, 1, 2]

[49/48, 25/24, 21/20, 8/7]
[1, 1, 2, 4]

[49/48, 21/20, 15/14, 8/7]
[2, 1, 1, 4]

[36/35, 28/27, 15/14, 7/6]
[2, 1, 2, 3]

[36/35, 25/24, 10/9, 7/6]
[3, 1, 1, 3]

[36/35, 21/20, 10/9, 5/4]
[2, 2, 3, 1]

[28/27, 21/20, 15/14, 6/5]
[2, 1, 3, 2]

[25/24, 21/20, 10/9, 8/7]
[1, 3, 1, 3]

[21/20, 16/15, 15/14, 7/6]
[1, 2, 3, 2]

[21/20, 15/14, 10/9, 8/7]
[3, 1, 2, 2]

9

[2401/2400, 50/49, 15/14, 8/7]
[2, 1, 2, 4]

[2401/2400, 50/49, 8/7, 6/5]
[2, 3, 2, 2]

[2401/2400, 16/15, 15/14, 5/4]
[1, 3, 4, 1]

[225/224, 36/35, 28/27, 6/5]
[2, 1, 3, 3]

[225/224, 36/35, 16/15, 7/6]
[2, 1, 3, 3]

[225/224, 16/15, 9/8, 7/6]
[2, 4, 1, 2]

[126/125, 25/24, 8/7, 7/6]
[2, 3, 3, 1]

[126/125, 15/14, 10/9, 9/8]
[2, 2, 4, 1]

[126/125, 15/14, 10/9, 7/6]
[2, 3, 3, 1]

[81/80, 28/27, 15/14, 6/5]
[1, 3, 3, 2]

[64/63, 50/49, 21/20, 5/4]
[2, 1, 4, 2]

[64/63, 21/20, 10/9, 9/8]
[2, 2, 2, 3]

[50/49, 49/48, 21/20, 8/7]
[1, 2, 2, 4]

[50/49, 21/20, 10/9, 8/7]
[1, 4, 2, 2]

[49/48, 36/35, 15/14, 8/7]
[3, 1, 1, 4]

[36/35, 28/27, 10/9, 9/8]
[2, 2, 2, 3]

[36/35, 25/24, 16/15, 7/6]
[3, 2, 1, 3]

[36/35, 21/20, 10/9, 9/8]
[2, 2, 4, 1]

[28/27, 16/15, 15/14, 9/8]
[2, 2, 2, 3]

[25/24, 21/20, 16/15, 8/7]
[2, 3, 1, 3]

10

[2401/2400, 64/63, 15/14, 8/7]
[2, 1, 4, 3]

[2401/2400, 16/15, 15/14, 7/6]
[1, 3, 5, 1]

[225/224, 64/63, 21/20, 5/4]
[1, 3, 4, 2]

[126/125, 81/80, 15/14, 10/9]
[2, 1, 2, 5]

[126/125, 49/48, 25/24, 8/7]
[2, 1, 3, 4]

[126/125, 21/20, 15/14, 10/9]
[2, 1, 3, 4]

[81/80, 64/63, 10/9, 7/6]
[3, 2, 3, 2]

[81/80, 36/35, 21/20, 10/9]
[1, 2, 2, 5]

[81/80, 28/27, 10/9, 8/7]
[3, 2, 3, 2]

[64/63, 25/24, 21/20, 8/7]
[1, 2, 4, 3]

[64/63, 21/20, 15/14, 7/6]
[2, 3, 3, 2]

[49/48, 36/35, 25/24, 8/7]
[3, 2, 1, 4]

[49/48, 36/35, 10/9, 5/4]
[2, 4, 3, 1]

[49/48, 16/15, 15/14, 6/5]
[2, 3, 4, 1]

[36/35, 28/27, 25/24, 7/6]
[4, 1, 2, 3]

[36/35, 21/20, 15/14, 10/9]
[2, 3, 1, 4]

[28/27, 21/20, 15/14, 8/7]
[2, 3, 3, 2]

[21/20, 16/15, 15/14, 10/9]
[3, 2, 3, 2]

11

[4375/4374, 36/35, 21/20, 3/2]
[2, 5, 3, 1]

[2401/2400, 50/49, 21/20, 8/7]
[2, 3, 2, 4]

[225/224, 126/125, 15/14, 10/9]
[1, 3, 2, 5]

[225/224, 28/27, 16/15, 9/8]
[2, 2, 4, 3]

[225/224, 21/20, 16/15, 7/6]
[3, 1, 5, 2]

[126/125, 28/27, 15/14, 5/4]
[2, 3, 5, 1]

[126/125, 25/24, 15/14, 10/9]
[3, 1, 3, 4]

[126/125, 25/24, 10/9, 8/7]
[3, 4, 1, 3]

[50/49, 49/48, 36/35, 8/7]
[1, 4, 2, 4]

[50/49, 36/35, 21/20, 10/9]
[1, 2, 4, 4]

[50/49, 28/27, 21/20, 6/5]
[3, 2, 4, 2]

[50/49, 21/20, 16/15, 7/6]
[3, 4, 2, 2]

[49/48, 36/35, 10/9, 9/8]
[2, 4, 4, 1]

[49/48, 16/15, 15/14, 9/8]
[2, 4, 4, 1]

[36/35, 28/27, 15/14, 9/8]
[2, 4, 2, 3]

[36/35, 25/24, 21/20, 10/9]
[3, 1, 3, 4]

12

[2401/2400, 225/224, 16/15, 4/3]
[1, 4, 6, 1]

[2401/2400, 49/48, 16/15, 15/14]
[1, 1, 4, 6]

[225/224, 36/35, 28/27, 7/6]
[2, 4, 3, 3]

[126/125, 28/27, 15/14, 7/6]
[2, 3, 6, 1]

[126/125, 25/24, 16/15, 8/7]
[3, 5, 1, 3]

[81/80, 64/63, 21/20, 10/9]
[3, 2, 2, 5]

[81/80, 49/48, 36/35, 10/9]
[1, 2, 4, 5]

[81/80, 36/35, 28/27, 10/9]
[3, 2, 2, 5]

[64/63, 21/20, 15/14, 10/9]
[2, 5, 3, 2]

[49/48, 21/20, 16/15, 15/14]
[2, 1, 4, 5]

[28/27, 21/20, 16/15, 15/14]
[2, 3, 2, 5]

[25/24, 21/20, 16/15, 15/14]
[2, 3, 4, 3]

13

[4375/4374, 126/125, 15/14, 10/9]
[1, 4, 5, 3]

[2401/2400, 225/224, 16/15, 5/4]
[1, 4, 7, 1]

[2401/2400, 64/63, 16/15, 15/14]
[2, 1, 3, 7]

[2401/2400, 64/63, 15/14, 9/8]
[2, 4, 4, 3]

[2401/2400, 50/49, 16/15, 15/14]
[2, 1, 4, 6]

[225/224, 126/125, 16/15, 10/9]
[3, 3, 2, 5]

[225/224, 21/20, 16/15, 10/9]
[3, 3, 5, 2]

[126/125, 50/49, 49/48, 8/7]
[2, 3, 4, 4]

[126/125, 50/49, 21/20, 10/9]
[2, 3, 4, 4]

[126/125, 28/27, 15/14, 9/8]
[2, 4, 6, 1]

[81/80, 28/27, 15/14, 8/7]
[3, 5, 3, 2]

[64/63, 50/49, 21/20, 7/6]
[2, 3, 6, 2]

[64/63, 25/24, 21/20, 9/8]
[4, 2, 4, 3]

[50/49, 28/27, 21/20, 8/7]
[3, 2, 6, 2]

[50/49, 21/20, 16/15, 10/9]
[3, 6, 2, 2]

[49/48, 36/35, 16/15, 15/14]
[3, 1, 4, 5]

[49/48, 36/35, 15/14, 10/9]
[3, 5, 1, 4]

[36/35, 28/27, 25/24, 9/8]
[4, 4, 2, 3]

14

[4375/4374, 49/48, 36/35, 3/2]
[2, 3, 8, 1]

[225/224, 49/48, 16/15, 6/5]
[4, 2, 7, 1]

[126/125, 64/63, 25/24, 8/7]
[4, 1, 6, 3]

[126/125, 36/35, 25/24, 10/9]
[3, 3, 4, 4]

[126/125, 28/27, 21/20, 15/14]
[2, 4, 1, 7]

[81/80, 64/63, 28/27, 10/9]
[5, 2, 2, 5]

[64/63, 28/27, 21/20, 15/14]
[2, 2, 5, 5]

[49/48, 36/35, 25/24, 10/9]
[3, 6, 1, 4]

[36/35, 28/27, 21/20, 15/14]
[2, 4, 3, 5]

15

[2401/2400, 225/224, 16/15, 7/6]
[1, 5, 8, 1]

[2401/2400, 50/49, 36/35, 8/7]
[4, 5, 2, 4]

[225/224, 49/48, 16/15, 9/8]
[4, 2, 8, 1]

[225/224, 36/35, 28/27, 9/8]
[2, 4, 6, 3]

[225/224, 25/24, 21/20, 16/15]
[3, 2, 3, 7]

[126/125, 81/80, 28/27, 15/14]
[2, 1, 5, 7]

[126/125, 28/27, 25/24, 15/14]
[3, 4, 1, 7]

[126/125, 25/24, 16/15, 15/14]
[3, 5, 4, 3]

[81/80, 50/49, 28/27, 6/5]
[4, 3, 6, 2]

[81/80, 28/27, 16/15, 15/14]
[3, 5, 2, 5]

[64/63, 50/49, 21/20, 10/9]
[2, 3, 8, 2]

[50/49, 49/48, 36/35, 10/9]
[1, 4, 6, 4]

[50/49, 25/24, 21/20, 16/15]
[3, 2, 6, 4]

[36/35, 25/24, 21/20, 16/15]
[3, 5, 3, 4]

16

[4375/4374, 126/125, 28/27, 15/14]
[1, 4, 3, 8]

[2401/2400, 64/63, 21/20, 15/14]
[2, 4, 3, 7]

[225/224, 126/125, 28/27, 15/14]
[1, 3, 5, 7]

[225/224, 64/63, 21/20, 7/6]
[3, 5, 6, 2]

[81/80, 64/63, 49/48, 10/9]
[5, 4, 2, 5]

[64/63, 49/48, 21/20, 15/14]
[4, 2, 5, 5]

[64/63, 25/24, 21/20, 15/14]
[4, 2, 7, 3]

17

[4375/4374, 36/35, 21/20, 5/4]
[3, 8, 5, 1]

[2401/2400, 126/125, 50/49, 8/7]
[4, 2, 7, 4]

[225/224, 49/48, 21/20, 16/15]
[5, 2, 1, 9]

[225/224, 28/27, 21/20, 16/15]
[5, 2, 3, 7]

[126/125, 64/63, 25/24, 9/8]
[4, 4, 6, 3]

[126/125, 50/49, 25/24, 10/9]
[6, 3, 4, 4]

[81/80, 36/35, 28/27, 15/14]
[3, 2, 7, 5]

[50/49, 49/48, 21/20, 16/15]
[5, 2, 6, 4]

[50/49, 28/27, 21/20, 16/15]
[5, 2, 8, 2]

[49/48, 36/35, 28/27, 15/14]
[3, 5, 4, 5]

18

[2401/2400, 225/224, 49/48, 16/15]
[1, 6, 1, 10]

[225/224, 126/125, 25/24, 16/15]
[3, 3, 5, 7]

[225/224, 64/63, 21/20, 10/9]
[3, 5, 8, 2]

[225/224, 49/48, 36/35, 16/15]
[5, 3, 1, 9]

[126/125, 36/35, 25/24, 16/15]
[3, 3, 8, 4]

[49/48, 36/35, 25/24, 16/15]
[3, 6, 5, 4]

19

[2401/2400, 225/224, 50/49, 16/15]
[2, 6, 1, 10]

[2401/2400, 50/49, 36/35, 10/9]
[4, 5, 6, 4]

[2401/2400, 50/49, 21/20, 16/15]
[2, 7, 6, 4]

[81/80, 64/63, 28/27, 15/14]
[5, 2, 7, 5]

[81/80, 50/49, 28/27, 8/7]
[6, 3, 8, 2]

[64/63, 50/49, 28/27, 21/20]
[2, 5, 2, 10]

[64/63, 50/49, 25/24, 21/20]
[4, 3, 2, 10]

[64/63, 36/35, 25/24, 21/20]
[4, 3, 5, 7]

[50/49, 36/35, 28/27, 21/20]
[5, 2, 4, 8]

[36/35, 28/27, 25/24, 21/20]
[7, 4, 5, 3]

20

[2401/2400, 225/224, 64/63, 16/15]
[2, 7, 1, 10]

[225/224, 81/80, 28/27, 16/15]
[5, 3, 5, 7]

21

[4375/4374, 36/35, 21/20, 9/8]
[4, 10, 6, 1]

[126/125, 50/49, 49/48, 10/9]
[6, 7, 4, 4]

[126/125, 50/49, 28/27, 21/20]
[2, 7, 4, 8]

[126/125, 50/49, 25/24, 16/15]
[6, 3, 8, 4]

[64/63, 50/49, 49/48, 21/20]
[4, 5, 2, 10]

[64/63, 49/48, 36/35, 15/14]
[4, 7, 5, 5]

22

[4375/4374, 126/125, 36/35, 15/14]
[4, 7, 3, 8]

[4375/4374, 49/48, 36/35, 5/4]
[3, 5, 13, 1]

[4375/4374, 36/35, 21/20, 15/14]
[4, 10, 7, 1]

[225/224, 64/63, 25/24, 21/20]
[3, 7, 2, 10]

[126/125, 36/35, 28/27, 25/24]
[3, 7, 4, 8]

[81/80, 64/63, 25/24, 21/20]
[3, 7, 5, 7]

[81/80, 36/35, 28/27, 25/24]
[3, 7, 7, 5]

[49/48, 36/35, 28/27, 25/24]
[3, 10, 4, 5]

23

[4375/4374, 50/49, 36/35, 21/20]
[4, 1, 10, 8]

[4375/4374, 36/35, 25/24, 21/20]
[4, 11, 1, 7]

[2401/2400, 64/63, 50/49, 21/20]
[2, 4, 7, 10]

[225/224, 126/125, 28/27, 16/15]
[8, 3, 5, 7]

[126/125, 64/63, 25/24, 15/14]
[7, 4, 9, 3]

[50/49, 49/48, 36/35, 16/15]
[5, 8, 6, 4]

24

[225/224, 64/63, 28/27, 21/20]
[5, 7, 2, 10]

[225/224, 36/35, 28/27, 21/20]
[5, 7, 9, 3]

25

[4375/4374, 126/125, 50/49, 15/14]
[4, 10, 3, 8]

[4375/4374, 81/80, 36/35, 21/20]
[5, 1, 12, 7]

[4375/4374, 81/80, 36/35, 16/15]
[5, 8, 5, 7]

[2401/2400, 126/125, 50/49, 10/9]
[4, 6, 11, 4]

[81/80, 50/49, 28/27, 16/15]
[8, 5, 10, 2]

26

[225/224, 64/63, 49/48, 21/20]
[5, 9, 2, 10]

[126/125, 64/63, 36/35, 25/24]
[7, 4, 3, 12]

[81/80, 64/63, 49/48, 15/14]
[5, 9, 7, 5]

[64/63, 49/48, 36/35, 25/24]
[4, 7, 10, 5]

27

[4375/4374, 49/48, 36/35, 9/8]
[4, 6, 16, 1]

[225/224, 81/80, 36/35, 28/27]
[5, 3, 7, 12]

[225/224, 49/48, 36/35, 28/27]
[5, 3, 10, 9]

[81/80, 50/49, 36/35, 28/27]
[8, 5, 2, 12]

[50/49, 49/48, 36/35, 28/27]
[5, 8, 10, 4]

29

[4375/4374, 49/48, 36/35, 15/14]
[4, 7, 17, 1]

[126/125, 81/80, 64/63, 25/24]
[7, 3, 7, 12]

[126/125, 81/80, 50/49, 28/27]
[2, 8, 7, 12]

[126/125, 64/63, 50/49, 25/24]
[10, 4, 3, 12]

[126/125, 50/49, 49/48, 16/15]
[6, 11, 8, 4]

[126/125, 50/49, 28/27, 25/24]
[10, 7, 4, 8]

[81/80, 64/63, 50/49, 28/27]
[10, 2, 5, 12]

[81/80, 64/63, 28/27, 25/24]
[10, 7, 7, 5]

30

[4375/4374, 126/125, 36/35, 25/24]
[4, 7, 11, 8]

[4375/4374, 81/80, 64/63, 16/15]
[5, 13, 5, 7]

[4375/4374, 49/48, 36/35, 25/24]
[4, 7, 18, 1]

[2401/2400, 225/224, 64/63, 21/20]
[2, 7, 11, 10]

[225/224, 126/125, 36/35, 28/27]
[8, 3, 7, 12]

31

[4375/4374, 50/49, 49/48, 36/35]
[4, 1, 8, 18]

[2401/2400, 50/49, 36/35, 16/15]
[8, 13, 6, 4]

[64/63, 50/49, 49/48, 36/35]
[4, 5, 12, 10]

32

[4375/4374, 81/80, 49/48, 36/35]
[5, 1, 7, 19]

[4375/4374, 81/80, 36/35, 28/27]
[5, 8, 12, 7]

[225/224, 126/125, 64/63, 25/24]
[3, 10, 7, 12]

33

[4375/4374, 126/125, 50/49, 21/20]
[4, 10, 11, 8]

34

[225/224, 81/80, 64/63, 28/27]
[5, 10, 7, 12]

35

[2401/2400, 50/49, 36/35, 28/27]
[8, 13, 10, 4]

36

[225/224, 64/63, 49/48, 36/35]
[5, 9, 12, 10]

[81/80, 64/63, 49/48, 25/24]
[10, 14, 7, 5]

37

[4375/4374, 81/80, 64/63, 21/20]
[5, 13, 12, 7]

[2401/2400, 126/125, 50/49, 16/15]
[8, 6, 19, 4]

[225/224, 126/125, 50/49, 28/27]
[8, 10, 7, 12]

[126/125, 50/49, 49/48, 28/27]
[10, 15, 8, 4]

39

[4375/4374, 2401/2400, 50/49, 36/35]
[4, 8, 9, 18]

41

[4375/4374, 126/125, 50/49, 25/24]
[4, 18, 11, 8]

[126/125, 64/63, 50/49, 49/48]
[10, 4, 15, 12]

[81/80, 64/63, 50/49, 49/48]
[10, 14, 5, 12]

43

[2401/2400, 64/63, 50/49, 36/35]
[12, 4, 17, 10]

44

[4375/4374, 81/80, 64/63, 28/27]
[5, 20, 12, 7]

[225/224, 126/125, 64/63, 28/27]
[15, 10, 7, 12]

45

[2401/2400, 126/125, 50/49, 28/27]
[8, 10, 23, 4]

46

[4375/4374, 126/125, 81/80, 36/35]
[12, 7, 8, 19]

[225/224, 81/80, 64/63, 49/48]
[5, 10, 19, 12]

49

[4375/4374, 126/125, 50/49, 49/48]
[4, 18, 19, 8]

51

[4375/4374, 81/80, 64/63, 49/48]
[5, 20, 19, 7]

53

[2401/2400, 126/125, 64/63, 50/49]
[12, 10, 4, 27]

[2401/2400, 81/80, 64/63, 50/49]
[12, 10, 14, 17]

54

[4375/4374, 225/224, 126/125, 36/35]
[12, 8, 15, 19]

56

[225/224, 126/125, 64/63, 49/48]
[15, 10, 19, 12]

57

[4375/4374, 2401/2400, 126/125, 50/49]
[4, 8, 18, 27]

60

[2401/2400, 225/224, 64/63, 36/35]
[12, 17, 21, 10]

65

[4375/4374, 126/125, 81/80, 64/63]
[12, 7, 27, 19]

[4375/4374, 126/125, 81/80, 50/49]
[12, 26, 8, 19]

70

[2401/2400, 225/224, 81/80, 64/63]
[12, 17, 10, 31]

73

[4375/4374, 225/224, 126/125, 50/49]
[12, 8, 34, 19]

80

[2401/2400, 225/224, 126/125, 64/63]
[12, 27, 10, 31]

92

[4375/4374, 225/224, 126/125, 64/63]
[12, 27, 34, 19]


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Message: 2891

Date: Sun, 30 Dec 2001 17:22 +0

Subject: Re: Optimal 5-Limit Generators for Dave

From: graham@xxxxxxxxxx.xx.xx

Gene:
> > > If 25/24 is a unison, then 6/5~5/4, and that is the basis of this 
> > temperament.

Dave:
> > Sure. But if we call this temperament "neutral thirds temperament" 
> > without qualification, this conflicts with the usage in Graham's 
> > catalog

Paul:
> Really? Why so? He also uses a generator of a neutral third in 
> his "neutral thirds temperament".

Yes, it looks okay to me.  To pick through my description

"Two neutral thirds make up a perfect fifth."

That's true for 6:5=~5:4

"The neutral thirds can be called 11:9."

But don't have to be.

"The `wolf third' to make up a 7-note scale
can be identified with 6:5 or 7:6 or neither."

And in this case neither.  

I'd call the temperament with the wolf third as 6:5 the "typical neutral 
third temperament" or the "meantone-like neutral third temperament" but 
that isn't the only way of doing it.  Taking meantone and dividing the 
fifths in two, but not calling the result an 11:9, would also be a neutral 
third temperament.  I'm not sure if this is what some of these examples 
are intending.

I should really update that description to specify that the scale be an 
MOS generated by the neutral third, or some permutation (like rast).


                     Graham


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Message: 2892

Date: Sun, 30 Dec 2001 23:59:37

Subject: Re: Optimal 5-Limit Generators for Dave

From: dkeenanuqnetau

--- In tuning-math@y..., graham@m... wrote:
> Paul:
> > Really? Why so? He also uses a generator of a neutral third in 
> > his "neutral thirds temperament".
> 
> Yes, it looks okay to me.  To pick through my description
...

I was referring to your catalog page which I thought did not allow for  
5 to map to 1 gen. In any case, when we're talking 5-limit 
temperaments I think we should have different names or qualifiers for 
the different mappings of the prime 5 when the prime 3 maps to 2 gens. 
I suggested "simple neutral thirds temperament" for the [2 1] mapping 
and "complex neutral thirds temperament" for the [2 8] mapping. What 
are the useful mappings when we go to 7-limit? It would be useful to 
have all these mappings listed on your catalog page, Graham.


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Message: 2893

Date: Sun, 30 Dec 2001 01:57:22

Subject: Hexatonic scales

From: genewardsmith

Here are the connected 5-limit hexatonic superparticular scales; "c" is the edge connectivity.

1--25/24--5/4--4/3--8/5--48/25
[25/24 6/5 16/15 6/5 6/5 25/24] c = 1

1--6/5--36/25--3/2--8/5--5/3
[6/5 6/5 25/24 16/15 25/24 6/5] c = 1

1--6/5--5/4--4/3--8/5--5/3
[6/5 25/24 16/15 6/5 25/24 6/5] c = 2

1--6/5--5/4--3/2--25/16--5/3
[6/5 25/24 6/5 25/24 16/15 6/5] c = 1

1--10/9--4/3--27/20--3/2--9/5
[10/9 6/5 81/80 10/9 6/5 10/9] c = 1

1--6/5--4/3--27/20--3/2--9/5
[6/5 10/9 81/80 10/9 6/5 10/9] c = 1


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Message: 2894

Date: Sun, 30 Dec 2001 02:38:06

Subject: Re: the unison-vectordeterminant relationship

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > ... you can see that the 1/6-comma meantone vector is has an
> > off-kilter relationship with 5-limit JI.  Two different linear
> > algebras simultaneous at work, right?  (I sure hope so...)
> 
> If you look at u^v, then it is linear in u *and* v, hence bilinear. 
>The same is true of a related doodad called the tensor product,

Ah, so this is related to the tensor product? Is there an ellipsoid 
associated with the wedge product??


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Message: 2895

Date: Mon, 31 Dec 2001 00:48:48

Subject: Re: Maple and graph theory

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> 
> > Maple has a graph theory package, which allows me to take a scale 
> >and compute its connectivity fairly easily.
> 
> Is this the same as Dave Keenan's connectivity?

Almost certainly not. I assume you are referring to where I say e.g. a 
meantone diatonic scale is 86% connected at the 9-limit. Which means 
that of the 21 possible dyads (considered octave-equivalently) 18 of 
them are (reasonable approximations of) 9-limit ratios.

I assume that Gene's measure gave the number of connected subgraphs 
that were disconnected from each other.


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Message: 2896

Date: Mon, 31 Dec 2001 20:33:09

Subject: Re: Some 7-limit superparticular pentatonics

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> Gene, are you
allowing 9-limit edges?

No, but it would be easy enough to do so. I'm already buried under the
varied possibilities as it is, though, and also want to explore
tempered scales using graph theory methods.

> Tomorrow, I'm going to try and tune these
> up and see what they sound like.

I was wondering also; I think I'll fiddle around on FTS with some of
these.


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Message: 2897

Date: Mon, 31 Dec 2001 01:29:40

Subject: Re: Maple and graph theory

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I assume that Gene's measure gave the number of connected subgraphs 
> that were disconnected from each other.

It's edge-connectivity--the number of edges which would need to be removed to make the graph disconnected.


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Message: 2898

Date: Mon, 31 Dec 2001 12:52:38

Subject: Re: coordinates from unison-vectors

From: monz

Hi J!


> From: unidala <JGill99@xxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, December 31, 2001 8:05 AM
> Subject: [tuning-math] Re: coordinates from unison-vectors
>
>
> ...
>
> J Gill: So your (dynamite presentation!) XLS Spreadsheet ...


Thanks for the positive feedback!


> ... will allow determination of mean-tone "coefficients" which
> "best fit" any 5-limit periodicity block?


Well... the spreadsheet *allows* determination of the meantone
coordinates and axis... but it still can't *make* the determination!
The user still has to do some of the work.

The spreadsheet automatically figures out where to close
the meantone chain, at (+/- 1/2 determinant) generators.  But
I haven't yet "coordinated" [groan pun intended] the mathematics
to determine the "best-fit" meantone automatically.  So far, you
can only try out different values and see what they look like.



> Not being very adept with this PB stuff you guys apply,


Have patience, J... I've spent the night working on a terrific
graphic which will go into a Dictionary entry webpage for
"Transformation".  That should clear up a lot of the fog for you.


> what are the mean-tone "coefficients" which represent
> a "best-fit" for Ellis' "Duodene", with 12 pitches at:
>
> 1/1--16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8
> and
> 16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8--2/1
>
> and for
> 1/1--16/15--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--9/5--15/8
> and
> 16/15--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--9/5--15/8--2/1
>
>
> Curiously,  J Gill



Well, as I said, I don't have mathematics to do the job, so
I just try a few by eye and intuition, and see what looks good.

First of all, the first two of your Duodenes are equivalent to
each other on my lattice, because it's 2-dimensional for
prime-factors 3 and 5, with 2 ignored.

Secondly, I could use only that first pair, because the
second pair are 7-limit, making the lattice 3-dimensional,
and my spreadsheet doesn't handle that ... yet ...



So:


Ellis's Duodene is a 12-tone set defined by the unison-vectors
[0 3] and [4 -1], as Paul shows in his "Gentle Introduction to
Fokker Periodicity Blocks, part 2":
A gentle introduction to Fokker periodicity blocks, part 2, *



Here's my spreadsheet comparing the Duodene with two different
meantones: -2/9-comma and -1/4-comma:
Yahoo groups: /tuning-math/files/monz/duodene.xls *


And a graphic of the -1/4-comma meantone lattice:
Yahoo groups: /tuning-math/files/monz/ *(0,3) (4,-1) PB
and -1-4cmt.gif

(or without dealing with the broken link, just go to
Yahoo groups: /tuning-math/files/monz/ *
and download the file named "(0,3) (4,-1) PB and -1-4cmt.gif".)


Legend:

* the pink outlines the parallelogram boundaries of the
    periodicity-block according to the shape defined by the
    pair of unison-vectors,

* the blue plots the coordinates of the twelve 5-limit JI
    pitch-classes of Ellis's "Duodene" which lie within the
    boundaries of the periodicity-block,

* the yellow plots the more-or-less arbitrarily chosen
    meantones, whose chains are closed according to the
    interval spanned by one of the unison-vectors,

* the green connects each meantone pitch-class with its
    closest JI relative within the periodicity-block,
    closeness measured in pitch-height.

(I drew those green lines in by hand... so if you change
the meantone in the spreadsheet, the lines won't adjust.)


Note that this periodicity-block has three pitch-classes
which fall right on the eastern boundary: (2,-1), (2,0)
and (2,1).  All three of these thus have alternates a comma
lower -- that's would be -[4,-1] unison-vector--, which
would place them on the western boundary: (-2,0), (-2,1),
and (-2,2), respectively.

Also, since (-2,2) and (2,1) happen to fall right on
northwest and northeast *corners* of the boundary (respectively),
they also have lower alternates at the distance of the *other*
unison-vector, which would be -[0 3], which would place the
alternates at (-2,-1) and (2,-2), respectively.

With a total of 17 possible pitch-classes which may be used
to define this periodicity-block, the only meantone which goes
right down the middle of all of them is -1/4-comma.  That's
why I chose to use that for the graphic.




-monz









_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 2899

Date: Mon, 31 Dec 2001 03:22:03

Subject: Some 10 note 22 et scales

From: genewardsmith

We can also use the assoicated graph to analyze scales other than RI
scales; here is the connectivity of the scales having eight steps of
size 2 and two steps of size 3 in the 22-et:

c = 6

2222322223

c = 5

2222232223

c = 4

2222223223

c = 3

2222222323 and 2222222233

No surprises here, but there might be other things people think would
be worth analyzing.


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