Tuning-Math Digests messages 3225 - 3249

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Message: 3225

Date: Mon, 14 Jan 2002 23:17:31

Subject: Re: metric visualization

From: paulerlich

By the way, Kees, by "norm" I assume you meant the 2-norm . . . but 
that would be the Euclidean norm . . . would the appropriate taxicab 
norm be, instead, the maximum of the entries in the vector? Or what?


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Message: 3226

Date: Mon, 14 Jan 2002 23:52:10

Subject: Re: algorithm sought

From: dkeenanuqnetau

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> Anyway, Gene, what I was thinking... under the
> dyadic definition, all n-limit chords must be
> connected on the n-limit lattice, and must have
> a Hahn-diameter of 1.  See:
> 
> Music (and Music Theory) *
> 
> There ought to be a geometric way to find these
> structures...

Does this have to work for temperaments or only rational scales?


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Message: 3228

Date: Tue, 15 Jan 2002 06:23:05

Subject: Re: algorithm sought

From: clumma

>>Paul H. applied it to temperaments, and scales.  I'm using it
>>to define n-limit chords (rational only).  Did I make a mistake?
>>Part of the difficulty for me is, the smallest ASSs are 9-limit,
>>and that requires more than 3 dimensions.
>
>I would work in 3-dimensions for the 9-limit, and just make 3 half
>the size of 5 or 7. In other words, 
> 
> ||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc)
> 
>would be the length of 3^a 5^b 7^c. Everything in a radius of 2 of
>anything will be consonant.

Thanks, Gene.  I _really_ can't visualize this, but perhaps it
will provide a general method for finding the chords I seek.
Would everything still hold if I used 4 dimensions and kept all
edges the same length?  By gods, I can't figure out where you're
getting the coefficients here.  And what are the double pipes?
Not abs. -- there's a sqrt on the other side...  I confess I
don't know the distance formula for triangular plots.  I could
derrive it with trig. . . nope, it's a mess, 'cause there are many
different triangles involved in the different diagonals.  So I
guess I would use the standard Euclidean distance, but I need to
know how to get delta(x) and delta(y) off a triagular lattice.

-Carl


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Message: 3229

Date: Tue, 15 Jan 2002 21:42:53

Subject: [tuning] Re: badly tuned remote overtones

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> I don't recall ever getting a response to this.
> Still interested ...
> 
> 
> > From: monz <joemonz@y...>
> > To: <tuning@y...>; <tuning-math@y...>
> > Sent: Friday, January 11, 2002 2:04 PM
> > Subject: [tuning-math] Re: [tuning] Re: badly tuned remote 
overtones
> >
> >
> > 
> > First, I'd like to start this post off with a link to my
> > "rough draft" of a lattice of the periodicity-block Gene
> > calculated for Schoenberg's theory:
> > 
> > Internet Express - Error 404 *
pblock.gif
> > 
> > This shows the 12-tone periodicity-block (primarily 3- and 5-
limit,
> > with one 11-limit pitch), and its equivalent p-block cousins at
> > +/- each of the four unison-vectors.
> > 
> > 
> > Now to respond to Paul...
> > 
> > 
> > > From: paulerlich <paul@s...>
> > > To: <tuning@y...>
> > > Sent: Friday, January 11, 2002 12:47 PM
> > > Subject: [tuning] Re: badly tuned remote overtones
> > >
> > >
> > > You seem to be brushing some of the unison vectors you had 
> > > previously reported, and from which Gene derived 7-, 5-, and 2-
tone 
> > > periodicity blocks, under the rug. 
> > 
> > 
> > Ah ...  so then this, from Gene: ...
> > 
> > > From: genewardsmith <genewardsmith@j...>
> > > To: <tuning-math@y...>
> > > Sent: Wednesday, December 26, 2001 3:25 PM
> > > Subject: [tuning-math] Re: Gene's notation & Schoenberg lattices
> > >
> > > ... This matrix is unimodular, meaning it has determinant +-1.
> > > If I invert it, I get
> > >
> > > [ 7 12  7 -2  5]
> > > [11 19 11 -3  8]
> > > [16 28 16 -5 12]
> > > [20 34 19 -6 14]
> > > [24 42 24 -7 17]
> > >
> > 
> > ... actually *does* specify "7-, 5-, and 2-tone periodicity 
blocks".
> > Yes?

I thought he specified those in another post.


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Message: 3230

Date: Tue, 15 Jan 2002 23:10:41

Subject: Re: [tuning] Re: badly tuned remote overtones

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, January 15, 2002 5:09 PM
> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > > I think one would *have* to include a 5-limit "enharmonic
> > > > unison-vector" here, since Schoenberg explicitly equated A#=Bb,
> > > > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent
> > > > stuff.
> > > 
> > > Did he do this explicitly within any of the 'constructions
> > > of unison vectors' you gleaned from him?
> > 
> > 
> > Well, not specifically *this* interval.  But according to his
> > notational usage, *any* of the 5-limit enharmonicities should apply.
> 
> Right, but . . . did he apply any of them explicitly within any of 
> the 'constructions of unison vectors' you gleaned from him? 
> Otherwise, you're just "assuming the answer".


I wanted to attempt a more rigorous answer to this.

Back in July, when I started all this about Schoenberg, I wrote:


Yahoo groups: /tuning-math/messages/516?expand=1 *
> Message 516
> From: monz  <joemonz@y...> 
> Date: Wed Jul 18, 2001 5:16am
> Subject: lattices of Schoenberg's rational implications
>
>
> ...
>
> Schoenberg then extends the diagram to include the
> following overtones:
>
>  fundamental  partials
> 
>      F         2...12, 16
>      C         2...11
>      G         2...12


Here, I will further adapt Schoenberg's diagram to make
his explanation as clear as possible, by adding the 
partial-numbers and the fundamentals, which are the two
factors which when multiplied together give the relative
frequency-number of each note.  The fundamentals are
F = 4, C = 6, G = 9.
 
                                      d = 12*9 = 108
                                      c = 11*9 =  99
                                      b = 10*9 =  90
                                      a =  9*9 =  81
                                      g =  8*9 =  72
                    f = 11*6 = 66
  f = 16*4 = 64
                                     (f =  7*9 =  63)
                    e = 10*6 = 60
                    d =  9*6 = 54     d =  6*9 =  54
  c = 12*4 = 48     c =  8*6 = 48
                                      b =  5*9 =  45
  b = 11*4 = 44
                   (bb=  7*6 = 42)
  a = 10*4 = 40
  g =  9*4 = 36     g =  6*6 = 36     g =  4*9 =  36
  f =  8*4 = 32
                    e =  5*6 = 30
 (eb=  7*4 = 28)
                                      d =  3*9 =  27
  c =  6*4 = 24     c =  4*6 = 24
  a =  5*4 = 20
                    g =  3*6 = 18     g =  2*9 =  18
  f =  4*4 = 16
  c =  3*4 = 12     c =  2*6 = 12
  f =  2*4 =  8


>        (eb)            (bb)
>  c   d   e   f   g   a   b   c   d   e   f   g   a   b   c   d
>                        [44]            [64]
>        (28)            (42)            [66]
> 24  27  30  32  36  40  45  48  54  60  63  72  81  90  99 108
>
> 
> ...
> 
> The partial-numbers are also given for the resulting scale
> at the bottom of the diagram, showing that 7th/F (= eb-28)
> is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker
> than 5th/G (= b-45).
> 
> Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C
> (= f-66) are all weaker still, thus I have included them in
> square brackets.  These overtones are not even mentioned by
> Schoenberg.


These are all the unison-vectors implied by Schoenberg's diagram:

E  5*6=30 : Eb 4*7=28  =  15:14
B 11*4=44 : Bb 7*6=42  =  22:21
B  5*9=45 : B 11*4=44  =  45:44
B  5*9=45 : Bb 7*6=42  =  15:14
F 16*4=64 : F  7*9=63  =  64:63
F 11*6=66 : F 16*4=64  =  33:32
F 11*6=66 : F  7*9=63  =  22:21
A  9*9=81 :(A 20*4=80) =  81:80
C 11*9=99 :(C 24*4=96) =  33:32 

(The high "A" and "C" in parentheses are not explicitly indicated
by Schoenberg, but may be inferred from his theory.)

So the only 5-limit unison-vector indicated here is the 81:80
syntonic comma, and even that is only inferred but not stated.
Its applicability to his theory, as well as that of other
5-limit UVs, must be inferred from a careful study of other
explanations in _Harmonielehre_, as I indicated in my last post.

The 15:14 arises only in connected with the notes Schoenberg
himself placed in parentheses.  The other UVs are explicitly
indicated by Schoenberg.



-monz



 



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Message: 3231

Date: Tue, 15 Jan 2002 00:17:42

Subject: Re: algorithm sought

From: clumma

>Does this have to work for temperaments or only rational scales?

Paul H. applied it to temperaments, and scales.  I'm using it
to define n-limit chords (rational only).  Did I make a mistake?
Part of the difficulty for me is, the smallest ASSs are 9-limit,
and that requires more than 3 dimensions.

-Carl


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Message: 3232

Date: Tue, 15 Jan 2002 03:00:45

Subject: Re: Gene's relative connectedness

From: Pierre Lamothe

(PL) (GWS)
  > Do a connectedness of a scale relative to S, be a property of the scale or a
  > property of the relation between the scale and S?

  The scale and S.
So our concepts are not conflictual since the contiguity is a property of a mode in itself.
  There can be more than one path according to the definitions as given.
  Another way to define it is to define the graph of a scale relative to S, and
  then it is connected iff the graph is connected
It's not easy to see that sense from your definition since a graph is defined by a set of nodes and
a set of vertices. It seems the scale would be the set of nodes while S would be the set of vertices.
However in a graph, one vertice link two nodes, but it seems you relie your nodes with a chain of
vertices, implying other nodes, outside the scale, to rely nodes. If I understand the connectedness
in a graph, that corresponds to a possible way between any two nodes, using a unique vertice
between two nodes.
  The question of infinity is another one; however the rational numbers are an
  infinite field, and in fact any ordered field (for which you were giving some
  of the axioms a while back) is infinite
I use only finite sets. In the harmoid frame, I work with the finite classes mod <2> and a set
of intervals representing these classes (first octave or centered octave or pivots). Even if a
relation remains valid at infinity, I never use something requiring the existence of elements
outside the perceptible domain. I can plunge a gammoid structure in a group, for instance,
to show the link with the periodicity block, but I have access at all that from operations in a
finite set.

You could say that using logarithm implies the infinity of the real field. Yes, but I don't use the
operative properties of that field, restricting to Z-module, in which I restrict to classes mod <2>,
and finally I restrict to a finite area well-defined around the unison.

Pierre



[This message contained attachments]


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Message: 3233

Date: Tue, 15 Jan 2002 20:55:15

Subject: Re: [tuning] Re: badly tuned remote overtones

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, January 15, 2002 5:09 PM
> Subject: [tuning-math] [tuning] Re: badly tuned remote overtones
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > 
> > > From: paulerlich <paul@s...>
> > > To: <tuning-math@y...>
> > > Sent: Tuesday, January 15, 2002 3:42 PM
> > > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones
> > >
> > >
> > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > > > 
> > > > I think one would *have* to include a 5-limit "enharmonic
> > > > unison-vector" here, since Schoenberg explicitly equated A#=Bb,
> > > > C#=Db, D#=Eb, etc., the usual 12-EDO enharmonically-equivalent
> > > > stuff.
> > > 
> > > Did he do this explicitly within any of the 'constructions
> > > of unison vectors' you gleaned from him?
> > 
> > 
> > Well, not specifically *this* interval.  But according to his
> > notational usage, *any* of the 5-limit enharmonicities should apply.
> 
> Right, but . . . did he apply any of them explicitly within any of 
> the 'constructions of unison vectors' you gleaned from him? 
> Otherwise, you're just "assuming the answer".


Well, on p 176 of _Harmonielehre_ (p 155 of the Carter translation),
Schoenberg illustrates the "Circle of 5ths", and explicitly notates
the equivalences Cb=B, Gb=F#, and Db=C# for the major keys, and
ab=g#, eb=g#, and bb=a# for the minor keys that are +5, +6,
and +7 "5ths" (respectively) from the origin C-major/a-minor.

This is in Chapter 9, "Modulation", and all thru his discussion
of modulation Schoenberg assumes the enharmonic equivalence of
keys like this.  This is before he ever gets into anything about
the enharmonic equivalence of implied 5-limit harmonies.

He never explicitly says that a "5th" is always to be
interpreted as a 3:2, but his explanation of the basic tones
of the scale as overtones of the subdominant, dominant, and
tonic implies that he's thinking of "5ths" as either 3-limit
intervals or meantone generators, or both.  And in the
"Table of the Circle of Fifths for C Major (a minor)", on
the following page, he lists the keys next to each other going
in opposite directions around the circle, so that one column
has only the "sharp" keys and the other only the "flat" keys.
To me, this implies even more strongly a conception based on
a generating "5th", whether it's Pythagorean or meantone.
And again, the enharmonic equivalences are explicitly stated.


If meantone is assumed instead of Pythagorean, then the
enharmonic equivalence illustrated in Schoenberg's
"Circle of 5ths" is the diesis 128:125 = [7  0 -3].
So I can see why you might consider *this* to be the
"missing link" unison-vector.

So, to answer your question directly: Schoenberg equates
pairs of enharmonically-equivalent pitches, but no, he
never explicitly mentions whether those pairs of pitches
are derived via Pythagorean or meantone tuning.  So yes,
I'm assuming certain intervals as unison-vectors, based
as much as possible on the pitch-relationships explicitly
detailed by Schoenberg.  But I would venture to say, based
on what he wrote in _Harmonielehre_, that he expected *both*
the Pythagorean comma *and* the diesis (and all of their
combinations) to be tempered out.


> > 
> > > And anyway, why not 128:125? Seems simpler . . .
> > 
> > 
> > OK, Paul, I tried 128:125 in place of 2048:2025, and the
> > inverse I get is:
> > 
> >    [ 12  7 12  0  -9 ]
> >    [ 19 11 19  0 -14 ]
> >    [ 28 16 28  0 -21 ]
> >    [ 34 20 34 -1 -26 ]
> >    [ 41 24 42  0 -31 ]
> > 
> > 
> > So you're right ... this still shows the inconsistent
> > mapping to 11 in h12(11)=41, g12(11)=42.  Naturally, since
> > I only replace one row of the UV-matrix, there's only
> > one column of the inverse that's different (see? ... I really
> > *am* learning this stuff!), and that's the last column.
> >
> > ...
> > 
> > If I assume what is probably the most basic case, and
> > plug the Pythagorean comma into that row,
> 
> ??? Why is that the most basic case?


This is my thinking:  Schoenberg's theory certainly assumes
all the "traditional" enharmonic equivalence of the 12-EDO scale
-- by "traditional", I mean all of the enharmonic equivalences
that may arise in the 3- and 5-limits.

To my mind, the 3-limit (linear, 1-D) is both historically and
conceptually more basic than 5-limit (planar, 2-D).  The 
notational difference between a "sharp" and what later became
its enharmonically equivalent "flat", ocurred first in Pythagorean
tuning.  And so, along this line of reasoning, the Pythagorean
comma is historically and conceptually a more basic enharmonicity
than any of the 5-limit examples.  However, as implied above, I
will also grant the possibility that Schoenberg may have intended
the diesis as a unison-vector, and will examine that case below
as well.


Also, I understand Gene's "notation" a little better now.
So, taking this particular matrix as an example,


      2  3  5  7 11       unison-vector    ~cents

   [ -2  2  1  0 -1 ]  =      45:44      38.90577323
   [-19 12  0  0  0 ]  =  531441:524288  23.46001038
   [ -5  1  0  0  1 ]  =      33:32      53.27294323
   [  6 -2  0 -1  0 ]  =      64:63      27.2640918
   [ -4  4 -1  0  0 ]  =      81:80      21.5062896


inverse

   [ 12  -7  12  0  12 ]
   [ 19 -11  19  0  19 ]
   [ 28 -16  28  0  27 ]
   [ 34 -20  34 -1  34 ]
   [ 41 -24  42  0  41 ]


So, for an example of how the unison-vector maps to
a homomorphism, the matrix describing the mapping
of 45:44 to h12 is:


   [ 12 ]  [ -2  2  1  0  -1 ]
   [ 19 ]  
   [ 28 ]
   [ 34 ]
   [ 41 ]

which translates into

    (12*-2)+(19*2)+(28*1)+(34*0)+(41*-1)
  =   -24  +  38  +  28  +   0  +  -41
  =   1


So when I look at how all the unison-vectors map to
the homomorphisms, I get:

     homomorphism
  h12 -h7 h12 h0 h12   unison-vector

  [  1  0  0  0  0 ]      45:44
  [  0  1  0  0  0 ]  531441:524288
  [  0  0  1  0  0 ]      33:32
  [  0  0  0  1  0 ]      64:63
  [  0  0  0  0  1 ]      81:80


So now let me try to get this straight.  This matrix is
telling us that one of three mappings to 12-EDO may be
chosen, in which we distinguish either 45:44, 33:32, or
81:80 as pairs of distinct notes.  Correct?


Plugging 128:125 into the 2nd row instead of the Pythagorean
comma, a look at the mapping of unison-vectors gives us
exactly the same matrix as above.  (But of course, this
time the last column of the inverse gives a 9-EDO rather
than 12-EDO mapping, so that there are two 12-EDO mappings
this time rather than three.)

So assuming 128:125 to be a unison-vector, we still may choose
between either of two 12-EDO mappings, in which we distinguish
either 45:44 or 33:32.  With 128:125 as a unison-vector, along
with the others we use here, 12-EDO *always* tempers out the
syntonic comma 81:80.  With the Pythagorean Comma as a UV instead,
12-EDO may or may not temper out the syntonic comma, depending
on which homomorphism is chosen.


> 
> > So, now it seems that I've found the inconsistency in
> > Schoenberg's mapping of 5 as well.
> 
> Only if you assume the Pythagorean comma, right?


Right -- that's the only example I've found so far
which changes the mapping of 5.

Gene's PB (using 56:55 as a UV) found an inconsistent
mapping to 7, and he and I have both found several which
map 11 inconsistently.



> You need to include the prime-factor 2 for PB calculations
> too, if you're to weed out cases of torsion.


Ahh! ... now *that's* a useful little tidbit!! Thanks!


 
> Any of the PBs that give you a determinant of 12, if all the unison 
> vectors are tempered out, implies 12-tET. Geometrically, this will be 
> modeled by a torus or hyper-torus . . . can you make out the 
> inflatable torus model in the photocopy of the Hall article I sent 
> you (sorry the photocopy didn't come out so good -- check your 
> library for a better version)?


The picture in the Hall article is pretty hard to make out ...
but there's an identical diagram of a "Chicken-wire Torus"
in "Parsimonious Graphs: A Study in Parsimony, Contextual
Transformations, and Modes of Limited Transposition" by Jack
Douthett and Peter Steinbach, in _Journal of Music Theory_ 42:2
(Fall 1998), on p 248.  So I understand how it works and can
see it to some degree on this diagram.  Of course, the actual
*physical* model Hall used is preferable ... where can I get one?!



-monz



 



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Message: 3234

Date: Tue, 15 Jan 2002 21:47:47

Subject: [tuning] Re: badly tuned remote overtones

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> I also never got replies on my questions here, and
> am still waiting.  I'm particularly curious about
> how 56/55 was added as a unison-vector.  Thanks.
> 
> 
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Friday, January 11, 2002 1:13 AM
> > Subject: [tuning-math] [tuning] Re: badly tuned remote overtones
> >
> >
> > Hi Paul and Gene,
> >
> >
> >
> > > From: paulerlich <paul@s...>
> > > To: <tuning@y...>
> > > Sent: Thursday, January 10, 2002 10:10 AM
> > > Subject: [tuning] Re: badly tuned remote overtones
> > >
> > >
> > > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > The periodicity-blocks that Gene made from my numerical 
analysis
> > > > of Schoenberg's 1911 and 1927 theories are a good start.
> > >
> > > Well, given that most of the periodicity blocks imply not 12-
tone,
> > > but rather 7-, 5-, and 2-tone scales, it strikes me that 
Schoenberg's
> > > attempted justification for 12-tET, at least as intepreted by 
you,
> > > generally fails. No?
> >
> >
> >
> > I originally said:
> >
> >
> > > From: monz <joemonz@y...>
> > > To: <tuning-math@y...>
> > > Sent: Tuesday, December 25, 2001 3:44 PM
> > > Subject: [tuning-math] lattices of Schoenberg's rational 
implications
> > >
> > >
> > > Unison-vector matrix:
> > >
> > > 1911 _Harmonielehre_ 11-limit system
> > >
> > > ( 1  0  0  1 ) = 33:32
> > > (-2  0 -1  0 ) = 64:63
> > > ( 4 -1  0  0 ) = 81:80
> > > ( 2  1  0 -1 ) = 45:44
> > >
> > > Determinant = 7

Well there you go. This shows that Schoenberg's UVs imply a 7-tone 
system, not a 12-tone system as you claim.

> > > ... <snip> ...
> > >
> > > But why do I get a determinant of 7 for the 11-limit system?
> > > Schoenberg includes Bb and Eb as 7th harmonics in his 
description,
> > > which gives a set of 9 distinct pitches.  But even when
> > > I include the 15:14 unison-vector,

In place of which one above?

> > > I still get a determinant
> > > of -7.  And if I use 16:15 instead, then the determinant
> > > is only 5.

15:14 and 16:15 are both clearly semitones. Why would you use them as 
UVs?

> >
> >
> > > From: genewardsmith <genewardsmith@j...>
> > > To: <tuning-math@y...>
> > > Sent: Wednesday, December 26, 2001 12:27 AM
> > > Subject: [tuning-math] Re: lattices of Schoenberg's rational
> implications
> > >
> > >
> > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > Can someone explain what's going on here, and what candidates
> > > > may be found for unison-vectors by extending the 11-limit 
system,
> > > > in order to define a 12-tone periodicity-block?  Thanks.
> > >
> > > See if this helps;
> > >
> > > We can extend the set {33/32,64/63,81/80,45/44} to an
> > > 11-limit notation in various ways, for instance
> > >
> > > <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]
> > >
> > > where g7 differs from h7 by g7(7)=19.
> >
> >
> > Gene, how did you come up with 56/55 as a unison-vector?

Because it works to get you 12 as one of the five resulting 
cardinalities.

> > Why did I get 5 and 7 as matrix determinants for the
> > scale described by Schoenberg, but you were able to
> > come up with 12?

Only by replacing 33/32 with 56/55.


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Message: 3235

Date: Tue, 15 Jan 2002 00:26:53

Subject: Re: algorithm sought

From: clumma

> Paul H. applied it to temperaments, and scales.  I'm using it
> to define n-limit chords (rational only).  Did I make a mistake?

To put it another way: every note in the chord must be connected
to every other note by exactly one lattice link.  In the brute-
force method, the huge majority of chords will later fail the
test.  Why not generate them directly by searching the lattice?
There's got to be some graph theory somewhere that will do this.

-Carl


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Message: 3236

Date: Tue, 15 Jan 2002 09:56:26

Subject: more graph theory terminology

From: clumma

It seems that undirected graphs with diameter 1 are considered
"complete" graphs.  It's easy to generate these at any card k.
They all have edge connectivity k-1.  So the problem of finding
all k-card chords in the n-limit may be equivalent to finding
all the orientations of the complete order-k graph in the n-
limit lattice.

And this may be equivalent to finding all the 1-diameter order-k
subgraphs of the n-limit lattice (when the lattice is a directed
graph).  Or maybe directedness isn't what we want to use to
differentiate the edges of the lattice.  There's also a concept
called "weighting" -- weighted graphs are called "networks".

-Carl


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Message: 3237

Date: Tue, 15 Jan 2002 21:48:01

Subject: Re: more graph theory terminology

From: clumma

>>> So it seems a 1-diameter subgraph is also called a clique? 
>> 
>>Unless the author defines clique to be maximal. The cliques of a  
>>graph give us consonant chords in the sense I've been using it on 
>>the masses of asses thread.
> 
>Look up "saturated" in Monz' dictionary -- perhaps this is
>relevant?

The maximal cliques of the n-limit lattice are the n-limit
saturated chords, I think.  My problem is to find all the
order-k cliques in the lattice.  [Hope I'm not making any
errors here -- I'm an utter neophyte with the graph theory.]

-Carl


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Message: 3238

Date: Tue, 15 Jan 2002 00:39:50

Subject: Re: algorithm sought

From: genewardsmith

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

> Paul H. applied it to temperaments, and scales.  I'm using it
> to define n-limit chords (rational only).  Did I make a mistake?
> Part of the difficulty for me is, the smallest ASSs are 9-limit,
> and that requires more than 3 dimensions.

I would work in 3-dimensions for the 9-limit, and just make 3 half the sizeof 5 or 7. In other words, 

||3^a 5^b 7^c|| = sqrt(a^2 + 4b^2 + 4c^2 + 2ab + 2ac + 4bc)

would be the length of 3^a 5^b 7^c. Everything in a radius of 2 of anythingwill be consonant.


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Message: 3239

Date: Tue, 15 Jan 2002 09:58:23

Subject: Re: more graph theory terminology

From: clumma

BTW- On the topic of using edge connectivity to evaluate scales...
While it is impressive that connectivitiy is sufficient to select
the diatonic scale from all 7-tone meantones, the results may not
be so impressive when k is closer to card(n) (in the 5-limit,
card(n) is only 3, while k=7 for the diatonic scale).  After all,
we're not claiming that things like harmonics 6-12 are our
favorite 11-limit scales (actually, I like this scale a lot, but
I seem to be alone in this opinion around here...).  A more
important property of the diatonic scale may be that it has several
weakly-connected, complete subgraphs.  As weakly connecteded as
they can be, and still be so numerous and in-tune, I could argue.
Of course this is nothing new; we've been using "chord coverage"
for years, and chords/notes is practically what this list is about.
But if we can put this stuff into graph-theory terms, we may be
able to use the many powerful existing tools out there to soup-up
our searches.

-Carl


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Message: 3240

Date: Tue, 15 Jan 2002 21:53:04

Subject: [tuning] Re: badly tuned remote overtones

From: paulerlich

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

> OK, right ... I understand all that.
> 
> But where did it come from?  Is it the result of adding
> or subtracting two of the already-existing unison-vectors?
> That wouldn't work, would it? ... because all the vectors
> in the matrix have to be independent.

Any vector that is _not_ the result of adding or subtracting two of 
the already-existing unison-vectors would work to create what Gene  
calls a "notation" (but is nothing like a musical notation anyone's 
ever seen before). In this case, he made a choice (56/55) that makes 
the 12-tone system come out nice in this "notation".


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Message: 3241

Date: Tue, 15 Jan 2002 00:53:19

Subject: Re: algorithm sought

From: clumma

>>Paul H. applied it to temperaments, and scales.  I'm using it
>>to define n-limit chords (rational only).  Did I make a mistake?

Turns out _diameter_ is already graph-theory terminology, and it
is the term we want.

I'm afraid I didn't keep the URL for the source of:

"Let G be a graph and v be a vertex of G. The eccentricity of the
vertex v is the maximum distance from v to any vertex. That is,
e(v)=max{d(v,w):w in V(G)}."

"The diameter of G is the maximum eccentricity among the vertices
of G. Thus, diameter(G)=max{e(v):v in V(G)}."

The "radius" of G is the minimum eccentricity.

-Carl


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Message: 3242

Date: Tue, 15 Jan 2002 10:35:08

Subject: Re: more graph theory terminology

From: genewardsmith

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

> But if we can put this stuff into graph-theory terms, we may be
> able to use the many powerful existing tools out there to soup-up
> our
searches.

You might also reverse the process, and try to see what a graph theory
concept means in scale terms. Things such as the chromatic number,
domination number, genus and so forth strike me as interesting in that
way.


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Message: 3243

Date: Tue, 15 Jan 2002 21:57:29

Subject: Re: algorithm sought

From: paulerlich

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

> >For the 5-limit, it is
> > 
> > ||3^a 5^b|| = sqrt(a^2 + ab + b^2)
> > 
> > For the 7-limit
> > 
> > ||3^a 5^b 7^c|| = sqrt(a^2 + b^2 + c^2 + ab + ac + bc)
> 
> Is there a post where the derivation for this is given?
> 
> >Beyond that we need to decide if 3 stays the same size as 5, 7,
> >and 11, or is half as long.
> 
> You can expect me to become completely confused when dealing
> with prime limits.

Gene is talking about odd limits, just as we are. He's just 
attempting to capture them in a Euclidean lattice with prime axes. 
I'm concerned that this won't always work -- Gene, perhaps you could 
post the implied lengths for

1:3
1:5
3:5
1:7
3:7
5:7
1:9
5:9
7:9
1:11
3:11
5:11
7:11
9:11

etc. (up to whatever odd limit you could see yourself being 
interested in).


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Message: 3244

Date: Tue, 15 Jan 2002 10:45:11

Subject: Re: more graph theory terminology

From: clumma

>>But if we can put this stuff into graph-theory terms, we may be
>>able to use the many powerful existing tools out there to soup-up
>>our searches.
> 
>You might also reverse the process, and try to see what a graph
>theory concept means in scale terms. Things such as the chromatic
>number, domination number, genus and so forth strike me as
>interesting in that way.

I bet!  There seeems to be no shortage of terms in this field.
I didn't actually run across domination number or genus.  I
did see chromatic number, but didn't stop to read the def.  I'll
go back and get these -- thanks for the pointers.

So it seems a 1-diameter subgraph is also called a clique?  Anyway,
the problem of finding n-limit chords seems related to something
called the Clique Problem.  Naturally, the mathematicians are more
interested in wether problems can be done or not than (the usually
much easier problem of) actually finding the answer. . .

"Algorithms for the Maximum Clique Problem
A clique in a graph is a set of vertices which are pairwise
adjacent.  The CLIQUE problem is to determine given a graph
G and an integer k, whether G has a k-clique. Although this
problem is NP-complete, several practical algorithms exist."

...'but we're not going to tell you what they are...'

-------

Other promising leads I haven't checked yet:

http://www1.ics.uci.edu/~eppstein/pubs/Epp-TR-94-25.pdf - Ok *

"Approximating Minimum-Size k-Connected Spanning Subgraphs via
Matching", Joseph Cheriyan, Ramakrishna Thurimella.

-Carl


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Message: 3247

Date: Tue, 15 Jan 2002 11:11:15

Subject: Re: more graph theory terminology

From: genewardsmith

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

> So it seems a
1-diameter subgraph is also called a clique? 

Unless the author defines clique to be maximal. The cliques of a graph
give us consonant chords in the sense I've been using it on the masses
of asses thread.

> A clique in a graph is a set of vertices which are pairwise
> adjacent.  The CLIQUE problem is to determine given a graph
> G and an integer k, whether G has a k-clique. Although this
> problem is NP-complete, several practical algorithms exist."

Graph theory is unfortunately a great source of NP-complete problems.
We might look for a graph theory package which does this--Mathematica
has a bigger one than Maple, so perhaps I should check that out.


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Message: 3249

Date: Tue, 15 Jan 2002 03:13:12

Subject: Re: Gene's relative connectedness

From: genewardsmith

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:

> or scale and degrees have another sense where it could exist more than one
> path between the first and the last degrees of the scale.

There can be more than one path according to the definitions as given.
Another way to define it is to define the graph of a scale relative to S, and then it is connected iff the graph is connected, for which see

http://mathworld.wolfram.com/connectedgraph.html *


> How to believe that a "relative connectedness" could convey an exact sense, when
> it seems that that conveys about nothing? 

A scale can for example be connected in the 7-limit but not in the 
5-limit, so it conveys something.

> Do a connectedness of a scale relative to S, be a property of the scale or a property
> of the relation between the scale and S?

The scale and S.

> I suppose Gene seeks to adapt, in his simplest way, my concept on contiguity. 

No, I seek to apply standard mathematical concepts of path-connectedness ingraphs to this particular situation.

> A gammier mode is not relatively connected but absolutely connected sincethe unique
> set A for which it has sense to refer the connectedness is strictly determined by the
> gammier structure itself. 

That could be done with RI scales, but is in general too restrictive for myintended applications.

> Why
I would want to introduce infinity, like your simplest way, while my
way is so short?

The question of infinity is another one; however the rational numbers
are an infinite field, and in fact any ordered field (for which you
were giving some of the axioms a while back) is infinite.
In any event, I want to consider more than RI scales.


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