Tuning-Math Digests messages 1550 - 1574

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Message: 1550

Date: Wed, 05 Sep 2001 23:03:45

Subject: Re: Tenney's harmonic distance

From: genewardsmith@xxxx.xxx

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

> You better believe it! So, any comments on the questions I asked?

I'll look at it again, but I have some questions also:

(1) Can you define harmonic entropy in terms of your taxicab metric, 
or if not in any terms you like?

(2) Do you know how to retune a midi file in such a way that the 
pitches are set to anything you choose?


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Message: 1551

Date: Wed, 05 Sep 2001 23:22:00

Subject: Re: Question for Gene

From: genewardsmith@xxxx.xxx

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

> In particular, I'm assuming a city-block or taxicab metric. Is Kees 
> observing that in his final lattice? It looks like he isn't.

Kees is talking about circles and transforming as if in a Euclidean 
space, so you aren't on the same wavelength.

> What else can you say?

I'm not sure what your triangular lattice metric is. A taxicab needs 
two lines to run along; you can make these 120 degrees to each other 
but you can't get an array of equilateral triangles out of it.

By "lattice", mathematicians usually mean one of two things. The 
first has to do with partial orderings and need not concern us, the 
second defines a lattice as a discrete subgroup of R^n whose quotient 
group is compact. I'm not always sure what people mean when they say 
lattice in this neighborhood.


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Message: 1552

Date: Wed, 05 Sep 2001 01:14:22

Subject: Re: Distance measures cut to order

From: genewardsmith@xxxx.xxx

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

> One of the most interesting mappings done by Wilson is his mapping
> onto a Penrose tiling, treating it as a two dimensional
> representation of a 5 dimensional space.

Wowsers!

  When nines or fifteens
> are treated as independent axes from threes and fives,
> interesting 'wormholes' in the lattice start to appear, where
> alternative representations of the same pitch class occur in
> surprisingly different contexts.

These wormholes will appear no matter what metric you use. They might 
be thought of as universal commas--3^2 is "approximated" by 9.


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Message: 1553

Date: Thu, 06 Sep 2001 18:38:19

Subject: Re: about hypothesis and theorem

From: genewardsmith@xxxx.xxx

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

I was looking at your 25/24 and 27/20 example again, and it seems to 
me your objection that my conditions on validiy are too weak is well-
taken. I get 

    [ 5]
v = [ 7]
    [11]

for the corresponding val, and 1-6/5-4/3-3/2-5/3-(2) for the block. 
The trouble is, the scale steps are not in order! We have v(1)=0, 
v(6/5)=1, v(4/3)=3, v(3/2)=2, v(5/3)=4, and it seems we should not 
allow such a beast. Perhaps requiring that the scale steps be in 
order for a set to be valid would be enough.


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Message: 1554

Date: Thu, 06 Sep 2001 18:59:25

Subject: Re: Theorem Paul

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., genewardsmith@j... wrote:

> If {n_i} are a set of generators for the kernel of a valid val v 
(or 
> in other words, if they generate the dual group to the group 
> generated by v) then we call the set valid. 

This needs to be changed to the following:

If n={n_i} are a set of generators for the kernel of a valid val v, 
and if B is the block defined by n and octave equivalence, and if the 
elements of B b_i are placed in ascending order by ascending values 
of v(b_i), then we will call the set valid.

The reason why the new condition is essential is that we are creating 
MOS by finding something close to the v(2)-et. That is a sort of 
super-MOS, with only one step size and everything as smooth as 
possible. However, if the steps are not in order it is not a super 
MOS at all, and we don't *want* to get close to it!


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Message: 1555

Date: Thu, 06 Sep 2001 15:46:22

Subject: Re: about hypothesis and theorem

From: Pierre Lamothe

In post 979 <genewardsmith@j...> wrote:

<< We find that h_4 has the property
   h_4(25/24)=0 and h_4(27/20)=1 >>

My question concerned an homomorphism in G = <2 3 5> Z^3
such that H(25/24) = 0 and H(27/20) = 0.

Using X = (x,y,z) this homomorphism is

  H(X) = 5x + 8y + 14z

and its projection in G/<2>Z is

  H(X mod 2) = (3y + 4z) mod 5

-----
 
The validity condition used in the theorem appears
independant of the primes, requiring only positive vals.

So, in G = <2 3 5> Z^3, forget the basis <2 3 5> and
replace it by B = <2 p1 p2> where p1 and p2 are unknown.

The unison vectors are now v = (i,3,-1) and u = (j,-1,2)
which were 27/20 and 25/24 in the basis <2 3 5>. Here
i and j are unknown, but the periodicity remains 5 and
the represention modulo 2 of the kernel (class 0 or
sublattice generated by u and v) is the same. What may
change is only the ordering of classes within the block.


        0         0         0      
    0         *         0         0
0         0     X X *         0    
      0         0 X X     0        
  0         0         *         0  
        0         0         0      
    0         0         0         0


The homomorphism, which was H(X) = 5x + 8y + 14z, is now

           [x  i  j]
H(X) = det [y  3 -1] = 5x - (2i + j)y - (i + 3j)z
           [z -1  2]

where i and j correspond to the "modality" of the unison
vectors. In the basis <2 3 7> this set of unison vectors
is perfectly valid and the intervals between the elements
of the block are precisely the complete slendro gammier.

I could have chosen a more "pathological" (skewed) case.
This one underline the problem with a weak conception
using periodicity block.

In the basis <2 3 5> the classes modulo 5 of the intervals
between the elements of the block are identical to the
<2 3 7> case with

H(X mod 2) = (3y + 4z) mod 5

        0         0         0      
    0         *         0         0
0         * 3 1 4 2 *         0    
      0     4 2 0 3 1     0        
  0         * 3 1 4 2 *         0  
        0         *         0      
    0         0         0         0

while these intervals correspond precisely to the Zarlino
gammier which is heptatonic and not pentatonic. Where's the
problem?

  [ I will neglect in the following the skewness
    of the mesh determined by a particular set of
    unison vectors for a given homomorphism. I
    want to focus on ordering and one can suppose
    it's the simplest block, so having the minimal
    complexity product or sonance sum of vectors. ]

Any homomorphism determines a partial ordering structure in
a lattice corresponding to its classes. Indeed, each vector
X is "labeled" by H(X) and the set of vectors is partially
ordered by the total order (... -3 -2 -1 0 1 2 3 ...) in Z.

For each "label" (or class) there exist an infinity of
vectors (or intervals) giving a dense recovering of all the
octaves. So this infinite ordering has nothing to do with
the ordering of the "size" (width) of the intervals (even
if it serves to find temperaments).

The algebra of classes has sense only to give consistency.
What is required in JI is a corresponding partial algebra of
intervals in an minuscule domain around the unison where it
remains possible to perceive difference in sonance quality.

So here's an essential condition in the use of unison vectors
and periodicity block:

   ---------------------------------------------------------
   The order of classes has to correspond to order of widths
   for the intervals of the chosen (supposed minimal) block,
   and consequently for the intervals between these elements. 
   ---------------------------------------------------------

Comparing the order in the "pathological" case with <2 3 5>
and the valid case with <2 3 7> we have

  0    1     2     3     4 

  1   9/8  15/8   3/2   5/4
  1   5/3   4/3  10/9  16/15
  1   8/5   9/5  16/15  6/5

  1   9/8  21/16  3/2   7/4
  1   7/6   4/3  12/7  16/9
  1   8/7   9/7  32/21 12/7


If the ordering structure is not considered, the pathological
structure is isomorph to the valid one (for the composition
of the intervals). It's a CS but non ordered. If we add width
ordering to these structures it's no longer isomorph.

-----

Using the 13 elements intervals generated by the "pathological"
block, one can use my methods deriving from gammier theory to
find the corresponding valid set of unison vectors.

Considering the ordered set of these intervals (width order),
we have simply to find the atoms of the set which are those that 
cannot be factorized in "inferior" elements distinct of unison
within this set.

These atoms are here 16/15, 10/9 and 9/8. If this set of 13
intervals is consistent, we will find an homomophism such that

  H(16/15) = H(10/9) = H(9/8) = 1

which is effectively

  H(X) = 7x + 11y + 16z

giving

  H(X mod 2) = (4x + 2z) mod 7


      0             0            0 
0             *             0      
        *   1 4 2 6   *             0
  0         6 3 0 4 1         0    
          *   1 4 2 6   *          
    0             *             0
            0             0 
      0             0

The simplest mesh is determined by 81/80 and 25/24

      0             0            0 
0             *             0      
        0       2 6 3 *             0
  0             0 4 1 5       0    
          0             *          
    0             0             0
            0             0 
      0             0

Finally, translating the block to fit within our
13 intervals we have the well-known Zarlino mode.

      0             0            0 
0             *             0      
        *     5 2 6   *             0
  0           3 0 4 1         0    
          *             *          
    0             *             0
            0             0 
      0             0


Pierre


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Message: 1556

Date: Thu, 06 Sep 2001 20:46:10

Subject: Re: Tenney's harmonic distance

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > You better believe it! So, any comments on the questions I asked?
> 
> I'll look at it again, but I have some questions also:
> 
> (1) Can you define harmonic entropy in terms of your taxicab 
metric, 
> or if not in any terms you like?

Look over the harmonic entropy list (are you subscribed yet)? It's a 
continuous function of interval size, has local minima at the simple 
ratios, is conceptually very simple . . .
> 
> (2) Do you know how to retune a midi file in such a way that the 
> pitches are set to anything you choose?

Lots of people should be able to help you with this. Herman Miller?


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Message: 1557

Date: Thu, 06 Sep 2001 20:50:37

Subject: Re: Question for Gene

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > In particular, I'm assuming a city-block or taxicab metric. Is 
Kees 
> > observing that in his final lattice? It looks like he isn't.
> 
> Kees is talking about circles and transforming as if in a Euclidean 
> space, so you aren't on the same wavelength.

What about the previous lattices on that page?
> 
> > What else can you say?
> 
> I'm not sure what your triangular lattice metric is. A taxicab 
needs 
> two lines to run along; you can make these 120 degrees to each 
other 
> but you can't get an array of equilateral triangles out of it.

Well, they're not equilateral triangles, but they are triangles. The 
metric is the shortest path along the edges of this triangular graph. 
Is it not correct to call that a "taxicab metric"?
> 
> By "lattice", mathematicians usually mean one of two things. The 
> first has to do with partial orderings and need not concern us, the 
> second defines a lattice as a discrete subgroup of R^n whose 
quotient 
> group is compact. I'm not always sure what people mean when they 
say 
> lattice in this neighborhood.

Sir, there is an accepted definition of "lattice" that is used in 
geometry and crystallography. Every point, and its local connections, 
is congruent to every other point and its local connections . . . 
something like that. We had this discussion a long time ago on the 
tuning list.


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Message: 1558

Date: Thu, 06 Sep 2001 20:50:57

Subject: Re: Tenney's harmonic distance

From: genewardsmith@xxxx.xxx

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

> Look over the harmonic entropy list (are you subscribed yet)? It's 
a 
> continuous function of interval size, has local minima at the 
simple 
> ratios, is conceptually very simple . . .

I looked at it; I think you need to upload something to the files 
which defines what the group is about by defining harmonic entropy.


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Message: 1559

Date: Thu, 06 Sep 2001 20:53:37

Subject: Re: about hypothesis and theorem

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:
> 
> > Could you show how the hypothesis, the definitions, the 
conditions 
> of
> > validity and the theorem would be applied in this case? Could you 
> exhibit a
> > generator and a scale?
> 
> We find that h_4 has the property h_4(25/24)=0 and h_4(27/20)=1. We 
> then look at vals of the form t*h_5 + h_4, and when t=1 we get
> 
>     [ 9]
> g = [13]
>     [20].
> 
> Note that this is *not* h_9, which has coordinate values 9, 14 and 
21.
> However, 7/5 is a semiconvergent to 13/9, 11/5 is a semiconvergent 
to 
> 20/9 and for that matter 1/5 is a semiconvergent to 2/9. We get a 
> scale of pattern 22221, 5 steps in a 9-et. It may not do a very 
good 
> job of representing your "pathological" block, but then 27/20 is 
not 
> much of a comma. If you want to exclude this kind of thing we need 
to 
> change the statement of the theorem, but then we must ask what, 
> exactly, people want to prove.

I think we have to add the condition that the JI block, pre-
tempering, is CS. My proof doesn't work otherwise.


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Message: 1560

Date: Thu, 06 Sep 2001 20:57:51

Subject: Re: Tenney's harmonic distance

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Look over the harmonic entropy list (are you subscribed yet)? 
It's 
> a 
> > continuous function of interval size, has local minima at the 
> simple 
> > ratios, is conceptually very simple . . .
> 
> I looked at it; I think you need to upload something to the files 
> which defines what the group is about by defining harmonic entropy.

There are a couple of posts which tell you how to calculate harmonic 
entropy . . . let me know if those are clear.


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Message: 1561

Date: Thu, 06 Sep 2001 22:07:34

Subject: Re: about hypothesis and theorem

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:
> 
> In post 979 <genewardsmith@j...> wrote:
> 
> << We find that h_4 has the property
>    h_4(25/24)=0 and h_4(27/20)=1 >>
> 
> My question concerned an homomorphism in G = <2 3 5> Z^3
> such that H(25/24) = 0 and H(27/20) = 0.
> 
> Using X = (x,y,z) this homomorphism is
> 
>   H(X) = 5x + 8y + 14z

Actually, this is the homomorphism for 48/25 and 27/20. We can find 
the right one by taking the determinant of

[ x  y  z]
[-2  3 -1]
[-3 -1  2],

which is 5x + 7y + 11z.

> Any homomorphism determines a partial ordering structure in
> a lattice corresponding to its classes. Indeed, each vector
> X is "labeled" by H(X) and the set of vectors is partially
> ordered by the total order (... -3 -2 -1 0 1 2 3 ...) in Z.

I just told Paul this definition of lattice we needed worry about and 
now you go and use it. :)

> So here's an essential condition in the use of unison vectors
> and periodicity block:

>    ---------------------------------------------------------
>    The order of classes has to correspond to order of widths
>    for the intervals of the chosen (supposed minimal) block,
>    and consequently for the intervals between these elements. 
>    ---------------------------------------------------------

This seems to be what I have just proposed.

> Using the 13 elements intervals generated by the "pathological"
> block, one can use my methods deriving from gammier theory to
> find the corresponding valid set of unison vectors.

Where is gammier theory described?

Welcome back, Pierre!


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Message: 1562

Date: Thu, 06 Sep 2001 22:25:01

Subject: Re: Tenney's harmonic distance

From: genewardsmith@xxxx.xxx

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

> There are a couple of posts which tell you how to calculate 
harmonic 
> entropy . . . let me know if those are clear.

Which posts?


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Message: 1563

Date: Thu, 06 Sep 2001 22:28:58

Subject: Re: Tenney's harmonic distance

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > There are a couple of posts which tell you how to calculate 
> harmonic 
> > entropy . . . let me know if those are clear.
> 
> Which posts?

350 . . . follow the thread from there.


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Message: 1564

Date: Thu, 06 Sep 2001 03:26:37

Subject: about hypothesis and theorem

From: Pierre Lamothe

Hi Paul and tuning-math members

I was surprised to find intense (and abstract) activity on the List after
my vacation. It takes a while before I have leisure to read all that. I
regret to have not the possibility to participate. However I would like
simply to ask a question permitting to see it misses probably a condition.

Let u and v be the vectors 25/24 and 27/20 in the lattice <2 3 5> Z^3 whose
generic element is (2^x)(3^y)(5^z). The vectors u and v determine (with the
octave) the "pathologic" periodicity block <1 9/8 5/4 3/2 15/8> supposed
valid (in the theorem) since it corresponds to the homomorphism

   H(x,y,z) = 5x + 8y + 14z

Could you show how the hypothesis, the definitions, the conditions of
validity and the theorem would be applied in this case? Could you exhibit a
generator and a scale?

Pierre


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Message: 1565

Date: Thu, 06 Sep 2001 07:39:23

Subject: Re: Theorem Paul

From: genewardsmith@xxxx.xxx

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

> Now do you have a quick way of determining the generator of the 
> linear temperament, given n-1 commatic unison vectors?

Let's see if this helps:

Recall that a notation for the note group N_p was a k-tuple of vals 
[u_1, ..., u_k], where k = pi(p) is the number of primes up to p, and 
where the kxk square matrix we get by writing the vals as column 
vectors is unimodular, meaning it has determinant +-1. We may call 
this the *val matrix* for the notation; corresponding to it is a 
*basis matrix* which is the matrix inverse of the val matrix. The 
rows of the basis matrix are the basis notes of the notation, and it 
may also be written as a k-tuple of rational numbers
(q_1, ..., q_k) where if q_i = [e_1, e_2, ..., e_k] we also write it 
multiplicitively as the rational number q_i= 2^e_1 * 3^e_2 ... p^e_k.

We then have for any prime r <= p

r = q_1^u_1(r) * q_2^u_2(r) * ... q_k^u_k(r),

so that anything which can be written as the product of the first k 
primes can also be written as the product of q_1, ..., q_k; that is, 
both are a basis for the note group N_p.

I just downloaded Graham's midiconv program, and he is doing this 
sort of thing in his tun files. For instance, in 12from31.tun we find 
the matrix

[-3 -1  2]
[ 7  0 -3]
[-4  4 -1]

which is a basis matrix (since it is unimodular.) Inverting it we get 
the val matrix

[12  7 3]
[19 11 5]
[28 16 7],

which is the notation [h_12, h_7, h_3]. If you look at Graham's file 
you will see he is using this notation.

Every note in N_5 can be expressed in terms of this notation as

q = (25/24)^h_12(q) * (128/125)^h_7(q) * (81/80)^h_3(q)

just as it can also be written

q = 2^v_2(q) * 3^v_3(q) * 5^v_5(q),

where v_2, v_3 and v_5 are the 2-adic, 3-adic and 5-adic valuations 
of number theory.

Suppose now we want to temper out 81/80, so that we will write the 
approximation to q, ~q, as

~q = a^h_12(q) * b^h_7(q).

Finding a basis for this temperament means the same as tuning the 
above basis, which we may do in various ways, e.g., least squares. If 
we like we may assume ~2 = 2, in which case we really need to specify 
only one value, since the other than be found from

2  =  a^h_12(2) * b^h_7(2) = a^12 * b^7.

To take another example, consider the basis matrix defined by the 
5-tuple (176/175, 385/384, 8019/8000, 441/440, 540/539), which in 
matrix form is

[ 4  0 -2 -1  1]
[-7 -1  1  1  1]
[-6  6 -3  0  1]
[-3  2 -1  2 -1]
[ 2  3  1 -2 -1].

The inverse of this matrix is

[72   58 -31   53  46]
[114  92 -49   84  73]
[167 135 -72  123 107]
[202 163 -87  149 129]
[249 201 -107 183 159],

which is the notation [h_72, h_58, -h_31, h_53, h_46]. If we remove 
any one element from the basis 5-tuple, and take octave equivalence 
in its place, we get a JI block whose number of notes is abs(h(2)) 
for the val corresponding to the basis element we removed. For 
instance, by taking out 540/539, which is in the kernel of all the 
vals but h_46, which has instead h_46(540/539)=1, we get a block of 
46 notes. We may temper this in various ways by removing other 
val/basis pairs, getting equal, linear etc. temperaments. Thus for 
instance by tuning  ~q = a^h_72(q) (for instance, in the usual way!) 
we get the 46 block expressed in the 72-et. If we tune 
~q = a^h_72(q) * b^h_53(q), we get a linear temperament, and so forth.

We also have for example that ker(h_72) is generated by all the basis 
vectors except 176/175, where h_72(176/175)=1. Just as each val is 
associated to the group it generates (of rank one) and hence to the 
dual group, i.e. the kernel, of corank one (in this case, that would 
be rank four), every basis note q_i generates a rank one group, whose 
dual group null(q_i) is of corank one (in this case four again.) 

While null(q_i) is of corank one and has an infinity of elements, if 
we list only valid vals of the form u_n for integers n we get a 
finite list, which is an interesting thing to consider for any 
comma-like interval. For instance, 128/125 is associated in this way 
to multiples of 3 through 42, excluding h_6 which is invalid. In the 
same way, 25/24 is associated to 3,4,7,10,13 and 17; and 81/80 to 
5,7,12,19,26,31,43,45,50,55,67,69,74,81,88,98,105 and 117. If we 
place some limit based on a measure of goodness when we do this we of 
course can get an even smaller list.


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Message: 1566

Date: Thu, 06 Sep 2001 21:01:31

Subject: Re: about hypothesis and theorem

From: Pierre Lamothe

In post 988 <genewardsmith@j...> wrote:

<< 
   I just told Paul this definition of lattice we needed
   worry about and now you go and use it. :)
>>

In this context it's just funny it looks it's me who picked
the idea of another concerning the ordering condition. But
generally I began to think it's a problem for me to diffuse
ideas very slowly in a such forum. I'm now hesitating to
post on many subject, waiting to have time to write first in
my website.

I had begun, for example, many graphical studies (in vectorial
form) about JI relations relatively to MIRACLE, Canasta and
Blackjack scales. These images criticize, for example, ideas
like the use of an absolute convexity on a linear temperament.
A such convexity has only to reflect a multilinear convexity
and I consider there is a flaw around the conception of the
scales mentioned. Do I have to show now what I have and discuss
that or to wait I will have time to write an article where I will
attack vigourously with these ideas? I don't know yet. I judge
only I want to be credited for my works.

-----

About the homomorphism I had calculated.

I have always the good functions but errors in calculation. :)

I don't know where I picked the wrong values. I had prepared
many examples before to choose the best one illustrating the
necessity to use the ordering condition. I copied bad. Using
my formula 

              [x  i  j]
   H(X) = det [y  3 -1] = 5x - (2i + j)y - (i + 3j)z
              [z -1  2]

obviously, with u = 27/20 and v = 25/24, we have 

   i = -2 and j = -3

   H(X) = 5x + 7y + 11z

   H(X mod 2) = (2y + z) mod 5

This homomorphism determines the same sublattice as the first
one and is simply inversed relatively to it (so not detected)

   (x,y,z) --> 5x + 8y + 14z

   (x,y,z) mod 2 --> (3y + 4z) mod 5

Near the unison we have

    1                   4
  3 0 2  rather than  2 0 3
    4                   1

-----

<genewardsmith@j...> wrote:

<< Where is gammier theory described? >>

It's a typical example of the dissemination of my ideas. I
did'nt take time to write a condensed paper. You could find
snatches only in French on my website or a bit in bad English
on some of my hundred posts on the tuning lists. Just seek for 
"Pierre" or "Lamothe" or "Lamonthe".

-----

My visit on the list was only to talk about this ordering
condition I had already mentionned in the appendice of a
precedent message to J. Gill. This appendice talked about
another condition concerning the complexity ordering which
is assured when using only integers in the basis.


Pierre


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Message: 1567

Date: Thu, 06 Sep 2001 08:20:11

Subject: Re: about hypothesis and theorem

From: genewardsmith@xxxx.xxx

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

> Could you show how the hypothesis, the definitions, the conditions 
of
> validity and the theorem would be applied in this case? Could you 
exhibit a
> generator and a scale?

We find that h_4 has the property h_4(25/24)=0 and h_4(27/20)=1. We 
then look at vals of the form t*h_5 + h_4, and when t=1 we get

    [ 9]
g = [13]
    [20].

Note that this is *not* h_9, which has coordinate values 9, 14 and 21.
However, 7/5 is a semiconvergent to 13/9, 11/5 is a semiconvergent to 
20/9 and for that matter 1/5 is a semiconvergent to 2/9. We get a 
scale of pattern 22221, 5 steps in a 9-et. It may not do a very good 
job of representing your "pathological" block, but then 27/20 is not 
much of a comma. If you want to exclude this kind of thing we need to 
change the statement of the theorem, but then we must ask what, 
exactly, people want to prove.


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Message: 1568

Date: Thu, 06 Sep 2001 21:14:18

Subject: Re: Question for Gene

From: Pierre Lamothe

In post 984 <genewardsmith@j...> wrote:

<<

> By "lattice", mathematicians usually mean one of two things. The 
> first has to do with partial orderings and need not concern us, the 
> second defines a lattice as a discrete subgroup of R^n whose quotient 
> group is compact. I'm not always sure what people mean when they say 
> lattice in this neighborhood.

and Paul respond:

<<

Sir, there is an accepted definition of "lattice" that is used in 
geometry and crystallography. Every point, and its local connections, 
is congruent to every other point and its local connections . . . 
something like that. We had this discussion a long time ago on the tuning
list.

>>


We don't have this problem in French for we use "treillis" in the fist
sense of partial ordering where two elements have always inferior and
superior "bornes", and "réseau" in the sense of discrete Z-module.

Personally, I use both the "treillis" (in melodic representation) and the
"réseaux" (in harmonic representation). I have the problem to be understood
with the unique term "lattice".


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Message: 1570

Date: Fri, 07 Sep 2001 18:35:54

Subject: Re: Distance measures cut to order

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> 
> > One of the most interesting mappings done by Wilson is his mapping
> > onto a Penrose tiling, treating it as a two dimensional
> > representation of a 5 dimensional space.

Was this posted a while ago? I don't see this post in today's 
archives.
> 
> Wowsers!

Well, it's a two dimensional _slice_ through a 5-dimensional space, 
as you probably know . . .
> 
>   When nines or fifteens
> > are treated as independent axes from threes and fives,
> > interesting 'wormholes' in the lattice start to appear, where
> > alternative representations of the same pitch class occur in
> > surprisingly different contexts.
> 
> These wormholes will appear no matter what metric you use. They 
might 
> be thought of as universal commas--3^2 is "approximated" by 9.

These are not what I call "wormholes".


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Message: 1571

Date: Fri, 07 Sep 2001 20:56:15

Subject: Re: about hypothesis and theorem

From: Paul Erlich

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

> I had begun, for example, many graphical studies (in vectorial
> form) about JI relations relatively to MIRACLE, Canasta and
> Blackjack scales. These images criticize, for example, ideas
> like the use of an absolute convexity on a linear temperament.
> A such convexity has only to reflect a multilinear convexity
> and I consider there is a flaw around the conception of the
> scales mentioned. Do I have to show now what I have and discuss
> that or to wait I will have time to write an article where I will
> attack vigourously with these ideas? I don't know yet. I judge
> only I want to be credited for my works.

I am very interested in any such criticisms, and welcome them warmly, 
but as you know, all theoretical edifices must have at their 
foundation, in my opinion, _perceptual_ conditions, not purely 
_mathematical_ ones. In my view, the operation of creating a "good" 
periodicity block, and then tempering out some or all of its defining 
intervals, is an eminently natural musical operation, logically prior 
to "higher-level" musical considerations such as the choice of a 
tonic, etc. All concepts, such as pitch, interval, etc., are 
considered to be perceptual entities from the beginning, and are 
only "mathematized" as necessary for ease in manipulation. In your 
gammier theory, by contrast, I am unable so far to discern any such 
foundation; instead I see some appeal to perceptual properties of 
intervals, applied seemingly incongruously to pitches, as well as an 
appeal to outmoded and ahistorical just rationalizations of various 
world scale systems.

Please understand this this is only my opinion and nothing could 
benefit us more than a hearty exchange of conflicting viewpoints.

Personally, I think Blackjack, let along Canasta, have too many notes 
to be heard and conceptualized in their entirety, the way diatonic 
scales and their Middle-Eastern cousins are, and perhaps my decatonic 
scales can be. But for the problem at hand, which was to provide 
Joseph Pehrson with a manageable subset of 72-tET to tune up on his 
keyboard, that would provide a maximum number of audibly just 
harmonies, they can't be beat, and I don't think the gammier theory 
will have much in addition to say on this question.


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Message: 1572

Date: Fri, 07 Sep 2001 23:02:52

Subject: Re: Distance measures cut to order

From: genewardsmith@xxxx.xxx

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

> Was this posted a while ago? I don't see this post in today's 
> archives.

Yahoo waited a few days before letting it appear.


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Message: 1573

Date: Fri, 07 Sep 2001 19:20:06

Subject: Re: about hypothesis and theorem

From: Pierre Lamothe

Paul

Sorry, I don't want to begin a discussion now. Maybe it will be possible to
compare our "perceptual foundation" in future. Until date you can keep
confortably :) the opinion I use maths with less sense about music than
what is in discussion on the tuning lists.

I leave words for a while. I give you only some images. It's made for the
eyes.

"Que ceux qui ont des yeux pour voir . . ."

<gammoids *>

Pierre


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Message: 1574

Date: Fri, 07 Sep 2001 02:05:02

Subject: Re: Tenney's harmonic distance

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:

> I used to do it by hand, setting the pitch bend in Cakewalk, but 
now I
> mainly use Graham Breed's Midiconv program, which puts in the pitch 
bends
> according to a scale file you can edit.

How do get Midiconv to input a midi file and output an ascii file of 
pitch values which I can edit?


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