about hypothesis and theorem

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

--- 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.

--- 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.

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

--- 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.

--- 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!

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

--- 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.

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 . . ."

<http://www.aei.ca/~plamothe/sys72/ib1215.htm>

Pierre