Math models (about Hypothesis)

I was not available at moment the thread Hypothesis was active. I had begun
to write this post but I had urgent tasks forcing me to stop. It is not
complete but since the List will probably terminate and since I miss yet
time I post that unfinished.

I would like to use the Paul's conjecture as starting point to show the
necessity for fine tuned definitions if we want to progress in mathematical
way.

I would have difficulty to express mathematically the given conjecture for
I don't know the mathematical definition used for almost all the terms.

-----------------------
distributional evenness
-----------------------

I begin with this definition appearing in the Monzo site.

<< distributional evenness

The scale has no more than two sizes of interval
in each interval class. >>

What follows is not a criticism of the Monzo's definition but only some
arguments to show that the conjecture has not really a mathematical form.

There exist maybe a light ambiguity with << no more than two >> but the
hard problem would be rather with the link between _scale_ and _class_ terms.

Forgetting for a moment the term scale, let us look at << no more than two
>> as if the property could be applied to a simple set. Supposing that,

<< . . . has no more than two sizes of interval
in each interval class. >>

might be mathematically written

A partition of a set S by an equivalence relation R
is _distributionally even_ if for any class x in S/R

0 < Card x < 3

where it is only assumed each class has either one or two elements.

Thus the partition {{a},{b},{c}} of a set {a,b,c} would be distributionally
even. For Paul, is that sense allowed?

Besides, << no more than two >> implies that the case

"for any x, Card x = 2"

is only a possibility among others. So, if the partition {{a},{b},{c}} is
not _distributionally even_, what rule is used to determine allowed
combinations of classes with one or two elements? Is the form << no more
than >> used only to allow the class of unison having only one interval
while all others would have two?

I imagine this light ambiguity is easy to clarified. The next one seems
more serious.

---------------
class and scale
---------------

Is a _scale_ anything else, for Paul, than an ordered finite set of reals.
If not so, what are the essential conditions? Has a _scale_ an essential
link, or only an optional one, with periodicity block and also with
properties often mentioned like consonance possibilities?

Are interval classes in a scale something added to that scale by an
external equivalence relation or are classes "sui generis" in sense that
the classes follows (by a general principle used for any scale) from the
intervals itself given as a whole?

In the following set and partitions

S = {1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9}

P1 = {{1}, {9/8, 32/27}, {4/3}, {3/2}, {27/16, 16/9}}

P2 = {{1}, {9/8}, {32/27}, {4/3}, {3/2}, {27/16}, {16/9}}

P3 = {{1}, {4/3, 27/16}, {9/8, 16/9}, {32/27, 3/2}}

how the definitions of scale, class and distributionally evenness should be
applied?

As you may see, I'm far from knowing clearly what this simple definition
mathematically means, but it could be worst with other definitions. My
discussions with Paul on the Tuning List shows we could differ on
periodicity block and srutis and probably on unison vectors and steps.

-----------------
periodicity block
-----------------

*** TONE ***

I will use here the term _tone_ in the strict sense
of an interval of the first octave as representing
its class modulo 2.

For me, a normal set of unison vectors in a discrete tones Z-module
determine a periodicity block refering to a canonical set of tones
determined by an oriented object with origin : the hyperparallelopiped
defined by the given unison vectors sharing a common origin.

This canonical set of tones corresponds both to

- tones inside the hyperparallelopiped having unison
as origin and

- tones on the unison vectors

(*** so neither ambiguity nor double counting ***).

Since the origin of the unison vectors is never inside the
hyperparallelopiped, so keeping the same shape but changing the origin for
another vertices don't correspond (most cases) to a simple translation of
the block in the lattice. One tone, minimally, would be changed compared to
the translated canonical set.

Besides, since Paul and Dave seems to refer rather at periodical shapes, I
would add that a same shape may correspond to distinct sets of unison
vectors. If you know only the tile you don't know forcely the tiling.

Another thing seems mysterious comparing our uses of periodicity block. For
me, a system build from the periodicity block concept implies all modes in
the system have the same amount of tones or degrees. If we use only one
block we may have only one mode. Using N blocks surrounding unison we
obtain for each D classes possibly N intervals. It remains possible after
that to use appropriate temperament (removing commatic vectors or keeping
integrally the structure).

But it seems Paul and Monz would start with periodicity blocks having too
tones, to be interpreted as degrees of a musical mode, in their approach of
the Hindu system where the amount of degrees is 7 (SA RI GA ...) rather
than 22, and it seems it is a temper process that serves to . . . (I don't
understand, so I have some questions).

If you use only one block, which have by definition only one element in
each of the D classes determined by the periodicity, what is the sense of
the presumed new classes which would appear after a temper treatment? Is
it like that you obtain seven classes in the Hindu system from a block
having a 22 or more periodicity? Do you have then an explicit epimorphism
transforming an interval in its degree value?

----- As I said, I cut here for I have not much time to write.
However, the questions already written may have interest.

Pierre

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:
>
> I would like to use the Paul's conjecture as starting point to show
the
> necessity for fine tuned definitions if we want to progress in
mathematical
> way.

I thank you for that.
>
> I would have difficulty to express mathematically the given
conjecture for
> I don't know the mathematical definition used for almost all the
terms.
>
>
> -----------------------
> distributional evenness
> -----------------------
>
> I begin with this definition appearing in the Monzo site.
>
> << distributional evenness
>
> The scale has no more than two sizes of interval
> in each interval class. >>
>
> What follows is not a criticism of the Monzo's definition but only
some
> arguments to show that the conjecture has not really a mathematical
form.
>
> There exist maybe a light ambiguity with << no more than two >> but
the
> hard problem would be rather with the link between _scale_ and
_class_ terms.
>
> Forgetting for a moment the term scale, let us look at << no more
than two
> >> as if the property could be applied to a simple set. Supposing
that,
>
> << . . . has no more than two sizes of interval
> in each interval class. >>
>
> might be mathematically written
>
> A partition of a set S by an equivalence relation R
> is _distributionally even_ if for any class x in S/R
>
> 0 < Card x < 3
>
> where it is only assumed each class has either one or two elements.
>
> Thus the partition {{a},{b},{c}} of a set {a,b,c} would be
distributionally
> even. For Paul, is that sense allowed?

I'm not following . . . this is too abstract.
>
> Besides, << no more than two >> implies that the case
>
> "for any x, Card x = 2"
>
> is only a possibility among others. So, if the partition {{a},{b},
{c}} is
> not _distributionally even_, what rule is used to determine allowed
> combinations of classes with one or two elements? Is the form << no
more
> than >> used only to allow the class of unison having only one
interval
> while all others would have two?

No . . . for example, the scale LssssLssss has only one size of class
0 _and_ only one size of class 5.

>
> Is a _scale_ anything else, for Paul, than an ordered finite set of
reals.

For these purposes, that is an adequate definition.
>
> Are interval classes in a scale something added to that scale by an
> external equivalence relation or are classes "sui generis" in sense
that
> the classes follows (by a general principle used for any scale)
from the
> intervals itself given as a whole?

The latter.
>
> In the following set and partitions
>
> S = {1, 9/8, 32/27, 4/3, 3/2, 27/16, 16/9}
>
> P1 = {{1}, {9/8, 32/27}, {4/3}, {3/2}, {27/16, 16/9}}
>
> P2 = {{1}, {9/8}, {32/27}, {4/3}, {3/2}, {27/16}, {16/9}}
>
> P3 = {{1}, {4/3, 27/16}, {9/8, 16/9}, {32/27, 3/2}}
>
> how the definitions of scale, class and distributionally evenness
should be
> applied?

I don't understand what you're doing with the partitions.

>
> -----------------
> periodicity block
> -----------------
>
>
> *** TONE ***
>
> I will use here the term _tone_ in the strict sense
> of an interval of the first octave as representing
> its class modulo 2.
>
>
> For me, a normal set of unison vectors in a discrete tones Z-module
> determine a periodicity block refering to a canonical set of tones
> determined by an oriented object with origin : the
hyperparallelopiped
> defined by the given unison vectors sharing a common origin.

To me, the periodicity block need not be a hyperparallelopiped. In
three dimensions, for example, the other useful shapes for a
periodicity block are the hexagonal prism and the rhombic
dodecahedron. These can be obtained from the parallelopiped by
transposing certain tones by certain unison vectors.
>
> This canonical set of tones corresponds both to
>
> - tones inside the hyperparallelopiped having unison
> as origin and

I'd prefer to define my 'canonical' periodicity block as the one with
the most consonances . . . if that is well-defined . . . rather than
the hyperparallelopiped. However, if it aids mathematical progress,
I'd be happy to stick with the hyperparallelopiped.
>
> - tones on the unison vectors

This I don't understand.
>
> (*** so neither ambiguity nor double counting ***).

Right . . .

>
> Besides, since Paul and Dave seems to refer rather at periodical
shapes, I
> would add that a same shape may correspond to distinct sets of
unison
> vectors. If you know only the tile you don't know forcely the
tiling.

True . . . due to the discrete nature of the lattice.
>
> Another thing seems mysterious comparing our uses of periodicity
block. For
> me, a system build from the periodicity block concept implies all
modes in
> the system have the same amount of tones or degrees.

Me too . . . if we mean the same thing by 'modes'.

> If we use only one
> block we may have only one mode.

Hmm . . . that doesn't make sense to me . . . if there are N tones,
there are N modes, since each tone can be taken as the 'tonic'. Of
course, characteristic dissonances within the scale may tend
to 'point' to some tonics more strongly than others . . . but the
entire hypothesis exists in a pre-tonal framework, where the choice
of a tonic is _not_ yet made.
>
> But it seems Paul and Monz would start with periodicity blocks
having too
> tones,

Too tones?

> to be interpreted as degrees of a musical mode, in their approach of
> the Hindu system where the amount of degrees is 7 (SA RI GA ...)
rather
> than 22, and it seems it is a temper process that serves to . . .
(I don't
> understand, so I have some questions).

Please, ask.
>
> If you use only one block, which have by definition only one
element in
> each of the D classes determined by the periodicity, what is the
sense of
> the presumed new classes which would appear after a temper
treatment?

Dissonances are converted into consonances.

> Is
> it like that you obtain seven classes in the Hindu system from a
block
> having a 22 or more periodicity?

I don't think Monz or I have done anything like that . . . is there a
particular message you're referring to?

Anyhow, Pierre, I'm glad you're trying to help, and I hope you'll be
able to contribute some mathematics and co-author a paper with me and
others on this list . . . perhaps future responses should be directed
to the Tuning list?

Hi Paul,

I will comment here very shortly so I forget

<< perhaps future responses should be directed to the
Tuning list? >>

for this time. I would like mainly to express that these quotes

<< the scale LssssLssss has only one size of class 0
_and_ only one size of class 5. >>

<< if there are N tones, there are N modes, since each
tone can be taken as the 'tonic' >>

indicate to me there exist a common ground on which a math discussion may
be fertile. As I understood that, your term _scale_ here is tightly related
to terms _step_ and _class_ obtained by rotation (I will discuss later
about class, partition and equivalence relation) and the last term refers
implicitely to the space of intervals subtended by a scale S and the matrix
S\S showing the interval interconnection.

All that implies what I name _macrotonal_ properties, qualifying so
properties that could'nt be applied to an isolated interval for it is
related to the specific configuration of an interval set.

When I saw how so much terms were used to qualify intervals (mathematically
defined) but so few to qualify structured interval sets, and when I saw the
same term scale used to qualify tausends of interval sets having almost
nothing (mathematically) in common, I was not highly motivated to face
that. Your answers give me a better motivation.

It remains only to find time. It will happen.

Pierre

In an exchange of Pierre and Paul...

> > << distributional evenness
> >
> > The scale has no more than two sizes of interval
> > in each interval class. >>
> >
> > What follows is not a criticism of the Monzo's definition but only
> some
> > arguments to show that the conjecture has not really a mathematical
> form.
> >
> > There exist maybe a light ambiguity with << no more than two >> but
> the
> > hard problem would be rather with the link between _scale_ and
> _class_ terms.

The way I tend to think of things (musically directed) is that
a scale is the set of distinct intervals and a mode is a specific
rotation of that set. Therefor, if a two dimensional matrix were
made of all the rotations (modes), viewing a column (class) would
have no more than two intervals. Perhaps there is a matrix operation
that can be done to fully define this property.

Paul said...

>
> No . . . for example, the scale LssssLssss has only one size of class
> 0 _and_ only one size of class 5.
>

which I believe is referring to what I would term the specific
rotation (mode) of the set that I would psuedo-formulate

( L s^4 )^2

(where ^ is a repeat operator).

Note that in my view, class(1) = { L : s }
class(2) = { L+s, : s+s }

where I don't care about the order of the intervals in the
classes.

If you haven't already, mathematicians, bail me out at this point
please (and when it gets strict you'll have to make up your mind
whether to work in the linear frequency domain or log, just for
consistency, etc, etc).

Bob Valentine

Robert C Valentine wrote:

<< The way I tend to think of things (musically directed)
is that a scale is the set of distinct intervals and a
mode is a specific rotation of that set. Therefor, if a
two dimensional matrix were made of all the rotations
(modes), viewing a column (class) would have no more than
two intervals. >>

It's almost what I presumed from the Paul's answers. These structured
notions invite to algebraic treatment. I don't begin inside this post and
give only here an example showing how to transform the implied notions in
mathematical objects like matrices and lattices.

d(M) = (s L)(0 1 1 1 1 0 1 1 1 1) == "LssssLssss"
(1 0 0 0 0 1 0 0 0 0)

o o o o o
o o o o o . . . .
o . . . .

-----

As alternative to the matrix where the classes would appear in columns, I
would use however the interval matrix where the classes appear in
diagonals, for that is the fundamental object from which the other notions
are derived and on which we have to ask questions about the implied musical
representation.

[ ... Is the space of intervals spanned by a scale simply a space for
modulation and transposition or also a space for harmony (I mean that the
total chromatic of the scale would be also acceptable as a chord)? How to
interpret in that context the historical treatment of the triton? Would
there exist subspaces matching more adequately varied choices of harmony in
time? ... ]

I forget for the moment the meaning and work only on mathematical
structures. We could not talk of harmony anyway without to specify the
basis (s L) but I want to show eventually we can do sophisticated maths
without to know the values of s and L.

-----

I would propose to call

_formal scale_

that category of scale defined by an elementary chain of

_formal steps_

where the term "formal" means that the value of the steps, like (s L) in a
bivalent system, are not determined, and where the important thing about
the concept is the link with the

_formal classes_

obtained by rotation (of the tonic).

Since neither pitch height values (Hz) nor interval values (cents) are
given, the formal scale concept would be obviously almost empty without
this implicit reference at such formal classes. A formal scale has very few
determined microtonal properties (we know only unison and octave -
supposing we talk about octaviant system) but very much macrotonal properties.

I imagine someone somewhere has already formalized something about that but
I presume the algebraic properties implied by such concepts perhaps have
not been yet well explicitated.

It will be a good opportunity to show here the utility of an abstract
theory like the one defining the chordoid structure. The interval set, at
this level, has not to be numerical values, neither frequency ratios nor
octave ratios. The simple set of coordinate vectors representing the
intervals, in the step generator acting as a basis, may have a chordoid
structure. More, I think to define a new object, the f-gammier or "formal
gammier", using only such coordinates. It appears to me easy to define and
handle a such formal structure since almost conditions of gammoids are
already implicit.

I will explain that later. For the moment, I would like only to discuss
about the _class_ concept applied to formal scale. In precedent discussion
with Robert Walker about propriety and Paul Erlich about constant structure
I had problem with the use of the term class, for instance in Monzo's
definition. I recognize that this current use matches the sense of
"equivalence class" in mathematics, but only if we interpret the term class
as a formal class.

If values are given to (s L) then the formal classes don't become forcely
numerical classes. For instance, the formal space S\S corresponding to the
formal scale S == LLsLLLs have true formal classes, since there exist a
true formal partition. If we use s = 4 and L = 9, we have true (tempered)
classes for the same reason. With s = 256/243 and L = 9/8, we have yet
(rational) classes. However with s = 1 and L = 2, there exist an ambiguity
: the presumed classes 3 and 4 have in common the value 6. Thus there not
exist a partition and the term class, in mathematical sense of "equivalence
class", is not applicable. So we have to distinguish that about the concept
of constant structure which corresponds to the concept of congruity
condition in gammoids.

We have seen that a formal property is not conserved for any set of values.
Now I would like to show that another formal property, the symmetry, is
conserved and has consequence for any set of values. In the example of
LssssLssss, there exist no possibility to give rational values (with
multiplication) to (s L) for class 5 has to be sqrt(2). The corresponding
property, for any set of integer values (with addition) in (s L), is there
exist a subspace which is a group. I refer here to the distinction between
symmetric s-gammier (which have a such group as part) and assymetric
a-gammier in "Pourquoi les gammes naturelles ne sont-elles pas symétriques
?" on my web page.

The concrete things will follow. Hoping someone interested to dig in that
direction.

Pierre