CS implies EPIMORPHISM

Maybe I was not sufficiently clear the precedent times I wrote about that.

Let S = (f0, f1, .., fn) be a scale, i.e. and ordered set of separate frequencies where fn is the octave of the tonic f0.

Let (Tij) be the matrix of intervals Tij = fj / fi (modulo octave) whose content is the set T of all possible intervals x (of the first octave) within S or any derived scale S' (horizontal or vertical line in the matrix) with another tonic in S and/or possibly another direction (dual).

The CS property means : if any interval x appears twice or more in the matrix, its positions belong to the same diagonal.

That implies : the set of \ diagonals is a valid partition of the set T (the intersection of any two classes being empty).

Consequently, let
D : T --> diag
be the surjective mapping of any interval Tij in its unique possible diagonal d, enumerated from 0 (the tonic diagonal) to n-1.The diagonal d represents the degree of the interval in these scales (or amount of steps from the tonic), each interval belonging to a distinct diagonal.

The point : D is a congruence, i.e. an equivalence relation (since the partition is valid) which is also a morphism, since in a such ordered matrix
D(xy) = D(x) + D(y)
Now, since the name of a morphism as surjection is EPIMORPHISM, may I conclude, as many times before, that
CS in a scale implies EPIMORPHISM
Ok, it's not sufficient to insure it's a good scale : it's only epimorphic.

However, it's epimorphic and CS means nothing else than it's epimorphic.

(... and there is no need to restrict to rationals for that.)

Finally, epimorphism don't imply periodicity block or convexity but only condition for that. For instance, the scale
1, 9/8, 5/4, 25/18, 40/27, 128/81, 16/9, 2
has the same steps (16/15, 10/9, 9/8) than the Zarlino scale and consequently the same epimorphism D (giving the diagonals) which, applied to coordinates (x,y,z) of its intervals, may be written
D(x,y,z) = 7x + 11y + 16z (mod 7)
so the same unison vectors, including 81/80 and 25/24, whose wedge product is [7,11,16], etc.
But it's clearly not a convex periodicity block. The convex hull of that scale in the lattice <3 5>(y,z) has 5 holes.

Pierre

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

This raises a facinating possibility, but I can't see that it works. Taking the diatonic scale in 12-et as an example, the group generated by the notes of the scale is the 12-et; there is no morphism from here to 7-et.

Gene wrote:

<< This raises a facinating possibility, but I can't see that it works. Taking the diatonic scale in 12-et as an example, the group generated by the notes of the scale is the 12-et; there is no morphism from here to 7-et. >>

As I already said, 12-et lost the underlying 5-limit diatonic structure. The scale 0 2 4 5 7 9 11 (mod 12) is inconsistent in itself and worth only as a blurred image of the consistent underlying JI. It's precisely what reveals its non-epimorphic property ( 6 would belong to degree 3 and 4 ) or, in other words, its non-CS property ( 6 subtended by 3 or 4 steps ).

On the other hand, the diatonic scale 0 9 17 22 31 39 48 (mod 53) is also an image of the Zarlino scale but consistent in itself, in other words, having the CS, or epimorphic, or congruity property (the concept I used in my theories).

Look. In 53-et, class 1 = ( 5, 8, 9 ) == ( 16/15, 10/9, 9/8 ) and class 0 = ( 1, 3 ) == ( 81/80, 25/24 ).

In comparaison, in 12-et it would be, class 1 = ( 1, 2 ) and class 0 = ( 1 ), what is inconsistent by definition.

The vanishing of the comma 81/80, its splitting up in several little errors, is what is seeked normally by temperaments. In that case we have to understand that the result is necessarily a blurred image keeping not its underlying structure. When the stucture is keeped, as in 53-et, the same problems occurs like the comma drift.

Pierre

--- In tuning-math@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> Gene wrote:
>
> << This raises a facinating possibility, but I can't see that it
>works. Taking the diatonic scale in 12-et as an example, the group
>generated by the notes of the scale is the 12-et; there is no
>morphism from here to 7-et. >>
>
> As I already said, 12-et lost the underlying 5-limit diatonic
>structure. The scale 0 2 4 5 7 9 11 (mod 12) is inconsistent in
>itself and worth only as a blurred image of the consistent
>underlying JI.

i can't think of a single point of view from which i would agree with
this value judgment. rather, the underlying JI you give is defective,
for example its ii chord is very out-of-tune.

> It's precisely what reveals its non-epimorphic property ( 6 would
>belong to degree 3 and 4 ) or, in other words, its non-CS property (
>6 subtended by 3 or 4 steps ).

in 12-equal, you're right, it's not CS. but in 19-equal or 31-equal,
it is.

> Look. In 53-et, class 1 = ( 5, 8, 9 ) == ( 16/15, 10/9, 9/8 ) and
class 0 = ( 1, 3 ) == ( 81/80, 25/24 ).
>
> In comparaison, in 12-et it would be, class 1 = ( 1, 2 ) and class
0 = ( 1 ), what is inconsistent by definition.

"consistent" has got to be the most overloaded term on these
forums . . . :)

>
> The vanishing of the comma 81/80, its splitting up in several
>little errors, is what is seeked normally by temperaments. In that
>case we have to understand that the result is necessarily a blurred
>image keeping not its underlying structure. When the stucture is
>keeped, as in 53-et, the same problems occurs like the comma drift.

and the out-of-tune ii triad.

I wrote:
As I already said, 12-et lost the underlying 5-limit diatonic
structure. The scale 0 2 4 5 7 9 11 (mod 12) is inconsistent in
itself and worth only as a blurred image of the consistent
underlying JI.
Paul wrote:
i can't think of a single point of view from which i would agree with
this value judgment. rather, the underlying JI you give is defective,
for example its ii chord is very out-of-tune.
Probably the term worth leaved you to think it was a value judgment. I'm not musician and I leave to
musicians the care to appreciate musical aspects.

I wanted only to say that the diatonic scale in 12-et don't enclose (in the list of its numbers) structural
properties. You have to consider something else to reconstitute the structure.

By underlying JI, I mean, here, what is enclosed, for instance, in the wedge product result (7,11,16).

Beside, there exist also a macrotonal approach (using not the JI microtonal properties) reconstituting
the structure : the t-gammier ( 0 2 4 7 11). That structure is epimoph (CS) and naturally the interval 6,
the tritone, don't exist within it. But once yet, the consistence is exterior to the isolated mode itself.

The underlying JI refered is not the isolated Zarlino scale, so consonant ii chord (10/9 4/3 5/3) exists.

What follows is not a judgment or a position but only a reference. It's represented in a portion of the
Zarlino gammier (not the isolated Zarlino scale). If I well understood the experience of Pierre-Yves
Asselin ( Musique et tempérament ) the choice of intonation, a cappella, for i - vi - iv - ii - v - i was
ooXo
.oXXo

oXXo
.oXoo

oXoo
.XXoo

XXoo
.Xooo

oooX
.ooXX

ooXo
.oXXo
without drift, inserting ( spontaneoulsly? ) the comma between ii and v.

Paul wrote:
"consistent" has got to be the most overloaded term on these
forums . . . :)
I imagine. I used it in macrotonal sense of structural consistence, qualifying so the imbrication of the
elements rather than the individual (microtonal) properties. Is consistent an imbrication obeing to simple
universal principles

Pierre

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

> If I well understood the experience of Pierre-Yves
> Asselin ( Musique et tempérament ) the choice of intonation, a
cappella, for i - vi - iv - ii - v - i was
> ooXo
> .oXXo
>
> oXXo
> .oXoo
>
> oXoo
> .XXoo
>
> XXoo
> .Xooo
>
> oooX
> .ooXX
>
> ooXo
> .oXXo
> without drift, inserting ( spontaneoulsly? ) the comma between ii
and v.

i think a solution nearer to reality would use the vicentino's second
tuning (adaptive just intonation), so that the simultaenous intervals
are all just but the successive intervals are not. the comma will be
distributed among the successive intervals. this way, instead of the
disturbingly large full-comma shift in the intonation of the 2nd
scale degree as in the solution you cite above, we have (ideally)
four 1/4-comma shifts -- each just below the limen of melodic
discriminability.

what if the (rotated) progression occured in the dorian mode? would
your source, or you, advocate shifting the *tonic* or *1/1* by a full
comma in this way?

> I used it in macrotonal sense of structural consistence, qualifying
so the imbrication of the
> elements rather than the individual (microtonal) properties. Is
consistent an imbrication obeing to simple
> universal principles

what do the words "imbrication" and "obeing" mean?

Paul wrote:
i think a solution nearer to reality would use the vicentino's second
tuning (adaptive just intonation), so that the simultaenous intervals
are all just but the successive intervals are not. the comma will be
distributed among the successive intervals. this way, instead of the
disturbingly large full-comma shift in the intonation of the 2nd
scale degree as in the solution you cite above, we have (ideally)
four 1/4-comma shifts -- each just below the limen of melodic
discriminability.
I hoped your advice on the Asselin solution. I like such short and sweet answer.
what if the (rotated) progression occured in the dorian mode? would
your source, or you, advocate shifting the *tonic* or *1/1* by a full
comma in this way
I advocate nothing in the musical domain as such but perhaps a clear separation
between what is a matter for musicians and what is a matter for scientists -- even
if the same person may play often the two roles -- and then, for the scientific views
and discourses, I would advocate, for sure, logic, coherence, rigourousness, etc.

I don't believe M. Asselin had treated that question. I read that many years ago when
I worked in his firm.

Just like that, I ask me here what is the analog progression in dorian ? In the two exact
"dorian" translation, the first has no triad on the tonic, and the progression in the second
case seems rather to be i - iii - v ... Is it the case ?
...U
UXXXoooU
.XXXTooo
.UooooooU
.....U

...U
UooooooU
.oooTXXX
.UoooXXXU
.....U
I wrote:
I used it in macrotonal sense of structural consistence, qualifying so the
imbrication of the elements rather than the individual (microtonal) properties.
Is consistent an imbrication obeing to simple universal principles.
Paul wrote:
what do the words "imbrication" and "obeing" mean?
Imbrication qualifie (macrotonally, i.e. independently of individual properties) how the elements
are interwoven or interlinked or emmeshed. By obeing simple principles, I mean meet simple
structural (math) conditions or axioms. Epimorphism and convexity are such topological
conditions independant of microtonal metrics. For instance, one can easily enumerate all
epimorphisms which are homotope in 3D for 5, 6, 7, 8... degrees, without considering harmonic
possibilities.

Pierre

P.S.

If I had'nt lost my computer and programs, some months ago, I could begin to talk about
problems I resolved. For instance, the fundamental domain in 3D, (i.e. the convex hull of minimal
unison vectors) varies with the microtonal metrics, but the shape is always an hexagon, as the
figures above. In 2D, it's a segment.

What is the polytope series, giving that shape in subsequent dimensions ? One can calculate
easily (without computer) the amount of faces and cells, and the decomposition in cross polytopes.
I found the corresponding name for 4D and 5D : cuboctahedron and prismatodecachoron.

For the moment, I am in forced sabbatical. I have to borrow a computer for posting.

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

> Paul wrote:
> i think a solution nearer to reality would use the vicentino's
second
> tuning (adaptive just intonation), so that the simultaenous
intervals
> are all just but the successive intervals are not. the comma will
be
> distributed among the successive intervals. this way, instead of
the
> disturbingly large full-comma shift in the intonation of the 2nd
> scale degree as in the solution you cite above, we have (ideally)
> four 1/4-comma shifts -- each just below the limen of melodic
> discriminability.
> I hoped your advice on the Asselin solution. I like such short and
sweet answer.
> what if the (rotated) progression occured in the dorian mode?
would
> your source, or you, advocate shifting the *tonic* or *1/1* by a
full
> comma in this way
> I advocate nothing in the musical domain as such but perhaps a
clear separation
> between what is a matter for musicians and what is a matter for
scientists -- even
> if the same person may play often the two roles -- and then, for
the scientific views
> and discourses, I would advocate, for sure, logic, coherence,
rigourousness, etc.
>
> I don't believe M. Asselin had treated that question. I read that
many years ago when
> I worked in his firm.
>
> Just like that, I ask me here what is the analog progression in
dorian ? In the two exact
> "dorian" translation, the first has no triad on the tonic, and the
progression in the second
> case seems rather to be i - iii - v ... Is it the case ?
> ...U
> UXXXoooU
> .XXXTooo
> .UooooooU
> .....U
>
> ...U
> UooooooU
> .oooTXXX
> .UoooXXXU
> .....U

i'm confused as to what you mean. rotating the progression so as to
begin and end on ii -- ii-V-I-vi-ii -- should tell you what i'm
talking about (i hope). rewriting in terms of dorian functions, it's
i-IV-VII-v-i, a progression one can find many examples of in pop and
rock music.

what's your "scientific" assessment of this progression?

> I wrote:
> I used it in macrotonal sense of structural consistence,
qualifying so the
> imbrication of the elements rather than the individual
(microtonal) properties.
> Is consistent an imbrication obeing to simple universal
principles.
> Paul wrote:
> what do the words "imbrication" and "obeing" mean?
> Imbrication qualifie (macrotonally, i.e. independently of
individual properties) how the elements
> are interwoven or interlinked or emmeshed. By obeing simple
principles, I mean meet simple
> structural (math) conditions or axioms. Epimorphism and convexity
are such topological
> conditions independant of microtonal metrics. For instance, one can
easily enumerate all
> epimorphisms which are homotope in 3D for 5, 6, 7, 8... degrees,
without considering harmonic
> possibilities.
>
>
> Pierre
>
>
> P.S.
>
> If I had'nt lost my computer and programs, some months ago, I could
begin to talk about
> problems I resolved. For instance, the fundamental domain in 3D,
(i.e. the convex hull of minimal
> unison vectors) varies with the microtonal metrics, but the shape
is always an hexagon, as the
> figures above. In 2D, it's a segment.
>
> What is the polytope series, giving that shape in subsequent
dimensions ? One can calculate
> easily (without computer) the amount of faces and cells, and the
decomposition in cross polytopes.
> I found the corresponding name for 4D and 5D : cuboctahedron and
prismatodecachoron.
>
> For the moment, I am in forced sabbatical. I have to borrow a
computer for posting.

all sounds very interesting . . .

Maybe it would have been better I precise the generator order used to generate the matrices in my precedent
post. There was successively < 1 5 3 15 9 45 27 > and < 1 5 3 15 9 >.