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 >.