Tuning-Math Digests messages 5375 - 5399

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Message: 5375

Date: Mon, 21 Oct 2002 13:29:23

Subject: Re: Epimorphic

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:
>Not so far as I can see.

I haven't found a CS and non-epimorphic counterexample yet.

Manuel


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Message: 5378

Date: Mon, 21 Oct 2002 16:02:29

Subject: Re: Epimorphic

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1

Allright, but are there any monotonic examples?

Manuel


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Message: 5379

Date: Mon, 21 Oct 2002 16:22:00

Subject: Re: Epimorphic

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>By the way, if you decide to impliment the epimorphism feature, I'd 
suggest "Scale is epimorphic with val >---" or "Scale is epimorphic with 
mapping ---"

Yes, it will show that.

Manuel


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Message: 5381

Date: Mon, 21 Oct 2002 13:31:21

Subject: Re : CS implies EPIMORPHISM

From: Pierre Lamothe

Paul wrote:
  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 know very few things in music as such. I seeked an analogy of the progression used by Asselin
in which there was generally two common intervals between successive chords. Now, It seems
easy to propose a similitude for an intonation where the comma wouldn't be distributed. Naturally,
I don't advocate something here and it's not a "scientific" assessment, since it's above all a matter
of music. I use only tools to show a certain similarity with the Asselin solution. In such cases, it
would have nothing to do with a second "commatic" tonic.

First, rather than using, a "dorian" which would be, as seen in a precedent post, an exact translation
(9/8 or 10/9) of the Zarlino scale S = < 1 9/8 5/4 4/3 3/2 5/3 15/8 2 > in the space
  ...U
  UooooooU
  .oooTooo
  .UooooooU
  .....U
generated by the scale S or equivalently G =  < 1 3 5 9 15 27 45 > (said its harmonic generator),
one can use the unique "dorian" mode ( 2 1 2 2 2 1 2 ) in the Zarlino gammier, i.e. the restricted
space
  ...U
  UooooooU
  .oooTooo
  .UooooooU
  .....U
whose generator is the very low < 1 3 5 9 15 >. That scale is < 1 9/8 6/5 4/3 3/2 5/3 16/9 2 >.

(Note that the modes in a gammier are not restricted to horizontal and vertical lines).

The intervals 10/9 and 9/5 aren't here an alternative as melodic steps (while 10/9 was an alternative
between 1 and 5/4), but nothing prevent to use it as optional harmonic intervals, in link with consonance
and functions.

I use here conveniently the tonic D for there is no alteration. The scale is in red and the harmonic alternative
in black.
  ...U
  UoEB..oU
  .oCGDAEo
  .Uo..FCoU
  .....U
(Unhappily, the ideas here will remain hidden from now without the color using)

Before to restrict at this space, I will use the larger space to show more clearly which of the possible
alternative chord corresponds to a function : is it located in tonic, subdominant or dominant region. So,
I will use first the following matrix,  generated by < 1 3 5 9 15 27 45 >
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
for the choice of the chords, and then that simpler generated by < 1 3 5 9 15 >
  D.A.E
  .DFAC
  GBD.A
  .G.DF
  CEGBD
Locating first, all chord variants DFA, GBD, CEG, ACE in the larger matrix
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
It seems clear that the functional region are
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
  D.A.E.B
  .DFACEG
  GBD.A.E
  .G.DFAC
  CEGBD.A
  .C.G.DF
  FACEGBD
  -------
One can thus easily visualize that dorian progression in the matrix with these simplified views
  -----
  D.A.E
  .DFAC
  GBD.A
  .G.DF
  CEGBD
  -----
  D.A.E
  .DFAC
  GBD.A
  .G.DF
  CEGBD
  -----
  D.A.E
  .DFAC
  GBD.A
  .G.DF
  CEGBD
  -----
  D.A.E
  .DFAC
  GBD.A
  .G.DF
  CEGBD
  -----
  D.A.E
  .DFAC
  GBD.A
  .G.DF
  CEGBD
  -----
completed with these views in the Z-module < 2 5 > ZxZ
  ---------
  ...U
  UoEB..oU
  .oCGDAEo
  .Uo..FCoU
  .....U
  ---------
  ...U
  UoEB..oU
  .oCGDAEo
  .Uo..FCoU
  .....U
  ---------
  ...U
  UoEB..oU
  .oCGDAEo
  .Uo..FCoU
  .....U
  ---------
  ...U
  UoEB..oU
  .oCGDAEo
  .Uo..FCoU
  .....U
  ---------
  ...U
  UoEB..oU
  .oCGDAEo
  .Uo..FCoU
  .....U
  ---------
I imagine that the simultaneous comma shift on E and C could be more disturbingly than the
Asselin example, so requiring even more its distribution. I'm not a musician and can't appreciate that.


Pierre


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Message: 5384

Date: Mon, 21 Oct 2002 14:04:01

Subject: Re: Epimorphic

From: Pierre Lamothe

Gene wrote:
  Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
Do you consider that unordered list as a scale ? Need correction or reordering.


Pierre


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Message: 5385

Date: Mon, 21 Oct 2002 14:36:23

Subject: Re: CS implies EPIMORPHISM

From: Pierre Lamothe

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 >.


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Message: 5395

Date: Tue, 22 Oct 2002 01:45:48

Subject: Re: Epimorphic

From: Carl Lumma

>>Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
> 
>Allright, but are there any monotonic examples?

Why does this fail?

The stronger argument against CS /-> Epimorphic is
that CS doesn't require JI, as Gene pointed out.

-C.


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Message: 5397

Date: Tue, 22 Oct 2002 11:36:49

Subject: Re: Epimorphic

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Yes that scale is indeed not epimorphic. So I'll
add the additional integer test to the code.
So doesn't that need to be added to the definition
in Joe's dictionary too, since all components being
integer doesn't follow automatically from 
h(qn)=n for n = 1 .. (number of notes - 1)?

Manuel


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Message: 5399

Date: Tue, 22 Oct 2002 12:30:31

Subject: Re: Epimorphic

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

So I may conclude that the simplest example of a JI,
CS and non-epimorphic scale is this one: 1/1--4/1

Manuel


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