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Message: 5375 - Contents - Hide Contents

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: 5376 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 13:50:10

Subject: Re: Epimorphic

From: Gene Ward Smith

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:


> Gene wrote:


>> Not so far as I can see.
> 


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

Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
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Message: 5377 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 13:54:11

Subject: Re: Epimorphic

From: Gene Ward Smith

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:


> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
>> Gene wrote:


>>> Not so far as I can see.
>> 


>> I haven't found a CS and non-epimorphic counterexample yet.
> 
> Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1



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 ---"
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Message: 5378 - Contents - Hide Contents

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 - Contents - Hide Contents

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: 5380 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 16:24:48

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "monz" <monz@a...> wrote:


> 


>> From: "David C Keenan" <d.keenan@u...>
>> To: "George Secor" <gdsecor@y...>
>> Cc: <tuning-math@y...>
>> Sent: Wednesday, September 18, 2002 6:12 PM
>> Subject: [tuning-math] Re: A common notation for JI and ETs
>> 
>> At 06:19 PM 17/09/2002 -0700, George Secor wrote:


>>> From:  George Secor (9/17/02, #4626)
>>> 
>>> Neither 306 nor 318 are 7-limit consistent, so I don't see much 
point
>>> in doing these, other than they may have presented an 
interesting
>>> challenge.
>> 


>> Good point. Forget 318-ET, but 306-ET is of interest for being 
strictly
>> Pythagorean. The fifth is so close to 2:3 that even god can 


barely tell the


>> difference. ;-)
> 


> what an interesting coincidence!  i just noticed this bit because
> Dave quoted it in his latest post.
> 
> just yesterday, i "discovered" for myself that 306edo is a great
> approximation of Pythagorean tuning, and that one degree of it
> designates "Mercator's comma" (2^84 * 3^53), which i think makes
> it particularly useful to those who are really interested in
> exploring Pythagorean tuning.



You probably saw that we are notating 306; I was pleasantly surprised 
to find that our notation works out much better for it than I 
expected.  I didn't immediately notice that it's every other tone of 
612, which gives the ratios of 5 their due.  But I don't expect that 
we'll be trying to notate 612 any time soon (if ever) -- it would 
require a bunch of new symbols, and we would be forced to cope with 
2079:2080 (~0.83 cents) not vanishing, since we're representing 
5103:5120 (the difference between 63:64 and 80:81, the 7 and 5 
commas) and 351:352 (the difference between 32:33 and 1024:1053, the 
11 and 13 dieses) with the same symbol.

Just to let you know that there's a limit to our madness.

--George
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Message: 5381 - Contents - Hide Contents

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: 5382 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 17:53:56

Subject: NMOS

From: Gene Ward Smith

Has anyone paid attention to scales which have a number of steps a
multiple of a MOS? They inherit structure from the MOS, and using a
2MOS or a 3MOS seems like a good way to fill in those annoying gaps.
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Message: 5383 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 17:56:57

Subject: Re: Epimorphic

From: Gene Ward Smith

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:


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


> Allright, but are there any monotonic examples?



I dunno--define "monotonic". To me it means monotonically increasing, which this scale does.
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Message: 5384 - Contents - Hide Contents

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 - Contents - Hide Contents

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: 5386 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 22:50:46

Subject: [tuning] Re: Everyone Concerned

From: wallyesterpaulrus

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:


> --- In tuning-math@y..., "wallyesterpaulrus" 
<wallyesterpaulrus@y...> wrote:
> 


>>> You should remember that many self-respecting mathematicians 
would 
>>> not call something a lattice unless it inherited a group 
structure 
>>> from R^n.
>> 


>> any examples of one that doesn't?
> 


> The hexagonal tiling of chords in the 5-limit, for one.



well, it's still a lattice in the crystallographic sense, or even the 
geometric sense: a regular array of points, that is, an array of 
points in which every point has exactly the same relationships with 
its neighbors as any other point does with *its* neighbors.
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Message: 5388 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 23:14:07

Subject: approximating pythagorean with ETs (was: Re: A common notation for JI and ETs)

From: wallyesterpaulrus

--- In tuning-math@y..., "monz" <monz@a...> wrote:


> 


>> From: "David C Keenan" <d.keenan@u...>
>> To: "George Secor" <gdsecor@y...>
>> Cc: <tuning-math@y...>
>> Sent: Wednesday, September 18, 2002 6:12 PM
>> Subject: [tuning-math] Re: A common notation for JI and ETs
>> 
>> 
>> At 06:19 PM 17/09/2002 -0700, George Secor wrote:


>>> From:  George Secor (9/17/02, #4626)
>>> 
>>> Neither 306 nor 318 are 7-limit consistent, so I don't see much 
point
>>> in doing these, other than they may have presented an 
interesting
>>> challenge.
>> 


>> Good point. Forget 318-ET, but 306-ET is of interest for being 
strictly
>> Pythagorean. The fifth is so close to 2:3 that even god can 
barely tell
> the
>> difference. ;-)
> 
> 


> what an interesting coincidence!  i just noticed this bit because
> Dave quoted it in his latest post.
> 
> just yesterday, i "discovered" for myself that 306edo is a great
> approximation of Pythagorean tuning, and that one degree of it
> designates "Mercator's comma" (2^84 * 3^53), which i think makes
> it particularly useful to those who are really interested in
> exploring Pythagorean tuning.
> 
> see my latest additions to:
> Yahoo groups: /monz/files/dict/pythag.htm * [with cont.] 
> 
> 
> 
> -monz
> "all roads lead to n^0"



monz, approximating pythagorean with ETs is a particularly simple 
problem, mathematically.

the perfect fifth, in terms of the octave, is log(3/2)/log(2).

the continued fraction expansion of this number is

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(5 + 1/(2 + 1/(23 + 1/
(2 + 1/(2 + 1/(1 + 1/(1 + 1/(55 + 1/(1 + 1/(4 . . . ))))))))))))))))

we can evaluate this, cutting off the expansion after more and more 
terms, to get more and more accurate ET approximations of the fifth 
in terms of the octave:

0 + 1/(1 + 1/(1)) = 1/2

0 + 1/(1 + 1/(1 + 1/(2))) = 3/5

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2)))) = 7/12

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3))))) = 24/41

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1)))))) = 31/53 

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(5))))))) = 179/306

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(5 + 1/(2)))))))) = 
389/665

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(5 + 1/(2 + 1/
(23))))))))) = 9126/15601

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(5 + 1/(2 + 1/(23 + 1/
(2))))))))) = 18641/31867

and so on . . . that 55 in the expansion tells you that

0 + 1/(1 + 1/(1 + 1/(2 + 1/(2 + 1/(3 + 1/(1 + 1/(5 + 1/(2 + 1/(23 + 1/
(2 + 1/(2 + 1/(1 + 1/(1 + 1/(55)))))))))))))) = 6195184/10590737 is 
exceedingly good for its size . . .
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Message: 5389 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 23:19:02

Subject: Re: Re : CS implies EPIMORPHISM

From: wallyesterpaulrus

i'm sorry, but i can't understand your reply below. can anyone?

p.s. i don't think "i am not a musician" excuses one from having to 
consider the relevance of musical situations if one's theory claims 
to pertain to music.

p.p.s. keep up the good work and ignore my whining.

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


> 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: 5390 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 23:22:47

Subject: Re: NMOS

From: wallyesterpaulrus

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:



> Has anyone paid attention to scales which have a number of steps a 
>multiple of a MOS?



the torsional scales do!
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Message: 5391 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 23:23:58

Subject: Re: Epimorphic

From: wallyesterpaulrus

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:


> --- In tuning-math@y..., manuel.op.de.coul@e... wrote:
>>> Try 1/1--2700/2401--5/4--4/3--3/2--5/3--2401/2400--2/1
>> 


>> Allright, but are there any monotonic examples?
> 


> I dunno--define "monotonic". To me it means monotonically 
>increasing, which this scale does.



2401/2400 is not between 5/3 and 2/1, gene!
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Message: 5392 - Contents - Hide Contents

Date: Mon, 21 Oct 2002 23:25:45

Subject: Fwd: [tuning] Re: Everyone Concerned

From: wallyesterpaulrus

--- In tuning-math@y..., <Josh@o...> wrote:


> I'm not a mathematician.
> 
> If, at some point, the "lattice" wraps around
> and becomes redundant, how can we assign a single
> point of reference?



why would we want to?



> If it's not a lattice, then what is it?



i think it's still a lattice!
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Message: 5395 - Contents - Hide Contents

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

Subject: Re: Epimorphic

From: Carl Lumma


>> >ry 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: 5396 - Contents - Hide Contents

Date: Tue, 22 Oct 2002 09:20:01

Subject: Re: Epimorphic

From: Gene Ward Smith

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:


> Carl wrote:


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


> Right, I was thinking that all scales being both CS and RI 
> are epimorphic. Now we still need a watertight definition
> of epimorphic. I must have misunderstood Pierre's definition
> of it.



Joe's dictionary gives the definition I introduced:

epimorphic 

A
scale has the epimorphic property, or is epimorphic, if there is a val
h such that if qn is the nth scale degree, then h(qn)=n. The val h is
the characterizing val of the scale. 

[from Gene Ward Smith, Yahoo tuning-math message 2569 (Thu Jan 10,
2002 11:11 pm)
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Message: 5397 - Contents - Hide Contents

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: 5398 - Contents - Hide Contents

Date: Tue, 22 Oct 2002 04:59:31

Subject: Re: Epimorphic

From: Gene Ward Smith

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


> 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.



Is that what I wrote? It should be

1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1

Barely distinguishable from a well-known scale.
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Message: 5399 - Contents - Hide Contents

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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