Tuning-Math Digests messages 5407 - 5431

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Message: 5407

Date: Tue, 22 Oct 2002 12:47:17

Subject: Re: epimorphic

From: Pierre Lamothe

Carl wrote:
  Pierre didn't say that. He said Epimorphic -> CS, which is
  exactly what Gene said. Clearly CS /-> Epimorphic. See my
  post in this thread.
Sorry Carl, it's not what I said. I justly wrote the first post in the "CS implies EPIMORPHISM" thread
for I saw much ambiguities in many posts about CS and epimorphic.

Gene has defined epimorphic for a scale as 
  "... there is a val h such that if qn is the nth scale degree, then h(qn) = n "
and a val (in a context of rational tone group) as
  "... an homomorphism from the tone group to the integers.
How are interconnected, by definition, scale and tone group ? Subgroup, subset, periodicity block ??

I used simply the well-known mathematical term EPIMORPHISM as surjective morphism and I shown:
  If a scale S has the CS property then there exist an epimorphism D applying each interval x,
  in the space of all intervals spanned by S, onto an integer corresponding to its scale degree,
  not only in S but in any derived scale S' obtained by tonic rotation and/or duality. 
Epimorphism don't imply CS

I hope the following counterexample will suffice. It is well-known that the Zarlino scale is both CS and
epimorphic. The unison vectors 81/80 (about 22 cents) and 25/24 (about 70 cents) generate the kernel
of that epimorphism. It is very easy to transform that CS scale in a non-CS scale but respecting that
epimorphism.

The Zarlino degree 6 == 15/8 has approximately 1088 cents. It misses 112 cents to reach the octave.
If you multiply 15/8 by a combination of unison vectors like 81/80 and 25/24, you dont change the class
of the epimorphism, since class 6 + n (class 0) = class 6. If you add at 1088 cents, for instance, two
comma of 22 cents and a chroma of 70 cents, the result is a degree 6 which is about 2 cents over the
octave.

If you don't see immediately that a such scale forcely reordered
  0 2 204 386 498 702 884 1200
is non-CS, consider that 204 is subtended by 2 steps between 0 and 204 and only one step between
498 (4/3) and 702 (3/2).


Pierre


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Message: 5410

Date: Tue, 22 Oct 2002 06:39:10

Subject: Re: NMOS

From: Carl Lumma

>>>Has anyone paid attention to scales which have a number of
>>>steps a multiple of a MOS?
>> 
>> the torsional scales do!
> 
>I meant a chain of generators where the number of generators is
>a multiple of a number giving a MOS--or in other words, is a
>multiple of something arising from a semiconvergent.

How would the multiple property justify itself againt scales
that were two MOSs superposed at some other interval (besides
the comma)?

In the case of Messiaien, the octatonic scale is an NMOS.
And as pointed out here before, it becomes Blackwood's
decatonic in 15-tET.  For the interlaced diatonic scales in
24-tET, Paul has pointed out that this has excellent 7-limit
harmony in 26.  I forget at what interval this is, but I
don't think it's the comma.

But Paul's excellent decatonics in 22 are two pentatonic MOSs
apart by a non-comma (the half-octave).  In short, regular
double-period linear temperaments, or torsional ones, or
whatever (I haven't been following) is so far as I can see the
strongest constraint justified here.

-Carl


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Message: 5414

Date: Tue, 22 Oct 2002 10:40:15

Subject: Re: Epimorphic

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

Gene wrote:
>Is that what I wrote? It should be
>1/1--2700/2401--5/4-4/3--3/2--5/3--2401/1280--2/1

My routine claims this scale is epimorphic. Being
puzzled, I found out why. The val was [7, 11, 16, 19.5]
and not the [7, 11, 16, 20] which was printed out.
So, do the components need to be integer, if so, why?
You said that h7(2401/2400)=2, but 2401/2400 isn't in the
scale so this is irrelevant?

Manuel


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Message: 5416

Date: Tue, 22 Oct 2002 10:43:41

Subject: Re: Epimorphic

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

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.

Manuel


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Message: 5422

Date: Tue, 22 Oct 2002 01:40:22

Subject: Re: Epimorphic

From: Carl Lumma

>>It turns out the question was moot since Pierre showed that it's
>>equivalent to CS.
> 
>Not so far as I can see.

Pierre didn't say that.  He said Epimorphic -> CS, which is
exactly what Gene said.  Clearly CS /-> Epimorphic.  See my
post in this thread.

-Carl


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