scales and periodicity blocks (from tuning-math2)

>Periodicity blocks? They [glumma scales] aren't epimorphic/CS.

Well, I thought we decided that block = epimorphic + convex.
It looks to me like the glumma family is convex. (Paul, how
do you feel about the convexity condition in light of its
exclusion of the melodic minor?)

Not epimorphic, eh? From monz's tuning dictionary, I get that
a scale is epimorphic when there's a single val that can map
all its degrees to integers. I'm missing some details. Can
you (Gene) show how one of these scales is not "epimorphic"?

"epimorphic/CS" -- forgive me if we've been over this (I
can't find anything in the archives), but does epimorphic
equal constant structures?

>They really aren't very regular, which I presume is why you
>don't like them.

That's right, and I think Paul had decided that blocks whose
smallest 2nd was larger than their smallest unison vector
would be... something. Proper? CS? I forget.

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> Not epimorphic, eh? From monz's tuning dictionary, I get that
> a scale is epimorphic when there's a single val that can map
> all its degrees to integers. I'm missing some details. Can
> you (Gene) show how one of these scales is not "epimorphic"?

For a p-limit JI scale of n steps, you get n-1 linear equations in a number of unknowns equal to the number of primes up to p which must have a solution for the scale to be epimorphic. Take, for intance,
recta3c1: [1,15/14,7/6,6/5,5/4,9/7,7/5,3/2,8/5,12/7,7/4,15/8].

If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us the equation -a+b+c-d-1=0. Taking the equations for 15/14, 7/6, 6/5, and
5/4 we find there is no solution, and hence no epimorphic scale can start out 1--15/14--7/6--6/5--5/4...

> "epimorphic/CS" -- forgive me if we've been over this (I
> can't find anything in the archives), but does epimorphic
> equal constant structures?

Epimorphic ==> CS, clearly. For the other way I need to be sure I really understand exactly what is CS.

>>Not epimorphic, eh? From monz's tuning dictionary, I get that
>>a scale is epimorphic when there's a single val that can map
>>all its degrees to integers. I'm missing some details. Can
>>you (Gene) show how one of these scales is not "epimorphic"?
>
>For a p-limit JI scale of n steps, you get n-1 linear equations
>in a number of unknowns equal to the number of primes up to p
>which must have a solution for the scale to be epimorphic.

Primes -- what about the 9-limit? Wouldn't we need an unknown
for the 9-axis?

>Take, for intance, recta3c1:
>[1,15/14,7/6,6/5,5/4,9/7,7/5,3/2,8/5,12/7,7/4,15/8].
>
>If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us
>the equation -a+b+c-d-1=0.

You expect the taxicab distance from the unison to a scale member
to equal its position in the scale?

>Taking the equations for 15/14, 7/6, 6/5, and 5/4 we find there
>is no solution,

...and you expect a single set of values for the unknowns will
work for all scale members? The only difference between
equations will be the signs and the term representing the scale
position in question?

>hence no epimorphic scale can start out 1--15/14--7/6--6/5--5/4...

You mean no epimorphic scale _with p unknowns_, right?

>>"epimorphic/CS" -- forgive me if we've been over this (I
>>can't find anything in the archives), but does epimorphic
>>equal constant structures?
>
>Epimorphic ==> CS, clearly.

Hopefully it will be clear to me soon...

>For the other way I need to be sure I really understand
>exactly what is CS.

I wish I knew the language you're using, I could tell you!

Gene, your tools are so cool, it would be a shame if you
remain the only person around here who knows how to use
them... have you thought of writing a 'Gene's tools for
dummies' paper, that explains why these tools work in terms
of lattice geometry (how most musician folks around here
think of scales and blocks, I reckon)?

Maybe it would help if you explained how you came to think
epimorphic was a Good Thing (since things like CS and
Rothenberg propriety apparently weren't part of your
reasoning).

-Carl

> From: "Carl Lumma" <clumma@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, October 13, 2002 2:48 PM
> Subject: [tuning-math] Re: scales and periodicity blocks (from
tuning-math2)
>
>
> Gene, your tools are so cool, it would be a shame if you
> remain the only person around here who knows how to use
> them... have you thought of writing a 'Gene's tools for
> dummies' paper, that explains why these tools work in terms
> of lattice geometry (how most musician folks around here
> think of scales and blocks, I reckon)?
>
> Maybe it would help if you explained how you came to think
> epimorphic was a Good Thing (since things like CS and
> Rothenberg propriety apparently weren't part of your
> reasoning).

i made an initial attempt here:
/tuning-math/files/dict/genemath.htm

but Gene, Carl's right ... i'm another who would really
appreciate nice long-winded explanations, using lots of
English words to translate the cryptic mathmatical explanations
you give of your work.

-monz

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> Primes -- what about the 9-limit? Wouldn't we need an unknown
> for the 9-axis?

Odd limits aren't relevant here.

> >Take, for intance, recta3c1:
> >[1,15/14,7/6,6/5,5/4,9/7,7/5,3/2,8/5,12/7,7/4,15/8].
> >
> >If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us
> >the equation -a+b+c-d-1=0.
>
> You expect the taxicab distance from the unison to a scale member
> to equal its position in the scale?

No; the idea is that there is a val h such that if qn is the nth scale step, starting at 0, then h(qn)=n. If the val does not exist, the scale is not epimorphic.

> >Taking the equations for 15/14, 7/6, 6/5, and 5/4 we find there
> >is no solution,
>
> ...and you expect a single set of values for the unknowns will
> work for all scale members?

The definition is, there *is* a val.

The only difference between
> equations will be the signs and the term representing the scale
> position in question?
>
> >hence no epimorphic scale can start out 1--15/14--7/6--6/5--5/4...
>
> You mean no epimorphic scale _with p unknowns_, right?

No 7-limit epimorphic scale, I should have said.

> Maybe it would help if you explained how you came to think
> epimorphic was a Good Thing (since things like CS and
> Rothenberg propriety apparently weren't part of your
> reasoning).

I was never quite sure what CS meant,that's all. Epimorphic has a clear definition.

>>>If h=[a,b,c,d] is our val, then h(15/14)=1 for example gives us
>>>the equation -a+b+c-d-1=0.
>>
>>You expect the taxicab distance from the unison to a scale member
>>to equal its position in the scale?
>
>No; the idea is that there is a val h such that if qn is the nth
>scale step, starting at 0, then h(qn)=n. If the val does not exist,
>the scale is not epimorphic.

That's what the tuning dictionary says. But from the signs on
the equation above, it looks like you expect a to be an exponent
for 2, b for 3, c for 5, d for 7, in the factorization of 15/14.

>>...and you expect a single set of values for the unknowns will
>>work for all scale members?
>
> The definition is, there *is* a val.

Right. Why?

>>You mean no epimorphic scale _with p unknowns_, right?
>
>No 7-limit epimorphic scale, I should have said.

Ok.

>I was never quite sure what CS meant,that's all. Epimorphic has
>a clear definition.

How'd you come up with it?

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:

> >I was never quite sure what CS meant,that's all. Epimorphic has
> >a clear definition.
>
> How'd you come up with it?

It seemed like a clearly important property; after defining it, I noticed that it seemed to be very close to, if not identical with, CS. I'd go look up the definition right now and start the long-overdue analysis of the two but I can't seem to find Monzo's dictionary anymore.

> From: "Gene Ward Smith" <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, October 14, 2002 1:11 AM
> Subject: [tuning-math] Re: scales and periodicity blocks (from
tuning-math2)
>
>
> --- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
>
> > >I was never quite sure what CS meant,that's all. Epimorphic has
> > >a clear definition.
> >
> > How'd you come up with it?
>
> It seemed like a clearly important property; after defining it,
> I noticed that it seemed to be very close to, if not identical with,
> CS. I'd go look up the definition right now and start the
> long-overdue analysis of the two but I can't seem to find
> Monzo's dictionary anymore.

for now, it's here:

/tuning-math/files/dict/index.htm

-monz

>It seemed like a clearly important property; after defining it, I
>noticed that it seemed to be very close to, if not identical with,
>CS. I'd go look up the definition right now and start the long-
>overdue analysis of the two but I can't seem to find Monzo's
>dictionary anymore.

Hopefully, you'll see his note os to its current (temporary)
location. The definition at the top is what you want:

"A tuning system where each interval occurs always subtended by the
same number of steps."

-Carl

[monz wrote...]
>for now, it's here:
>/tuning-math/files/dict/index.htm

Heya Monz,

You did a good job of collecting the key thoughts from
that very ConfuSing thread. The only gotcha is that the
last quote on the page is by Paul, not me (the message
number is correct).

-Carl

--- In tuning-math@y..., "Carl Lumma" <clumma@y...> wrote:
> >It seemed like a clearly important property; after defining it, I
> >noticed that it seemed to be very close to, if not identical with,
> >CS. I'd go look up the definition right now and start the long-
> >overdue analysis of the two but I can't seem to find Monzo's
> >dictionary anymore.
>
> Hopefully, you'll see his note os to its current (temporary)
> location. The definition at the top is what you want:
>
> "A tuning system where each interval occurs always subtended by the
> same number of steps."

I don't see any reference to JI in this definition, so I don't think it means the same as epimorphic; certainly, however, any epimorphic scale will have this property.

>>"A tuning system where each interval occurs always subtended
>>by the same number of steps."
>
>I don't see any reference to JI in this definition, so I don't
>think it means the same as epimorphic; certainly, however, any
>epimorphic scale will have this property.

While Kraig's statement is very clear about being the complete
definition of CS (and thus epimorphic->CS but not CS->epimorphic),
I believe Erv came up with CS for subsets of the JI lattice. If
true, we're still finding new overlap between our steps and Erv's.

-Carl