JI 12-tone 7-limit epimorphic scales?

I started to look into the possibilities for 7-limit 12-tone JI scales. The
first one I tried turned out by coincidence to be epimorphic, which is what
I want, but it may not be in other respects as good as some other
possibilities. However it will be a lot of work to find good scales among
the many possibilities and I think that Gene and perhaps others on tuning
math might have some preliminary results that might answer questions such
as:

What are the "best" 12-tone 7-limit epimorphic scales? (I'm interested in
any sense of "best" that anyone else might want to entertain.)

Do any of these have good approximations of 11-limit intervals that are
actually useful for creating 11-limit chords? Or is this too much to ask of
a 12-tone scale?

Also: scala uses the term "JI epimorphic" I believe. Is this different
from just "epimorphic"?

Thanks for any help.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> What are the "best" 12-tone 7-limit epimorphic scales? (I'm
interested in
> any sense of "best" that anyone else might want to entertain.)

I'm having enough trouble trying to see how to classify all the good
lumma scales, meaning only answering this question up to equivalence
by 225/224. Carl has his collection of best scales, however, which I
imagine you know.

> Do any of these have good approximations of 11-limit intervals that are
> actually useful for creating 11-limit chords? Or is this too much
to ask of
> a 12-tone scale?

If you go through all 55 notes of Crystal Ball 2, you find a lot of
instances of approximations involving 385/384, 1375/1372 or 5632/5625,
which could be used in scales as small as 12 notes, certainly.
If we look just at 385/384, from 5/4 to 16/7 is approximately 11/6,
from 5/4 to 12/7 approximately 11/8, from 6/5 to 7/4 approximately
16/11, from 8/7 to 15/8 approximately 18/11, from 9/7 to 15/8
approximately 16/11, and from 12/7 to 15/8 approximately 12/11.

> Also: scala uses the term "JI epimorphic" I believe. Is this different
> from just "epimorphic"?

In theory you could consider mappings from something other than JI,
but it's hard to see, given intervals expressed in cents, how we are
supposed to express it uniquely in terms of, for instance,
225/224-planar and then use that to investigate if some scale is
225/224-planar epimorphic. The most obvious way to give the scale
precisely would be as a 5-limit scale, but then we are back to
JI-epimorphic. I think Manuel could simply leave the "JI" part out if
he liked.

on 8/10/04 12:23 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> What are the "best" 12-tone 7-limit epimorphic scales? (I'm
> interested in
>> any sense of "best" that anyone else might want to entertain.)
>
> I'm having enough trouble trying to see how to classify all the good
> lumma scales, meaning only answering this question up to equivalence
> by 225/224. Carl has his collection of best scales, however, which I
> imagine you know.

Yes, it sounds like what I need to to is get the most complete list of
7-limit 12-tone scales that I can find into my copy of scala, and start by
weeding out the ones that aren't epimorphic. I don't think scala has a way
to do a search like this automatically (with epimorphic as a criterion), so
it will be a lot of work.

I should also have said that in what I'm looking for I want both 3 and 5
dimensions represented in addition to 7.

Do you happen to know whether epimorphism for 7-limit scales of this type
depends on any basic features of lattice structure? I have suspected
intuitively that epimorphism would require a 3-dimensional lattice that is
basically rectilinear (or parallelopiped actually if the lattice is
triangular), i.e. in a 2x2x3 or 2x3x2 or 3x2x2 arrangement. Have you
discovered any relationships like this that can be counted on?

>> Do any of these have good approximations of 11-limit intervals that are
>> actually useful for creating 11-limit chords? Or is this too much
> to ask of
>> a 12-tone scale?
>
> If you go through all 55 notes of Crystal Ball 2, you find a lot of
> instances of approximations involving 385/384, 1375/1372 or 5632/5625,
> which could be used in scales as small as 12 notes, certainly.
> If we look just at 385/384, from 5/4 to 16/7 is approximately 11/6,
> from 5/4 to 12/7 approximately 11/8, from 6/5 to 7/4 approximately
> 16/11, from 8/7 to 15/8 approximately 18/11, from 9/7 to 15/8
> approximately 16/11, and from 12/7 to 15/8 approximately 12/11.

I think maybe I'll leave the 11-limit requirements as secondary until I
solve the first problem.

Thanks,
Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Yes, it sounds like what I need to to is get the most complete list of
> 7-limit 12-tone scales that I can find into my copy of scala, and
start by
> weeding out the ones that aren't epimorphic.

You also need to decide what to do with things like glumma, which is
epimorphic in a non-monotone ordering (permutation epimorphic, I was
calling that.)

> Do you happen to know whether epimorphism for 7-limit scales of this
type
> depends on any basic features of lattice structure? I have suspected
> intuitively that epimorphism would require a 3-dimensional lattice
that is
> basically rectilinear (or parallelopiped actually if the lattice is
> triangular), i.e. in a 2x2x3 or 2x3x2 or 3x2x2 arrangement.

No, an epimorphic scale can be more general than this, and usually is.

Have you
> discovered any relationships like this that can be counted on?

Fokker blocks are an easy way to construct epimorphic scales, but you
probably knew that already.

on 8/12/04 7:06 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Yes, it sounds like what I need to to is get the most complete list of
>> 7-limit 12-tone scales that I can find into my copy of scala, and
> start by
>> weeding out the ones that aren't epimorphic.
>
> You also need to decide what to do with things like glumma, which is
> epimorphic in a non-monotone ordering (permutation epimorphic, I was
> calling that.)

I think I don't want that. My idea of epimorphic included that the playing
position on a linear keyboard be the same for all appearances of the same
chord. If that is true I think it also implies that this matches the
typical 12-tone playing patterns for major and minor triads (where they
occur), which is also a desirable result.

>> Do you happen to know whether epimorphism for 7-limit scales of this
> type
>> depends on any basic features of lattice structure? I have suspected
>> intuitively that epimorphism would require a 3-dimensional lattice
> that is
>> basically rectilinear (or parallelopiped actually if the lattice is
>> triangular), i.e. in a 2x2x3 or 2x3x2 or 3x2x2 arrangement.
>
> No, an epimorphic scale can be more general than this, and usually is.

Even true for monotone epimorphic?

> Have you
>> discovered any relationships like this that can be counted on?
>
> Fokker blocks are an easy way to construct epimorphic scales, but you
> probably knew that already.

I guess I've been on the list a little too long now to be considered a
newbie. But no I haven't assimilated Fokker blocks yet. I'm still at the
stone knives stage. My intuition is ahead of me, which perhaps gives the
impression that I know things that I don't actually know yet, so to speak.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/12/04 7:06 PM, Gene Ward Smith <gwsmith@s...> wrote:

> I think I don't want that. My idea of epimorphic included that the
playing
> position on a linear keyboard be the same for all appearances of the
same
> chord. If that is true I think it also implies that this matches the
> typical 12-tone playing patterns for major and minor triads (where they
> occur), which is also a desirable result.

Well, no, it doesn't quite imply that. Glumma, in the val ordering,
would do that also, being a permutation epimorphic/constant structure
scale; however major triads would be 0-3-7 and minor triads 0-4-7. Of
course, this isn't an equal temperament and a 0-3-7 cannot be counted
on to be a major triad--on three occasions it is, and on three
occasions it is a 1-7/6-7/5 diminished triad, and on the other six
occasions various other chords.

> > No, an epimorphic scale can be more general than this, and usually is.
>
> Even true for monotone epimorphic?

Normally, yes. Sometimes you get a nice regular figure, such as (to
take some 5-limit examples) the duodene or the thirds scale, but this
is the exception.

on 8/13/04 12:48 AM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 8/12/04 7:06 PM, Gene Ward Smith <gwsmith@s...> wrote:
>
>> I think I don't want that. My idea of epimorphic included that the
> playing
>> position on a linear keyboard be the same for all appearances of the
> same
>> chord. If that is true I think it also implies that this matches the
>> typical 12-tone playing patterns for major and minor triads (where they
>> occur), which is also a desirable result.
>
> Well, no, it doesn't quite imply that. Glumma, in the val ordering,
> would do that also, being a permutation epimorphic/constant structure
> scale; however major triads would be 0-3-7 and minor triads 0-4-7. Of
> course, this isn't an equal temperament and a 0-3-7 cannot be counted
> on to be a major triad--on three occasions it is, and on three
> occasions it is a 1-7/6-7/5 diminished triad, and on the other six
> occasions various other chords.

It's surprising to me that de-ordering can result in increased structural
consistency. Are there any graphical illustrations anywhere of scales that
do this that make it clear how this can work?

-Kurt

on 8/12/04 7:06 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Do you happen to know whether epimorphism for 7-limit scales of this
> type
>> depends on any basic features of lattice structure? I have suspected
>> intuitively that epimorphism would require a 3-dimensional lattice
> that is
>> basically rectilinear (or parallelopiped actually if the lattice is
>> triangular), i.e. in a 2x2x3 or 2x3x2 or 3x2x2 arrangement.
>
> No, an epimorphic scale can be more general than this, and usually is.
>
> Have you
>> discovered any relationships like this that can be counted on?
>
> Fokker blocks are an easy way to construct epimorphic scales, but you
> probably knew that already.

A skim of Paul's "gentle introduction" doesn't seem to touch on what you
were talking about. Where is the best place to read about this?

So does the Fokker block approach generate epimorphic scales of the
non-monotonic variety? Or is it monotonic only (I hope)?

I looked up Fokker block and saw that it is a parallelepiped structure,
which then somehow reinforced my original (incorrect) intuition. Hmm.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> So does the Fokker block approach generate epimorphic scales of the
> non-monotonic variety? Or is it monotonic only (I hope)?

It can easily enough be non-monotonic; in fact the Dona Nobis Pacem
scale I used recently here is exactly that. Not only that, it's a good
example of where some 5-limit chords will mostly likely be assimilated
to a 7-limit interpretation, the sort of thing Carl was talking about.

> I looked up Fokker block and saw that it is a parallelepiped structure,
> which then somehow reinforced my original (incorrect) intuition. Hmm.

Of course, paralleopideds are not the only thing which works; various
other sorts of convex bodies will also. Simply detempering a MOS with
respect to a distance measure such as the Hahn norm works also as a
means of concocting epimorphic scales.