The glumma val

Glumma, or lumma_g which is what the Scala archive has it down for, is
quite a remarkable scale, with six tetrads--three major and three
minor. I note the lumma_g file says Carl invented it, but my
recollection is that I did. Whoever concocted it, it has another
interesting property--it is epumorphic according to Scala's definition
of epimorphic, but not with a standard val. Instead, glumma is
Scala-epimorphic with val <12 19 27 34|, which I hereby christen the
glumma val. I was not counting it as epimorphic since you need to take
the scale in a non-monotonic ordering, but Scala's definition is
interesting and useful, so I'm glad it is implemented.

The glumma val of course has the Pythagorean comma as its three-limit
comma; in the 5-limit, it has {125/108, 135/128} as a TM basis, and in
the 7-limit, {15/14, 64/63, 125/108}. This gives a comma sequence of
3^12/2^19, 135/128, 15/14. If we take the commas 15/14 and 64/63
together, we get the temperament with wedgie <<1 -3 -2 -7 -6 4||,
which is in the pelogic family, a relative of mavila and hexadecimal,
with a copop generator of 4/7 (which suggests there isn't much point
dealing with it as an actual temperament.) Glumma is a detempering of
the 12-note MOS for this scale, which in the cataloged mode is -4 to 7
fifths.

Glumma has a rectangular arrangement of tetrads in the cubic lattice
of tetrads, and when I first presented it I pointed out there was a
whole family of similar scales which you can get by symmetrical
lattice isogenies. I should check if any other of these are
Scala-epimorphic.

Here is the permuted version of glumma:

! glim.scl
Glumma arranged in <12 19 27 34| order
12
!
36/35
8/7
5/4
6/5
10/7
48/35
3/2
5/3
12/7
7/4
96/49
2

>Glumma, or lumma_g which is what the Scala archive has it down for, is
>quite a remarkable scale, with six tetrads--three major and three
>minor. I note the lumma_g file says Carl invented it, but my
>recollection is that I did.

Yep, you did, based on a scale I call stellhex.

>Glumma has a rectangular arrangement of tetrads in the cubic lattice
>of tetrads, and when I first presented it I pointed out there was a
>whole family of similar scales which you can get by symmetrical
>lattice isogenies. I should check if any other of these are
>Scala-epimorphic.

The one you liked of these was reca3c1.scl, which scala doesn't
call epimporphic.

All of them are highly irregular.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The one you liked of these was reca3c1.scl, which scala doesn't
> call epimporphic.

Scala calls none of them epimorphic other than glumma. It may be the
only permutation-epimorphic one among the twelve, but I would have to
check myself because being permutation-epimorphic, ie being epimorphic
with respect to a permuted ordering, does not seem to be what Scala
has implemented. I Hahn-reduced a chain of fifths according to the
kernel of <12 19 27 34|, and got thereby a scale mapped to 12 steps by
the glumma val, but Scala did not regard it as epimorphic.

It's a decent scale; aside from being permutation epimorphic, giving
it an unusual kind of regular structure, it has five major triads,
five minor triads, a major tetrad and a minor tetrad.

Here's the scale:

! rectoo.scl
Hahn-reduced circle of fifths via <12 19 27 34| kernel
12
!
10/9
8/7
6/5
5/4
4/3
3/2
25/16
8/5
5/3
7/4
9/5
2